research article stable numerical evaluation of...

Research ArticleStable Numerical Evaluation of Finite HankelTransforms and Their Application

Manoj P Tripathi12 B P Singh3 and Om P Singh1

1 Department of Mathematical Sciences Indian Institute of Technology Banaras Hindu University Varanasi 221005 India2Department of Mathematics Udai Pratap Autonomous College Varanasi 221002 India3 Department of Mathematics IEC College of Engineering amp Technology Greater Noida 201306 India

Correspondence should be addressed to Om P Singh singhomgmailcom

Received 1 June 2014 Revised 21 September 2014 Accepted 23 September 2014 Published 13 November 2014

Academic Editor Baruch Cahlon

Copyright copy 2014 Manoj P Tripathi et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

A new stable algorithm based on hat functions for numerical evaluation of Hankel transform of order ] gt minus1 is proposed inthis paper The hat basis functions are used as a basis to expand a part of the integrand 119903119891(119903) appearing in the Hankel transformintegralThis leads to a very simple efficient and stable algorithm for the numerical evaluation of Hankel transformThe novelty ofour paper is that we give error and stability analysis of the algorithm and corroborate our theoretical findings by various numericalexperiments Finally an application of the proposed algorithm is given for solving the heat equation in an infinite cylinder with aradiation condition

1 Introduction

Classically the Hankel transform 119865] of order ] of a function119891(119903) is defined by

119865] (119901) = intinfin

0

119903119891 (119903) 119869] (119901119903) 119889119903 119901 gt 0 (1)

As the Hankel transform is self-reciprocal its inverse is givenby

119891 (119903) = int

infin

0

119901119865] (119901) 119869] (119901119903) 119889119901 (2)

where 119869] is the ]th order Bessel function of first kindThis form of Hankel transform (HT) has the advantage

of reducing to the Fourier sine or cosine transform when ] =plusmn12TheHankel transform arises naturally in the discussionof problems posed in cylindrical coordinates and hence as aresult of separation of variables involving Bessel functionsSo it has found wide range of applications related to theproblems inmathematical physics possessing axial symmetry[1] But analytical evaluations of 119865](119901) are rare so numericalmethods are important

The usual classical methods like Trapezoidal rule Cotesrule and so forth connected with replacing the integrandby sequence of polynomials have high accuracy if integrandis smooth But 119903119891(119903)119869](119901119903) and 119901119865](119901)119869](119901119903) are rapidlyoscillating functions for large 119903 and 119901 respectively

To overcome these difficulties two different techniquesare available in the literature The first is the fast Hankeltransform as proposed by Siegman [2] Here by substitutionand scaling the problem is transformed in the space of thelogarithmic coordinates and the fast Fourier transform inthat space But it involves the conventional errors arisingwhen a nonperiodic function is replaced by its periodicextension Moreover it is sensitive to the smoothness offunction in that space The second method is based on theuse of Filon quadrature philosophy [3] In Filon quadraturephilosophy the integrand is separated into the productof an (assumed) slowly varying component and a rapidlyoscillating component In the context of Hankel transformthe former is 119903119891(119903) and the latter is 119869](119901119903) But the errorassociated with Filon quadrature philosophy is appreciablefor 119901 lt 1There are several extrapolationmethods developedin the eighties In particular the papers by Levin and Sidi[4ndash7] are relevant Levin and Sidi in 1981 [4] developed very

Hindawi Publishing CorporationInternational Journal of AnalysisVolume 2014 Article ID 670562 11 pageshttpdxdoiorg1011552014670562

2 International Journal of Analysis

general class of extrapolation methods for various types ofinfinite integrals whether oscillatory or not Sidi in [5ndash7]emphasized oscillatory integral involving Fourier andHankelTransforms among other more general cases In additionthe subject is dealt with in great detail in a book by Sidi [8Chapter 5]

Later in 2004 Guizar-Sicairos and Gutierrez-Vega [9]obtained a powerful scheme to calculate the HT of order] ge 0 by extending the zero order HT algorithm of Yu etal [10] to higher orders Their algorithm is based on theorthogonality properties of Bessel functions Postnikov [11]proposed for the first time a novel and powerful method forcomputing zero and first order HT by using Haar waveletsRefining the idea of Postnikov [11] Singh et al [12 13]obtained three efficient algorithms for numerical evaluationof HT of order ] gt minus1 using linear Legendre multi-wavelets Legendre wavelets and rationalized Haar waveletswhich were shown to be superior to the other mentionedalgorithms Recently Singh et al [14] and Pandey et al[15] tested their proposed algorithms on some examplesfor their stability with respect to noise 120572120579

119894where 120579

119894is a

uniform random variable in [minus1 1] added to the input signal119891(119903) But none of the above mentioned algorithms wereanalyzed theoretically for stability and no error analysis wasprovided

The lack of error and stability analysis in the papers citedabove motivated us for the present work In the presentpaper we use hat basis functions for approximation of119903119891(119903) and replace it by its approximation in (1) therebygetting an efficient and stable algorithm for the numericalevaluation of the HT of order ] gt minus1 Error and stabilityanalysis of the algorithm are given in Sections 4 and 5respectively and further corroborated by the numericalexperiments performed on the test functions in Section 6Test functions with known analytic HT are used with ran-dom noise term 120572120579

119894added to the input field 119891(119903) where

120579119894is a uniform random variable with values in [minus1 1]

to illustrate the stability and efficiency of the proposedalgorithm In Section 7 the algorithm is applied for solvingthe heat equation in an infinite cylinder with a radiationcondition

2 Hat Functions and TheirAssociated Properties

Hat functions are defined on the domain [0 1] These arecontinuous functions with shape of hats when plotted ontwo-dimensional planes The interval [0 1] is divided into 119899subintervals [119894ℎ (119894+1)ℎ] 119894 = 0 1 2 119899minus1 of equal lengthsℎ where ℎ = 1119899 The hat functionrsquos family of first (119899 + 1) hatfunctions is defined as follows [16]

1205950(119905) =

ℎ minus 119905

ℎ

0 le 119905 lt ℎ

0 otherwise

120595119894(119905)

=

119905 minus (119894 minus 1) ℎ

ℎ

(119894 minus 1) ℎ le 119905 lt 119894ℎ

(119894 + 1) ℎ minus 119905

ℎ

119894ℎ le 119905 lt (119894 + 1) ℎ 119894 = 1 2 119899 minus 1

0 otherwise

120595119899(119905) =

119905 minus (1 minus ℎ)

ℎ

1 minus ℎ le 119905 le 1

0 otherwise(3)

From the definition of hat functions it is obvious that

120595119894(119896ℎ) =

1 119894 = 119896

0 119894 = 119896

(4)

The hat functions 120595119895(119905) are continuous linearly independent

and are in 1198712[0 1]A function 119891 isin 1198712[0 1]may be approximated as

119891 (119905) cong

119894=119899

sum

119894=0

119891119894120595119894(119905) = 119891

01205950(119905) + 119891

11205951(119905)

+ 11989121205952(119905) + sdot sdot sdot + 119891

119899120595119899(119905)

(5)

The important aspect of using extended hat functions inthe approximation of function 119891(119905) lies in the fact that thecoefficients 119891

119894in (5) are given by

119891119894= 119891 (119894ℎ) for 119894 = 0 1 2 119899 where ℎ = 1

119899

(6)

3 Method of Solution

To derive the algorithm we first assume that the domainspace of input signal 119891(119903) extends over a limited region 0 le119903 le 119877 From physical point of view this assumption isreasonable due to the fact that the input signal 119891(119903) whichrepresents the physical field either is zero or has an infinitelylong decaying tail outside a disc of finite radius 119877 Thereforeinmany practical applications either the input signal119891(119903) hasa compact support or for a given 120576 gt 0 there exists a positivereal 119877 such that | intinfin

119877

119903119891(119903)119869](119901119903)119889119903| lt 120576 which is the case if119891(119903) = 119900(119903

120582

) where 120582 lt minus32 as 119903 rarr infin Hence in eithercase from (1) we have

119865] (119901) = int119877

0

119903119891 (119903) 119869] (119901119903) 119889119903 (7)

By scaling (7) may be written as

119865] (119901) = int1

0

119903119891 (119903) 119869] (119901119903) 119889119903 (8)

which is known as finite Hankel transform (FHT) [14 15 17]The FHT is a good approximation of the HT given by (1)Writing 119903119891(119903) = 119892(119903) in (8) we get

119865] (119901) = int1

0

119892 (119903) 119869] (119901119903) 119889119903 (9)

International Journal of Analysis 3

Using (5) and (6) we may approximate 119892(119903) as

119892 (119903) cong

119899

sum

119894=0

119892 (119894ℎ) 120595119894(119903) (10)

From (10) (9) may be written as

119865] (119901) cong int1

0

(

119899

sum

119894=0

119892 (119894ℎ) 120595119894(119903)) 119869] (119901119903) 119889119903

=

119899

sum

119894=0

119892 (119894ℎ) int

1

0

120595119894(119903) 119869] (119901119903) 119889119903

(11)

Using (3) (11) may be expressed as

119865] (119901) cong 119892 (0) intℎ

0

(

ℎ minus 119903

ℎ

) 119869] (119901119903) 119889119903

+

119899minus1

sum

119894=1

119892 (119894ℎ) [int

119894ℎ

(119894minus1)ℎ

(

119903 minus (119894 minus 1) ℎ

ℎ

) 119869] (119901119903) 119889119903

+int

(119894+1)ℎ

119894ℎ

((119894 + 1) ℎ minus 119903

ℎ

) 119869] (119901119903) 119889119903]

+ 119892 (119899ℎ) int

1

1minusℎ

(


ℎ

) 119869] (119901119903) 119889119903

(12)

It is noteworthy here that the integrals involved in (12) areevaluated by using the following formulae

int

119886

0

119869] (119905) 119889119905 = 2 lim119873rarrinfin

119873

sum

119899=0

119869]+2119899+1 (119886) Re (]) gt minus1 (13)

(see [18 p 333])

int

119886

0

1199051minus]119869] (119905) 119889119905 =

1

2]minus1 ]

minus 1198861minus]119869]minus1 (119886) (14)

(see [18 p 333])

int

119886

0

119905120583

119869] (119905) 119889119905

= 119886120583(] + 120583 + 1) 2

(] minus 120583 + 1) 2

times lim119873rarrinfin

119873

sum

119899=0

(] + 2119899 + 1) Γ (((] minus 120583 + 1) 2) + 119899)Γ (((] + 120583 + 3) 2) + 119899)

times 119869]+2119899+1 (119886) Re (] + 120583 + 1) gt 0

(15)

(see [19 p 480])

4 Error Analysis

Let the RHS of (10) be denoted by 119892119899(119903) that is

119892119899(119903) =

119899

sum

119894=0

119892 (119894ℎ) 120595119894(119903) (16)

Now replacing 119892(119903) by 119892119899(119903) in (9) we define an 119899th approxi-

mate 119865]119899(119901) of the FHT 119865](119901) as follows

Definition 1 An 119899th approximate finite Hankel transform of119891(119903) denoted by 119865]119899(119901) is defined as

119865]119899 (119901) = int1

0

119892119899(119903) 119869] (119901119903) 119889119903

=

119899

sum

119894=0

119892 (119894ℎ) int

1

0

120595119894(119903) 119869] (119901119903) 119889119903

(17)

Let 120576119899(119901) denote the absolute error between the FHT 119865](119901)

and its 119899th approximate 119865]119899(119901) then we have the following

Theorem2 If119892(119903) is approximated by the family of first (119899+1)hat functions as given in (10) then

(i) |119892(119895ℎ) minus 119892119899(119895ℎ)| = 0 for 119895 = 0 1 2 119899

(ii) |119892(119903) minus 119892119899(119903)| le (12119899

2

)|11989210158401015840

(119895ℎ)| + 119874(11198993

) for 119895ℎ lt119903 lt (119895 + 1)ℎ 119895 = 0 1 2 119899 minus 1

(iii) 120576119899(119901) = |119865](119901) minus 119865]119899(119901)| le 1198722119899

2

+ 119874(11198993

) where|11989210158401015840

(119895ℎ)| le 119872

Proof (i) From (16) and (4) the value of 119892119899(119903) at 119895th nodal

point 119903 = 119895ℎ 119895 = 0 1 2 119899 is given by 119892119899(119895ℎ) =

sum119899

119894=0119892(119894ℎ)120595

119894(119895ℎ) = 119892(119895ℎ)

So |119892(119895ℎ) minus 119892119899(119895ℎ)| = 0 for 119895 = 0 1 2 119899

(ii) If 119903 lies between two consecutive integer multiples ofℎ that is 119895ℎ lt 119903 lt (119895 + 1)ℎ 119895 = 0 1 2 119899 minus 1 then from(16) we have

119892119899(119903) =

119899

sum

119894=0

119892 (119894ℎ) 120595119894(119903)

= 119892 (119895ℎ) 120595119895(119903) + 119892 ((119895 + 1) ℎ) 120595

119895+1(119903)

= 119892 (119895ℎ) (

(119895 + 1) ℎ minus 119903

ℎ

)

+ 119892 (119895ℎ + ℎ) (

119903 minus 119895ℎ

ℎ

) (Using (3))

= 119892 (119895ℎ) minus 119895ℎ(

119892 (119895ℎ + ℎ) minus 119892 (119895ℎ)

ℎ

)

+ 119903(

119892 (119895ℎ + ℎ) minus 119892 (119895ℎ)

ℎ

)

= 119892 (119895ℎ) + (119903 minus 119895ℎ)(

119892 (119895ℎ + ℎ) minus 119892 (119895ℎ)

ℎ

)

(18)

As ℎ rarr 0 from (18) we obtain

119892119899(119903) cong 119892 (119895ℎ) + (119903 minus 119895ℎ) 119892

1015840

(119895ℎ) (19)By expanding 119892

119899(119903) in the form of Taylorrsquos series in the

powers of (119903 minus 119895ℎ) we have

119892 (119903) =

infin

sum

119896=0

(119903 minus 119895ℎ)119896

119896

119892(119896)

(119895ℎ) (20)


where 119892(119896) denotes the 119896th order derivative of 119892(119903) Using (19)and (20) the error between exact and approximate values of119892(119903) is given by

119892 (119903) minus 119892119899(119903) =

infin

sum

119896=2

(119903 minus 119895ℎ)119896

119896

119892(119896)

(119895ℎ)

=

(119903 minus 119895ℎ)2

2

11989210158401015840

(119895ℎ) + 119874 (119903 minus 119895ℎ)3

(21)

Since (119903 minus 119895ℎ) lt ℎ and 119899ℎ = 1 from (21) we get

1003816100381610038161003816119892 (119903) minus 119892

119899(119903)1003816100381610038161003816le

1

21198992

1003816100381610038161003816100381611989210158401015840

(119895ℎ)

10038161003816100381610038161003816+ 119874(

1

1198993) (22)

(iii) The absolute error 120576119899(119901) between exact FHT 119865](119901)

and 119899th approximate FHT 119865]119899(119901) is given by

120576119899(119901) =

10038161003816100381610038161003816119865] (119901) minus 119865]119899 (119901)

10038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

int

1

0

119892 (119903) 119869] (119901119903) 119889119903 minus int1

0

119892119899(119903) 119869] (119901119903) 119889119903

100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

119899minus1

sum

119895=0

int

(119895+1)ℎ

119895ℎ

(119892 (119903) minus 119892119899(119903)) 119869] (119901119903) 119889119903

10038161003816100381610038161003816100381610038161003816100381610038161003816

(23)

As |119869](119901119903)| le 1 we have

120576119899(119901) le

119899minus1

sum

119895=0

int

(119895+1)ℎ

119895ℎ

1003816100381610038161003816119892 (119903) minus 119892

119899(119903)1003816100381610038161003816119889119903

le

119899minus1

sum

119895=0

int

(119895+1)ℎ

119895ℎ

(

1

21198992

1003816100381610038161003816100381611989210158401015840

(119895ℎ)

10038161003816100381610038161003816+ 119874(

1

1198993))119889119903

(Follows from (22))

(24)

If |11989210158401015840(119895ℎ)| le 119872 then it follows that

120576119899(119901) le

119872

21198992

119899minus1

sum

119895=0

int

(119895+1)ℎ

119895ℎ

119889119903

+ 119874(

1

1198993)

119899minus1

sum

119895=0

int

(119895+1)ℎ

119895ℎ

119889119903

=

119872

21198992+ 119874(

1

1198993) as 119899ℎ = 1

(25)

5 Stability Analysis

In this section the stability of the proposed algorithm isanalyzed under the influence of noise In what follows theexact data function is denoted by 119891(119903) and the noisy datafunction119891120572(119903) is obtained by adding a randomnoise120572 to119891(119903)such that 119891120572(119903

119894) = 119891(119903

119894) + 120572120579

119894 where 119903

119894= 119894ℎ 119894 = 0 1 2 119899

119899ℎ = 30 and 120579119894is a uniform random variable with values in

[minus1 1] such that max |119891120572(119903119894) minus 119891(119903

119894)| le 120572 0 le 119894 le 119899 If 119903119891120572(119903)

is denoted by 119892120572(119903) and the approximate HT of the perturbedfunction is denoted by 119865120572]119899(119901) then from (17)

119865120572

]119899 (119901) =

119899

sum

119894=0

119892120572

(119894ℎ) int

1

0

120595119894(119903) 119869] (119901119903) 119889119903

=

119899

sum

119894=0

119894ℎ (119891 (119894ℎ) + 120572120579119894) int

1

0

120595119894(119903) 119869] (119901119903) 119889119903

(26)

From (26) and (17) we have

10038161003816100381610038161003816119865120572

]119899 (119901) minus 119865]119899 (119901)10038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum

119894=0

119892120572

(119894ℎ) int

1

0

120595119894(119903) 119869] (119901119903) 119889119903

minus

119899

sum

119894=0

119892 (119894ℎ) int

1

0

120595119894(119903) 119869] (119901119903) 119889119903

1003816100381610038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum

119894=0

119894ℎ (120572120579119894) int

1

0

120595119894(119903) 119869] (119901119903) 119889119903

1003816100381610038161003816100381610038161003816100381610038161003816

le

119899

sum

119894=0

119894ℎ1003816100381610038161003816120572120579119894

1003816100381610038161003816int

1

0

1003816100381610038161003816120595119894(119903)1003816100381610038161003816

1003816100381610038161003816119869] (119901119903)

1003816100381610038161003816119889119903

le

119899

sum

119894=0

119894ℎ1003816100381610038161003816120572120579119894

1003816100381610038161003816int

1

0

1003816100381610038161003816120595119894(119903)1003816100381610038161003816119889119903 (As 100381610038161003816

1003816119869] (119901119903)

1003816100381610038161003816le 1)

(27)

Let 120579119872= max|120579

119894| then substituting 120595

119894(119903) from (3) we get

10038161003816100381610038161003816119865120572

]119899 (119901) minus 119865]119899 (119901)10038161003816100381610038161003816

le 120579119872ℎ |120572|

times [

119899minus1

sum

119894=1

119894 (int

119894ℎ

(119894minus1)ℎ

(119903 minus (119894 minus 1) ℎ)

ℎ

119889119903

+int

(119894+1)ℎ

119894ℎ

((119894 + 1) ℎ minus 119903)

ℎ

119889119903)

+ 119899int

1

1minusℎ

(119903 minus (1 minus ℎ))

ℎ

119889119903]

= 120579119872|120572| [ℎ

2

119899minus1

sum

119894=1

119894 + 119899

ℎ2

2

] = 120579119872|120572| (

1198992

ℎ2

2

)

le|120572|

2

(As 120579119872le 1 119899ℎ = 1)

(28)

Thus we have proved that

Theorem 3 When the input signal 119891(119903) is corrupted withnoise 120572 the proposed algorithm reduces the noise at least bya factor of 12 in the output data 119865]119899(119901)


Table 1 Least square errors 119864119895(119901)2

in different examples for 119899 = 100

119864119895(119901)2

Example 1 Example 2 Example 3 Example 4 Example 510038171003817100381710038171198640(119901)

10038171003817100381710038172

72696119890 minus 005 11078119890 minus 016 49160119890 minus 006 69741119890 minus 005 66819119890 minus 006

10038171003817100381710038171198641(119901)

10038171003817100381710038172


10038171003817100381710038171198642(119901)

10038171003817100381710038172


6 Numerical Illustrations

The test problems included in this paper are solved withand without random perturbations (noises) to illustrate theefficiency and stability of proposed algorithm by choosingthree different values of noise 120572 as 120572

0= 0 120572

1= 0002 and

1205722= 0005 (as discussed in the beginning of Section 5) In the

following examples the errors119864119895(119901) 119895 = 0 1 2 are computed

and their graphs are sketched for 119899 = 100 where 119864119895(119901) =

(Exact FHT 119865](119901)minus approximate FHT 119865120572119895]119899(119901) obtained from(26) with random noise 120572

119895)

Further the parameter 119901 ranges between 0 and 30 in stepsof 02 Figures 3 and 6 depict the graph of |119865120572119895]119899(119901) minus 119865]119899(119901)|119895 = 1 2 for different test functions in Examples 1 and 2respectively These graphs are in conformity withTheorem 3For all the illustrations the computations are done in MAT-LAB 701 For different examples the least square errors119864119895(119901)2involved in computations of approximate FHT with

noises 120572119895 119895 = 0 1 2 are given in Table 1These are calculated

using the formula

10038171003817100381710038171003817119864119895(119901)

100381710038171003817100381710038172=radicsum119899

119894=0(119864119895(119901119894))

2

119899 + 1

(29)

where 119901119894is taken in steps of 02 in the range [0 30]

Example 1 Consider the function 119891(119903) = radic(1 minus 1199032) 0 le 119903 le1 given in [14 20] for which

1198651(119901) =

120587

1198692

1(1199012)

2119901

0 lt 119901 lt infin

0 119901 = 0

(30)

Numerical evaluation of 1198651(119901) has been achieved by Barakat

and Sandler [20] by using Filon quadrature philosophy butagain the associated error is appreciable for 119901 lt 1 whereasour method gives almost zero error in that range Equation(12) is used to obtain the 119899th approximate HT 119865

1119899(119901) The

comparison between exact HT 1198651(119901) and approximate HT

1198651119899(119901) is shown in Figure 1 In Figure 2 the errors 119864

0(119901)

1198641(119901) and119864

2(119901) for 119899 = 100 are plotted In Figure 3 |119865120572

1119899(119901)minus

1198651119899(119901)| is shown for noises 120572 = 0002 0005 This figure is in

conformity withTheorem 3

Example 2 (Sombrero function) A very important and oftenused function is the Circ function that can be defined as

Circ( 119903119886

) =

1 119903 le 119886

0 119903 gt 119886

(31)

0 5 10 15 20 25 30

0

002

004

006

008

01

012

014

016

018

p

Exac

t and

appr

oxim

ate F

HT

minus002

Figure 1 Exact HT 1198651(119901) (solid line) and approximate HT 119865

1119899(119901)

(dotted line) for 119899 = 100 Example 1

The zeroth order HT of Circ(119903119886) is the Sombrero function[13 21] that will be written as 119878

0(119901) with the following

analytical expression

1198780(119901) = 119886

21198691(119886119901)

119886119901

(32)

We use (12) to obtain the approximation for the FHT 1198650119899(119901)

In Figure 4 we sketch the error 1198640(119901) between exact HT

1198780(119901) and approximate FHT119865

0119899(119901) of Circ function (without

noise) for 119886 = 1 and 119899 = 10 It is evident that even forsuch a small value of 119899 the error is appreciably small Figure 5compares the errors 119864

1(119901) and 119864

2(119901) for 119899 = 100 Figure 6

shows the plot of |1198651205720119899(119901) minus 119865

0119899(119901)| for 120572 = 0002 and 120572 =

0005 with 119899 = 100

Example 3 Consider the function 119891(119903) = (2120587)[arccos(119903) minus119903radic(1 minus 119903

2)] 0 le 119903 le 1 given in [14 22] forwhich zeroth order

exact HT is

1198650(119901) = 2

1198692

1(1199012)

1199012

0 le 119901 lt infin (33)

It is a well-known pair which arises in optical diffractiontheory [23] The function 119891(119903) is known as optical transferfunction of an aberration-free optical system with a circularaperture and 119865

0(119901) is the corresponding spread function

Barakat and Parshall [22] evaluated1198650(119901) using Filon quadra-

ture philosophy but the associated error is again appreciablefor 119901 lt 1 whereas our method gives significantly small error


0 5 10 15 20 25 30

0

1

2

3

p

minus5

minus4

minus3

minus2

minus1

times10minus4

E0(p)E1(p)

andE2(p)

E0(p)

E1(p)

E2(p)

Figure 2 1198640(119901) 119864

1(119901) 119864

2(119901) for 119899 = 100 Example 1

0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

p

times10minus4

For noise 120572 = 0005

For noise 120572 = 0002

Figure 3 Plot of |1198651205721119899(119901)minus119865

1119899(119901)| for 120572 = 0002 and 120572 = 0005with

119899 = 100 Example 1

in that range In Figure 7 the error 1198640(119901) is shown Figure 8

is the comparison between 1198641(119901) and 119864

2(119901) for 119899 = 100

Example 4 Consider the function [24] 119891(119903) = 119903](1198862

minus 1199032

)120583

119867(119886 minus 119903) Re(]) gt minus1Re(120583) gt minus1 0 lt 119903 lt 119886 where119867(119903) is a unit step function

defined by

119867(119903) =

1 119903 ge 0

0 119903 lt 0

(34)

The ]th order HT of 119891(119903) is given by

119865] (119901) = 2]119886120583+]+1

119901minus120583minus1

120583 + 1 119869]+120583+1 (119886119901) (35)

Using the method of solution developed in Section 3 theapproximate HT 119865]119899(119901) has been calculated for ] = 32

0 5 10 15 20 25 30p

0

05

1

15

2

25

3

minus05

E0(p)

times10minus11

Figure 4 1198640(119901) for 119899 = 10 Example 2

0 5 10 15 20 25 30p

0

2

minus14

minus12

minus10

minus8

minus6

minus4

minus2

times10minus4

E1(p)

andE2(p)

E1(p)

E2(p)

Figure 5 1198641(119901) and 119864

2(119901) for 119899 = 100 Example 2

0 5 10 15 20 25 30p

0

02

04

06

08

1

12times10minus3

For noise 120572 = 0005

For noise 120572 = 0002


0119899(119901)| for 120572 = 0002 and 120572 = 0005with

119899 = 100 Example 2


0

01

02

03

04

05

06

07

08

09

1

0 5 10 15 20 25 30p

times10minus5

E0(p)

Figure 7 1198640(119901) for 119899 = 100 in Example 3

0

2

0 5 10 15 20 25 30p

times10minus4

E1(p)

andE2(p)

E1(p)

E2(p)

minus12

minus10

minus8

minus6

minus4

minus2

Figure 8 1198641(119901) and 119864

2(119901) for 119899 = 100 in Example 3

0

001

002

003

004

005

006

007

008

Exac

t and

appr

oxim

ate F

HT

0 5 10 15 20 25 30p

minus002

minus001

Figure 9 Exact HT 11986532(119901) (solid line) and approximate HT

11986532119899

(119901) (dotted line) for 120583 = 12 and 119899 = 100 Example 4

0

1

2

0 5 10 15 20 25 30p

minus4

minus3

minus2

minus1

times10minus4

E0(p)

E1(p)

E2(p)

E0(p)E1(p)

andE2(p)

Figure 10 1198640(119901) 119864

1(119901) 119864

2(119901) for 119899 = 100 in Example 4

0

05

1

0 5 10 15 20 25 30p

Erro

r (w

ithou

t noi

se )E0(p)

minus2

minus15

minus1

minus05

times10minus5


120583 = 12 and 119886 = 1 The comparison between 11986532119899

(119901) andexact FHT 119865

32(119901) is shown in Figure 9 Further in Figure 10

a comparison between the different errors 1198640(119901) 119864

1(119901) and

1198642(119901) is shown for 119899 = 100

Example 5 In this example we choose as a test function thegeneralized version of the top-hat function given as 119891(119903) =119903][119867(119903) minus119867(119903 minus 119886)] 119886 gt 0 and119867(119903) is the unit step function

defined by (34) Then

119865] (119901) =119869]+1 (119901)

119901

(36)

In [9] authors took 119886 = 1 and ] = 4 for numericalcalculations We take 119886 = 1 ] = 5 and observe that theassociated errors with and without random noises are quitesmall The error 119864

0(119901) (without noise) is shown in Figure 11


0

05

0 5 10 15 20 25 30p

minus3

minus25

minus2

minus15

minus1

minus05

times10minus4

2E1(p)

andE2(p)

2E1(p)

E2(p)

Figure 12 Comparison between 1198641(119901) and 119864

2(119901) for 119899 = 100 in

Example 5

0 02 04 06 08 1

0

02

04

06

08

1

12

r

minus02

f(r)

ua(r0)

andf(r)

ua(r 0)

Figure 13 The initial condition function 119891(119903) (solid red line) and119906119886(119903 0) (= lim

119905rarr0+119906119886(119903 119905)) (lowastblue)

0 01 02 03 04 05 06 07 08 09

0

1

2

3

4

r

minus3

minus2

minus1

times10minus3

E(r)

Figure 14 Error between 119891(119903) and 119906119886(119903 0)

0 02 04 06 08 1

0

02

04

06

08

1

12

r

Solu

tion

for d

iffer

ent t

minus02

u0 for t = 0

u1 for t = 001

u2 for t = 002

u3 for t = 004

Figure 15The various profiles of the solutions 119906119886(119903 119905) at fixed times

0 01 02 03 04 05 06 07 08 09 10

1

2

r

Abso

lute

erro

rtimes10minus4

For t = 0

For t = 001

For t = 002

For t = 004

Figure 16 The absolute error between 119906119886(119903 119905) and 119906(119903 119905) for

different 119905

0

05

1

0

05

1

0

02

04

06

08

1

rt

minus02

u(rt)

Figure 17 The solution 119906(119903 119905) for 0 lt 119905 lt 1 and 0 lt 119903 lt 1


0

05

1

0

05

10

02

04

06

08

1

rt

ua(rt)

Figure 18 The approximate solution 119906119886(119903 119905) using 119899th approximate

FHT 1198655119899(1205735119904) for 0 lt 119905 lt 1 and 0 lt 119903 lt 1

for 119899 = 100 In Figure 12 errors 1198641(119901) and 119864

2(119901) for 119899 = 100

are shown

7 Applications

In this section we solve heat equation in cylindrical coor-dinates inside an infinitely long cylinder of radius unity byusing the theory of finite Hankel transform (FHT) developedin the preceding pages The initial temperature is givenby 119891(119903) and radiation takes place at the surface into thesurrounding medium maintained at zero temperature Weseek a function 119906(119903 119905) where 119903 is radius and 119905 is time (119906 doesnot depend upon 120579 and 119905) in the following application Themathematical model of this problem is the diffusion equationin polar coordinates given by

1205972

119906 (119903 119905)

1205972119903

+

1

119903

120597119906 (119903 119905)

120597119903

minus

]2

1199032119906 (119903 119905)

=

120597119906 (119903 119905)

120597119905

(0 lt 119903 lt 1 0 lt 119905 lt infin)

(37)

with the following initial and boundary conditions

(i) as 119905 rarr 0+ 119906(119903 119905) rarr 119891(119903) = 119903

][119867(119903) minus 119867(119903 minus 1)]

where 0 le 119903 le 1 and119867(119903) is unit step function givenby (34)

(ii) as 119903 rarr 1minus (120597119906120597119903) + 119866119906 rarr 0 for each fixed 119905 gt 0

and 119866 gt 0

When 119906(119903 119905) denotes the temperature within the cylinder119866 gt 0means that heat is being radiated away from the surfaceof the cylinder

LetΩ]119903 denote the Bessel differential operator (1205972

1205972

119903) +

(1119903)(120597120597119903) minus (]21199032) Then differential equation (37) can bewritten as

Ω]119903119906 =120597119906

120597119905

(38)

Applying the ]th order finite Hankel transform operator toboth sides of the differential equation (38) and using (8) weget

int

1

0

119903Ω]119903119906119869] (120573]119904119903) 119889119903 = int1

0

119903 (

120597119906

120597119905

) 119869] (120573]119904119903) 119889119903 (39)

where 120573]119904 is the 119904th positive root of equation [24]

119866119869] (119911) + 1199111198691015840

] (119911) = 0 ] ge minus1

2

(40)

The LHS of (39) may be written as

int

1

0

119903 (

1205972

119906

1205972119903

+

1

119903

120597119906

120597119903

minus

]2

1199032119906) 119869] (120573]119904119903) 119889119903

= int

1

0

(

1205972

119906

1205972119903

) 119903119869] (120573]119904119903) 119889119903

+ int

1

0

(

120597119906

120597119903

) 119869] (120573]119904119903) 119889119903 minus ]2 int1

0

(

119906

119903

) 119869] (120573]119904119903) 119889119903

= lim119903rarr1minus

120597119906 (119903 119905)

120597119903

119869] (120573]119904) minus 120573]119904 int1

0

119903

120597119906

120597119903

1198691015840

] (120573]119904119903) 119889119903

minus ]2 int1

0

119906

119903

119869] (120573]119904119903) 119889119903

= minus119866119869] (120573]119904) lim119903rarr1minus119906 (119903 119905) minus 120573]119904 int

1

0

120597119906

120597119903

1199031198691015840

] (120573]119904119903) 119889119903

minus ]2 int1

0

119906

119903

119869] (120573]119904119903) 119889119903 (As 119903 997888rarr 1minus

120597119906

120597119903

+ 119866119906 997888rarr 0)

(41)

Since 120573]119904 is a root of 119866119869](119911) + 1199111198691015840

](119911) = 0 we have 119866119869](120573]119904) =minus120573]119904119869

1015840

](120573]119904) and thus

int

1

0

119903 (

1205972

119906

1205972119903

+

1

119903

120597119906

120597119903

minus

]2

1199032119906) 119869] (120573]119904119903) 119889119903

= 120573]1199041198691015840

] (120573]119904) lim119903rarr1minus119906 (119903 119905)

minus 120573]1199041198691015840

] (120573]119904) lim119903rarr1minus119906 (119903 119905)

+ 120573]119904 int1

0

119906 (1198691015840

] (120573]119904119903) + 120573]11990411990311986910158401015840

] (120573]119904119903)) 119889119903

minus ]2 int1

0

119906

119903

119869] (120573]119904119903) 119889119903

(42)

Using the fact that 119869](119909) is a solution of 1199092(11988921199101198891199092) +119909(119889119910119889119909) + (119909

2

minus ]2)119910 = 0 (42) may be simplified as

int

1

0

119903 (

1205972

119906

1205972119903

+

1

119903

120597119906

120597119903

minus

]2

1199032119906) 119869] (120573]119904119903) 119889119903

= minus1205732

]119904 int1

0

119906 (119903 119905) 119903119869] (120573]119904119903) 119889119903

(43)



minus 1205732

]119904 int1

0

119906 (119903 119905) 119903119869] (120573]119904119903) 119889119903

=

120597

120597119905

(int

1

0

119903119906 (119903 119905) 119869] (120573]119904119903) 119889119903)

(44)

If FHT int10

119906(119903 119905)119903119869](120573]119904119903) 119889119903 of 119906(119903 119905) is denoted by119880(120573]119904 119905)then (44) may be written as

120597119880 (120573]119904 119905)

120597119905

= minus1205732

]119904119880(120573]119904 119905) (45)

whose solution is given by

119880(120573]119904 119905) = 119860 (120573]119904) 119890minus1205732]119904119905 (46)

The constant of integration 119860(120573]119904) in (46) is determined byinitial condition Thus

119860 (120573]119904) = 119865] (120573]119904) = int1

0

119891 (119903) 119903119869] (120573]119904119903) 119889119903 (47)

Hence from (47) (46) becomes

119880 (120573]119904 119905) = 119865] (120573]119904) 119890minus1205732]119904119905 (48)

Therefore by inversion theorem [25] we have

119906 (119903 119905) = lim119873rarrinfin

119873

sum

119904=1

21205732

]119904119865] (120573]119904) 119890minus1205732]119904119905119869] (120573]119904119903)

(1205732

]119904 minus ]2) 1198692] (120573]119904) + 1205732]11990411986910158402] (120573]119904)

= lim119873rarrinfin

119873

sum

119904=1

21205732

]119904119865] (120573]119904) 119890minus1205732]119904119905119869] (120573]119904119903)

(1205732

]119904 minus ]2) 1198692] (120573]119904) + 11986621198692] (120573]119904)

(As 119866119869] (120573]119904) = minus120573]1199041198691015840

] (120573]119904))

(49)

So

119906 (119903 119905) = lim119873rarrinfin

119873

sum

119904=1

21205732

]119904119865] (120573]119904) 119890minus1205732]119904119905

(1205732

]119904 minus ]2 + 1198662)

119869] (120573]119904119903)

1198692

] (120573]119904) (50)

We want to prove that 119906(119903 119905) given by (50) is truly a solutionof (37) that satisfies the given initial and boundary conditionsTo achieve this we need the following well known estimates[17]

119865] (120573]119904) = 119874(1

12057332

]119904)

120573]119904 asymp 120587(119904 +1

4

)

1198692

] (120573]119904) asymp2

120587120573]119904

as 119904 997888rarr infin

(51)

Using the above estimates we see that series (50) and theseries obtained by applyingΩ]119903 and 120597120597119905 separately under thesummation sign of (50) converge uniformly on 0 lt 119903 lt 1 and119905 gt 0 Hence by applyingΩ]119903 minus 120597120597119905 on (50) we get

(Ω]119903 minus120597

120597119905

) 119906 (119903 119905)

= ( lim119873rarrinfin

119873

sum

119904=1

21205734

]119904119865] (120573]119904) 119890minus1205732]119904119905

(1205732

]119904 minus ]2 + 1198662)

sdot [

[

(1205732

]1199041199032

11986910158401015840

] (120573]119904119903) + 120573]1199041199031198691015840

] (120573]119904119903)

+ (1205732

]1199041199032

minus ]2) 119869] (120573]119904119903))

times (1205732

]1199041199032

1198692

] (120573]119904))minus1

]

]

) = 0

(52)

Hence we see that (50) satisfies the differential equation (37)Let us verify the boundary condition (ii) we have

lim119903rarr1minus[

120597119906 (119903 119905)

120597119903

+ 119866119906 (119903 119905)]


[

[

119873

sum

119904=1

21205732

]119904119865] (120573]119904) 119890minus1205732]119904119905

(1205732

]119904 minus ]2 + 1198662) 1198692] (120573]119904)

times (120573]1199041198691015840

] (120573]119904119903) + 119866119869] (120573]119904119903))]

]

(53)

and since the convergence is uniform we can take thelim119903rarr1minus inside the summation sign and arrive at the conclu-

sion as 120573]119904rsquos are the roots of the equation119866119869](119911)+1199111198691015840

](119911) = 0The initial condition (i) is already taken care of as we

evaluated the constant 119860(120573]119904) by using it Through Figures13ndash18 we establish the accuracy of the proposed methodFor numerical solution of differential equation (37) we havetaken ] = 5 and 119866 = 1 We have used MATLAB routineldquofzerordquo to find zeros of (40) and we have drawn Figures 13ndash18 by truncating series (50) at 119873 = 30 While evaluatingthe solution 119906(119903 119905) from (50) we have evaluated 119865

5(1205735119904) first

from its analytical expression given by (36) and denoted thesolution thus obtained by 119906(119903 119905) in Figures 13ndash18 and thenevaluating 119899th approximates FHT 119865

5119899(1205735119904) for 119899 = 100

using our proposed algorithm for evaluation of the FHT asgiven by (17) This solution is denoted by 119906

119886(119903 119905) in the above

mentioned figuresFigure 13 compares the given initial condition 119891(119903) with

119906119886(119903 119905) as 119905 rarr 0

+ and Figure 14 shows the correspondingerror 119864(119903) = 119906

119886(119903 0) minus 119891(119903) Figure 15 depicts the various

profiles of 119906119886(119903 119905) at fixed times 119905

119895= 0 1100 150 125

where the various profiles are denoted by 1199060 1199061 1199062 and

1199063 The absolute errors between 119906

119886(119903 119905) and 119906(119903 119905) at fixed

times 119905119895= 0 1100 150 125 are shown in Figure 16 As

the maximum possible error occurs in the neighborhood of


0 and 0001 we have restricted 119905 in (0 1] in Figures 17 and18 representing 119906(119903 119905) and 119906

119886(119903 119905) respectively and note that

they are in good agreement in the range

8 Conclusions

A new stable and efficient algorithm based on hat functionsfor the numerical evaluation of Hankel transform is proposedand analyzed Choosing hat basis functions to expand theinput signal 119903119891(119903) makes our algorithm attractive in theirapplication in the applied physical problems as they eliminatethe problems connected with the Gibbs phenomenon takingplace in [9 11] We have given for the first time (as per ourinformation) error and stability analysis which was one ofthe main drawbacks of the earlier algorithms and by variousnumerical experiments we have corroborated our theoreticalfindings Stability with respect to the data is restored andexcellent accuracy is obtained even for small sample intervaland high noise levels in the data From the various figuresit is obvious that the algorithm is consistent and does notdepend on the particular choice of the input signal Theaccuracy and simplicity of the algorithm provide an edgeover the many other algorithms Finally an application of theproposed algorithm is given for solving the heat equation inan infinite cylinder with a radiation condition

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] I N Sneddon The Use of Integral Transforms McGraw-Hill1972

[2] A E Siegman ldquoQuasi fastHankel transformrdquoOptics Letters vol1 no 1 pp 13ndash15 1977

[3] L Filon ldquoOn quadrature formula for trigonometric integralsrdquoProceedings of the Royal Society of Edinburgh vol 49 pp 38ndash471928-1929

[4] D Levin andA Sidi ldquoTwo new classes of nonlinear transforma-tions for accelerating the convergence of infinite integrals andseriesrdquo Applied Mathematics and Computation vol 9 no 3 pp175ndash215 1981

[5] A Sidi ldquoExtrapolation methods for oscillatory infinite inte-gralsrdquo Journal of the Institute of Mathematics and its Applica-tions vol 26 no 1 pp 1ndash20 1980

[6] A Sidi ldquoA user-friendly extrapolation method for oscillatoryinfinite integralsrdquoMathematics of Computation vol 51 no 183pp 249ndash266 1988

[7] A Sidi ldquoComputation of infinite integrals involving Besselfunctions of arbitrary order by the 119863-transformationrdquo Journalof Computational and Applied Mathematics vol 78 no 1 pp125ndash130 1997

[8] A Sidi Practical Extrapolation Methods Theory and Applica-tions vol 10 of Cambridge Monographs on Applied and Compu-tational Mathematics Cambridge University Press CambridgeUK 2003

[9] M Guizar-Sicairos and J C Gutierrez-Vega ldquoComputation ofquasi-discrete Hankel transforms of integer order for propagat-ing optical wave fieldsrdquo Journal of the Optical Society of AmericaA Optics Image Science and Vision vol 21 no 1 pp 53ndash582004

[10] L Yu M Huang M Chen W Chen W Huang and Z ZhuldquoQuasi-discrete Hankel transformrdquoOptics Letters vol 23 no 6pp 409ndash411 1998

[11] E B Postnikov ldquoAbout calculation of the Hankel transformusing preliminary wavelet transformrdquo Journal of Applied Math-ematics no 6 pp 319ndash325 2003

[12] V K Singh O P Singh and R K Pandey ldquoNumerical evalu-ation of the Hankel transform by using linear Legendre multi-waveletsrdquoComputer Physics Communications vol 179 no 6 pp424ndash429 2008

[13] V K Singh O P Singh and R K Pandey ldquoEfficient algorithmsto compute Hankel transforms using waveletsrdquo ComputerPhysics Communications vol 179 no 11 pp 812ndash818 2008

[14] O P Singh V K Singh and R K Pandey ldquoAn efficient and sta-ble algorithm for numerical evaluation of Hankel transformsrdquoJournal of Applied Mathematics amp Informatics vol 28 no 5-6pp 1055ndash1071 2010

[15] R K Pandey O P Singh and V K Singh ldquoA stable algorithmfor numerical evaluation of Hankel transforms using Haarwaveletsrdquo Numerical Algorithms vol 53 no 4 pp 451ndash4662010

[16] E Babolian and M Mordad ldquoA numerical method for solvingsystems of linear and nonlinear integral equations of the secondkind by hat basis functionsrdquo Computers amp Mathematics withApplications vol 62 no 1 pp 187ndash198 2011

[17] R S Pathak and O P Singh ldquoFinite Hankel transforms ofdistributionsrdquo Pacific Journal of Mathematics vol 99 no 2 pp439ndash458 1982

[18] A Erdelyi EdTables of Integral TransformsMcGraw-Hill NewYork NY USA 1954

[19] M Abramowitz and I A Stegun Handbook of MathematicalFunctions with Formulas Graphs and Mathematical TablesDover New York NY USA 1965

[20] R Barakat and B H Sandler ldquoEvaluation of first-order Hankeltransforms using Filon quadrature philosophyrdquo Applied Mathe-matics Letters vol 11 no 1 pp 127ndash131 1998

[21] J D Secada ldquoNumerical evaluation of the Hankel transformrdquoComputer Physics Communications vol 116 no 2-3 pp 278ndash294 1999

[22] R Barakat and E Parshall ldquoNumerical evaluation of the zero-order Hankel transform using Filon quadrature philosophyrdquoApplied Mathematics Letters vol 9 no 5 pp 21ndash26 1996

[23] J Gaskell Linear Systems Fourier Transforms and Opticschapter 11 John Wiley amp Sons New York NY USA 1978

[24] A D Poularika ldquoThe Hankel transformrdquo in The Handbook ofFormulas and Tables for Signal Processing CRC Press LLC BocaRaton Fla USA 1999

[25] G NWatsonTheory of Bessel Functions Cambridge UniversityPress Cambridge UK 2nd edition 1958

Submit your manuscripts athttpwwwhindawicom

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

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Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


general class of extrapolation methods for various types ofinfinite integrals whether oscillatory or not Sidi in [5ndash7]emphasized oscillatory integral involving Fourier andHankelTransforms among other more general cases In additionthe subject is dealt with in great detail in a book by Sidi [8Chapter 5]

Later in 2004 Guizar-Sicairos and Gutierrez-Vega [9]obtained a powerful scheme to calculate the HT of order] ge 0 by extending the zero order HT algorithm of Yu etal [10] to higher orders Their algorithm is based on theorthogonality properties of Bessel functions Postnikov [11]proposed for the first time a novel and powerful method forcomputing zero and first order HT by using Haar waveletsRefining the idea of Postnikov [11] Singh et al [12 13]obtained three efficient algorithms for numerical evaluationof HT of order ] gt minus1 using linear Legendre multi-wavelets Legendre wavelets and rationalized Haar waveletswhich were shown to be superior to the other mentionedalgorithms Recently Singh et al [14] and Pandey et al[15] tested their proposed algorithms on some examplesfor their stability with respect to noise 120572120579

119894where 120579

119894is a

uniform random variable in [minus1 1] added to the input signal119891(119903) But none of the above mentioned algorithms wereanalyzed theoretically for stability and no error analysis wasprovided

The lack of error and stability analysis in the papers citedabove motivated us for the present work In the presentpaper we use hat basis functions for approximation of119903119891(119903) and replace it by its approximation in (1) therebygetting an efficient and stable algorithm for the numericalevaluation of the HT of order ] gt minus1 Error and stabilityanalysis of the algorithm are given in Sections 4 and 5respectively and further corroborated by the numericalexperiments performed on the test functions in Section 6Test functions with known analytic HT are used with ran-dom noise term 120572120579

119894added to the input field 119891(119903) where

120579119894is a uniform random variable with values in [minus1 1]

to illustrate the stability and efficiency of the proposedalgorithm In Section 7 the algorithm is applied for solvingthe heat equation in an infinite cylinder with a radiationcondition

2 Hat Functions and TheirAssociated Properties

Hat functions are defined on the domain [0 1] These arecontinuous functions with shape of hats when plotted ontwo-dimensional planes The interval [0 1] is divided into 119899subintervals [119894ℎ (119894+1)ℎ] 119894 = 0 1 2 119899minus1 of equal lengthsℎ where ℎ = 1119899 The hat functionrsquos family of first (119899 + 1) hatfunctions is defined as follows [16]

1205950(119905) =

ℎ minus 119905

ℎ

0 le 119905 lt ℎ

0 otherwise

120595119894(119905)

=

119905 minus (119894 minus 1) ℎ

ℎ

(119894 minus 1) ℎ le 119905 lt 119894ℎ

(119894 + 1) ℎ minus 119905

ℎ

119894ℎ le 119905 lt (119894 + 1) ℎ 119894 = 1 2 119899 minus 1

0 otherwise

120595119899(119905) =


ℎ

1 minus ℎ le 119905 le 1

0 otherwise(3)

From the definition of hat functions it is obvious that

120595119894(119896ℎ) =

1 119894 = 119896

0 119894 = 119896

(4)

The hat functions 120595119895(119905) are continuous linearly independent

and are in 1198712[0 1]A function 119891 isin 1198712[0 1]may be approximated as

119891 (119905) cong

119894=119899

sum

119894=0

119891119894120595119894(119905) = 119891

01205950(119905) + 119891

11205951(119905)

+ 11989121205952(119905) + sdot sdot sdot + 119891

119899120595119899(119905)

(5)

The important aspect of using extended hat functions inthe approximation of function 119891(119905) lies in the fact that thecoefficients 119891

119894in (5) are given by

119891119894= 119891 (119894ℎ) for 119894 = 0 1 2 119899 where ℎ = 1

119899

(6)

3 Method of Solution

To derive the algorithm we first assume that the domainspace of input signal 119891(119903) extends over a limited region 0 le119903 le 119877 From physical point of view this assumption isreasonable due to the fact that the input signal 119891(119903) whichrepresents the physical field either is zero or has an infinitelylong decaying tail outside a disc of finite radius 119877 Thereforeinmany practical applications either the input signal119891(119903) hasa compact support or for a given 120576 gt 0 there exists a positivereal 119877 such that | intinfin

119877

119903119891(119903)119869](119901119903)119889119903| lt 120576 which is the case if119891(119903) = 119900(119903

120582

) where 120582 lt minus32 as 119903 rarr infin Hence in eithercase from (1) we have

119865] (119901) = int119877

0

119903119891 (119903) 119869] (119901119903) 119889119903 (7)

By scaling (7) may be written as

119865] (119901) = int1

0

119903119891 (119903) 119869] (119901119903) 119889119903 (8)

which is known as finite Hankel transform (FHT) [14 15 17]The FHT is a good approximation of the HT given by (1)Writing 119903119891(119903) = 119892(119903) in (8) we get

119865] (119901) = int1

0

119892 (119903) 119869] (119901119903) 119889119903 (9)



119892 (119903) cong

119899

sum

119894=0

119892 (119894ℎ) 120595119894(119903) (10)


119865] (119901) cong int1

0

(

119899

sum

119894=0

119892 (119894ℎ) 120595119894(119903)) 119869] (119901119903) 119889119903

=

119899

sum

119894=0

119892 (119894ℎ) int

1

0

120595119894(119903) 119869] (119901119903) 119889119903

(11)


119865] (119901) cong 119892 (0) intℎ

0

(

ℎ minus 119903

ℎ

) 119869] (119901119903) 119889119903

+

119899minus1

sum

119894=1

119892 (119894ℎ) [int

119894ℎ

(119894minus1)ℎ

(

119903 minus (119894 minus 1) ℎ

ℎ

) 119869] (119901119903) 119889119903

+int

(119894+1)ℎ

119894ℎ

((119894 + 1) ℎ minus 119903

ℎ

) 119869] (119901119903) 119889119903]

+ 119892 (119899ℎ) int

1

1minusℎ

(


ℎ

) 119869] (119901119903) 119889119903

(12)


int

119886

0

119869] (119905) 119889119905 = 2 lim119873rarrinfin

119873

sum

119899=0

119869]+2119899+1 (119886) Re (]) gt minus1 (13)

(see [18 p 333])

int

119886

0

1199051minus]119869] (119905) 119889119905 =

1

2]minus1 ]


(see [18 p 333])

int

119886

0

119905120583

119869] (119905) 119889119905

= 119886120583(] + 120583 + 1) 2

(] minus 120583 + 1) 2


119873

sum

119899=0

(] + 2119899 + 1) Γ (((] minus 120583 + 1) 2) + 119899)Γ (((] + 120583 + 3) 2) + 119899)

times 119869]+2119899+1 (119886) Re (] + 120583 + 1) gt 0

(15)

(see [19 p 480])

4 Error Analysis


119892119899(119903) =

119899

sum

119894=0

119892 (119894ℎ) 120595119894(119903) (16)




119865]119899 (119901) = int1

0

119892119899(119903) 119869] (119901119903) 119889119903

=

119899

sum

119894=0

119892 (119894ℎ) int

1

0

120595119894(119903) 119869] (119901119903) 119889119903

(17)




(i) |119892(119895ℎ) minus 119892119899(119895ℎ)| = 0 for 119895 = 0 1 2 119899

(ii) |119892(119903) minus 119892119899(119903)| le (12119899

2

)|11989210158401015840

(119895ℎ)| + 119874(11198993

) for 119895ℎ lt119903 lt (119895 + 1)ℎ 119895 = 0 1 2 119899 minus 1

(iii) 120576119899(119901) = |119865](119901) minus 119865]119899(119901)| le 1198722119899

2

+ 119874(11198993

) where|11989210158401015840

(119895ℎ)| le 119872



sum119899

119894=0119892(119894ℎ)120595

119894(119895ℎ) = 119892(119895ℎ)

So |119892(119895ℎ) minus 119892119899(119895ℎ)| = 0 for 119895 = 0 1 2 119899


119892119899(119903) =

119899

sum

119894=0

119892 (119894ℎ) 120595119894(119903)

= 119892 (119895ℎ) 120595119895(119903) + 119892 ((119895 + 1) ℎ) 120595

119895+1(119903)

= 119892 (119895ℎ) (

(119895 + 1) ℎ minus 119903

ℎ

)

+ 119892 (119895ℎ + ℎ) (

119903 minus 119895ℎ

ℎ

) (Using (3))

= 119892 (119895ℎ) minus 119895ℎ(

119892 (119895ℎ + ℎ) minus 119892 (119895ℎ)

ℎ

)

+ 119903(

119892 (119895ℎ + ℎ) minus 119892 (119895ℎ)

ℎ

)

= 119892 (119895ℎ) + (119903 minus 119895ℎ)(

119892 (119895ℎ + ℎ) minus 119892 (119895ℎ)

ℎ

)

(18)


119892119899(119903) cong 119892 (119895ℎ) + (119903 minus 119895ℎ) 119892

1015840

(119895ℎ) (19)By expanding 119892



119892 (119903) =

infin

sum

119896=0

(119903 minus 119895ℎ)119896

119896

119892(119896)

(119895ℎ) (20)



119892 (119903) minus 119892119899(119903) =

infin

sum

119896=2

(119903 minus 119895ℎ)119896

119896

119892(119896)

(119895ℎ)

=

(119903 minus 119895ℎ)2

2

11989210158401015840

(119895ℎ) + 119874 (119903 minus 119895ℎ)3

(21)


1003816100381610038161003816119892 (119903) minus 119892

119899(119903)1003816100381610038161003816le

1

21198992

1003816100381610038161003816100381611989210158401015840

(119895ℎ)

10038161003816100381610038161003816+ 119874(

1

1198993) (22)



120576119899(119901) =

10038161003816100381610038161003816119865] (119901) minus 119865]119899 (119901)

10038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

int

1

0

119892 (119903) 119869] (119901119903) 119889119903 minus int1

0

119892119899(119903) 119869] (119901119903) 119889119903

100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

119899minus1

sum

119895=0

int

(119895+1)ℎ

119895ℎ

(119892 (119903) minus 119892119899(119903)) 119869] (119901119903) 119889119903

10038161003816100381610038161003816100381610038161003816100381610038161003816

(23)

As |119869](119901119903)| le 1 we have

120576119899(119901) le

119899minus1

sum

119895=0

int

(119895+1)ℎ

119895ℎ

1003816100381610038161003816119892 (119903) minus 119892

119899(119903)1003816100381610038161003816119889119903

le

119899minus1

sum

119895=0

int

(119895+1)ℎ

119895ℎ

(

1

21198992

1003816100381610038161003816100381611989210158401015840

(119895ℎ)

10038161003816100381610038161003816+ 119874(

1

1198993))119889119903

(Follows from (22))

(24)


120576119899(119901) le

119872

21198992

119899minus1

sum

119895=0

int

(119895+1)ℎ

119895ℎ

119889119903

+ 119874(

1

1198993)

119899minus1

sum

119895=0

int

(119895+1)ℎ

119895ℎ

119889119903

=

119872

21198992+ 119874(

1

1198993) as 119899ℎ = 1

(25)



119894) = 119891(119903

119894) + 120572120579

119894 where 119903

119894= 119894ℎ 119894 = 0 1 2 119899



119894)| le 120572 0 le 119894 le 119899 If 119903119891120572(119903)


119865120572

]119899 (119901) =

119899

sum

119894=0

119892120572

(119894ℎ) int

1

0

120595119894(119903) 119869] (119901119903) 119889119903

=

119899

sum

119894=0

119894ℎ (119891 (119894ℎ) + 120572120579119894) int

1

0

120595119894(119903) 119869] (119901119903) 119889119903

(26)


10038161003816100381610038161003816119865120572

]119899 (119901) minus 119865]119899 (119901)10038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum

119894=0

119892120572

(119894ℎ) int

1

0

120595119894(119903) 119869] (119901119903) 119889119903

minus

119899

sum

119894=0

119892 (119894ℎ) int

1

0

120595119894(119903) 119869] (119901119903) 119889119903

1003816100381610038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum

119894=0

119894ℎ (120572120579119894) int

1

0

120595119894(119903) 119869] (119901119903) 119889119903

1003816100381610038161003816100381610038161003816100381610038161003816

le

119899

sum

119894=0

119894ℎ1003816100381610038161003816120572120579119894

1003816100381610038161003816int

1

0

1003816100381610038161003816120595119894(119903)1003816100381610038161003816

1003816100381610038161003816119869] (119901119903)

1003816100381610038161003816119889119903

le

119899

sum

119894=0

119894ℎ1003816100381610038161003816120572120579119894

1003816100381610038161003816int

1

0

1003816100381610038161003816120595119894(119903)1003816100381610038161003816119889119903 (As 100381610038161003816

1003816119869] (119901119903)

1003816100381610038161003816le 1)

(27)

Let 120579119872= max|120579


119894(119903) from (3) we get

10038161003816100381610038161003816119865120572

]119899 (119901) minus 119865]119899 (119901)10038161003816100381610038161003816

le 120579119872ℎ |120572|

times [

119899minus1

sum

119894=1

119894 (int

119894ℎ

(119894minus1)ℎ

(119903 minus (119894 minus 1) ℎ)

ℎ

119889119903

+int

(119894+1)ℎ

119894ℎ

((119894 + 1) ℎ minus 119903)

ℎ

119889119903)

+ 119899int

1

1minusℎ


ℎ

119889119903]

= 120579119872|120572| [ℎ

2

119899minus1

sum

119894=1

119894 + 119899

ℎ2

2

] = 120579119872|120572| (

1198992

ℎ2

2

)

le|120572|

2

(As 120579119872le 1 119899ℎ = 1)

(28)






119864119895(119901)2


10038171003817100381710038172


10038171003817100381710038171198641(119901)

10038171003817100381710038172


10038171003817100381710038171198642(119901)

10038171003817100381710038172




0= 0 120572

1= 0002 and





119895)



using the formula

10038171003817100381710038171003817119864119895(119901)

100381710038171003817100381710038172=radicsum119899

119894=0(119864119895(119901119894))

2

119899 + 1

(29)



1198651(119901) =

120587

1198692

1(1199012)

2119901


0 119901 = 0

(30)



1119899(119901) The



0(119901)

1198641(119901) and119864


1119899(119901)minus




Circ( 119903119886

) =

1 119903 le 119886

0 119903 gt 119886

(31)

0 5 10 15 20 25 30

0

002

004

006

008

01

012

014

016

018

p

Exac

t and

appr

oxim

ate F

HT

minus002


1119899(119901)





1198780(119901) = 119886

21198691(119886119901)

119886119901

(32)






1(119901) and 119864

2(119901) for 119899 = 100 Figure 6


0119899(119901)| for 120572 = 0002 and 120572 =

0005 with 119899 = 100



exact HT is

1198650(119901) = 2

1198692

1(1199012)

1199012

0 le 119901 lt infin (33)






0 5 10 15 20 25 30

0

1

2

3

p

minus5

minus4

minus3

minus2

minus1

times10minus4

E0(p)E1(p)

andE2(p)

E0(p)

E1(p)

E2(p)

Figure 2 1198640(119901) 119864

1(119901) 119864

2(119901) for 119899 = 100 Example 1

0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

p

times10minus4

For noise 120572 = 0005

For noise 120572 = 0002


1119899(119901)| for 120572 = 0002 and 120572 = 0005with

119899 = 100 Example 1



2(119901) for 119899 = 100


minus 1199032

)120583


defined by

119867(119903) =

1 119903 ge 0

0 119903 lt 0

(34)


119865] (119901) = 2]119886120583+]+1


120583 + 1 119869]+120583+1 (119886119901) (35)


0 5 10 15 20 25 30p

0

05

1

15

2

25

3

minus05

E0(p)

times10minus11


0 5 10 15 20 25 30p

0

2

minus14

minus12

minus10

minus8

minus6

minus4

minus2

times10minus4

E1(p)

andE2(p)

E1(p)

E2(p)

Figure 5 1198641(119901) and 119864

2(119901) for 119899 = 100 Example 2

0 5 10 15 20 25 30p

0

02

04

06

08

1

12times10minus3

For noise 120572 = 0005

For noise 120572 = 0002


0119899(119901)| for 120572 = 0002 and 120572 = 0005with

119899 = 100 Example 2


0

01

02

03

04

05

06

07

08

09

1

0 5 10 15 20 25 30p

times10minus5

E0(p)


0

2

0 5 10 15 20 25 30p

times10minus4

E1(p)

andE2(p)

E1(p)

E2(p)

minus12

minus10

minus8

minus6

minus4

minus2

Figure 8 1198641(119901) and 119864

2(119901) for 119899 = 100 in Example 3

0

001

002

003

004

005

006

007

008

Exac

t and

appr

oxim

ate F

HT

0 5 10 15 20 25 30p

minus002

minus001


11986532119899


0

1

2

0 5 10 15 20 25 30p

minus4

minus3

minus2

minus1

times10minus4

E0(p)

E1(p)

E2(p)

E0(p)E1(p)

andE2(p)

Figure 10 1198640(119901) 119864

1(119901) 119864

2(119901) for 119899 = 100 in Example 4

0

05

1

0 5 10 15 20 25 30p

Erro

r (w

ithou

t noi

se )E0(p)

minus2

minus15

minus1

minus05

times10minus5



(119901) andexact FHT 119865



1(119901) and

1198642(119901) is shown for 119899 = 100



119865] (119901) =119869]+1 (119901)

119901

(36)




0

05

0 5 10 15 20 25 30p

minus3

minus25

minus2

minus15

minus1

minus05

times10minus4

2E1(p)

andE2(p)

2E1(p)

E2(p)


2(119901) for 119899 = 100 in

Example 5

0 02 04 06 08 1

0

02

04

06

08

1

12

r

minus02

f(r)

ua(r0)

andf(r)

ua(r 0)


119905rarr0+119906119886(119903 119905)) (lowastblue)

0 01 02 03 04 05 06 07 08 09

0

1

2

3

4

r

minus3

minus2

minus1

times10minus3

E(r)


0 02 04 06 08 1

0

02

04

06

08

1

12

r

Solu

tion

for d

iffer

ent t

minus02

u0 for t = 0

u1 for t = 001

u2 for t = 002

u3 for t = 004


0 01 02 03 04 05 06 07 08 09 10

1

2

r

Abso

lute

erro

rtimes10minus4

For t = 0

For t = 001

For t = 002

For t = 004


different 119905

0

05

1

0

05

1

0

02

04

06

08

1

rt

minus02

u(rt)



0

05

1

0

05

10

02

04

06

08

1

rt

ua(rt)




2(119901) for 119899 = 100

are shown

7 Applications


1205972

119906 (119903 119905)

1205972119903

+

1

119903

120597119906 (119903 119905)

120597119903

minus

]2

1199032119906 (119903 119905)

=

120597119906 (119903 119905)

120597119905

(0 lt 119903 lt 1 0 lt 119905 lt infin)

(37)


(i) as 119905 rarr 0+ 119906(119903 119905) rarr 119891(119903) = 119903

][119867(119903) minus 119867(119903 minus 1)]



and 119866 gt 0



1205972

119903) +


Ω]119903119906 =120597119906

120597119905

(38)


int

1

0

119903Ω]119903119906119869] (120573]119904119903) 119889119903 = int1

0

119903 (

120597119906

120597119905

) 119869] (120573]119904119903) 119889119903 (39)


119866119869] (119911) + 1199111198691015840

] (119911) = 0 ] ge minus1

2

(40)


int

1

0

119903 (

1205972

119906

1205972119903

+

1

119903

120597119906

120597119903

minus

]2

1199032119906) 119869] (120573]119904119903) 119889119903

= int

1

0

(

1205972

119906

1205972119903

) 119903119869] (120573]119904119903) 119889119903

+ int

1

0

(

120597119906

120597119903

) 119869] (120573]119904119903) 119889119903 minus ]2 int1

0

(

119906

119903

) 119869] (120573]119904119903) 119889119903


120597119906 (119903 119905)

120597119903

119869] (120573]119904) minus 120573]119904 int1

0

119903

120597119906

120597119903

1198691015840

] (120573]119904119903) 119889119903

minus ]2 int1

0

119906

119903

119869] (120573]119904119903) 119889119903


1

0

120597119906

120597119903

1199031198691015840

] (120573]119904119903) 119889119903

minus ]2 int1

0

119906

119903

119869] (120573]119904119903) 119889119903 (As 119903 997888rarr 1minus

120597119906

120597119903

+ 119866119906 997888rarr 0)

(41)

Since 120573]119904 is a root of 119866119869](119911) + 1199111198691015840

](119911) = 0 we have 119866119869](120573]119904) =minus120573]119904119869

1015840

](120573]119904) and thus

int

1

0

119903 (

1205972

119906

1205972119903

+

1

119903

120597119906

120597119903

minus

]2

1199032119906) 119869] (120573]119904119903) 119889119903

= 120573]1199041198691015840

] (120573]119904) lim119903rarr1minus119906 (119903 119905)

minus 120573]1199041198691015840

] (120573]119904) lim119903rarr1minus119906 (119903 119905)

+ 120573]119904 int1

0

119906 (1198691015840

] (120573]119904119903) + 120573]11990411990311986910158401015840

] (120573]119904119903)) 119889119903

minus ]2 int1

0

119906

119903

119869] (120573]119904119903) 119889119903

(42)


2


int

1

0

119903 (

1205972

119906

1205972119903

+

1

119903

120597119906

120597119903

minus

]2

1199032119906) 119869] (120573]119904119903) 119889119903

= minus1205732

]119904 int1

0

119906 (119903 119905) 119903119869] (120573]119904119903) 119889119903

(43)



minus 1205732

]119904 int1

0

119906 (119903 119905) 119903119869] (120573]119904119903) 119889119903

=

120597

120597119905

(int

1

0

119903119906 (119903 119905) 119869] (120573]119904119903) 119889119903)

(44)

If FHT int10


120597119880 (120573]119904 119905)

120597119905

= minus1205732

]119904119880(120573]119904 119905) (45)


119880(120573]119904 119905) = 119860 (120573]119904) 119890minus1205732]119904119905 (46)


119860 (120573]119904) = 119865] (120573]119904) = int1

0

119891 (119903) 119903119869] (120573]119904119903) 119889119903 (47)


119880 (120573]119904 119905) = 119865] (120573]119904) 119890minus1205732]119904119905 (48)


119906 (119903 119905) = lim119873rarrinfin

119873

sum

119904=1

21205732

]119904119865] (120573]119904) 119890minus1205732]119904119905119869] (120573]119904119903)

(1205732

]119904 minus ]2) 1198692] (120573]119904) + 1205732]11990411986910158402] (120573]119904)


119873

sum

119904=1

21205732

]119904119865] (120573]119904) 119890minus1205732]119904119905119869] (120573]119904119903)

(1205732

]119904 minus ]2) 1198692] (120573]119904) + 11986621198692] (120573]119904)

(As 119866119869] (120573]119904) = minus120573]1199041198691015840

] (120573]119904))

(49)

So

119906 (119903 119905) = lim119873rarrinfin

119873

sum

119904=1

21205732

]119904119865] (120573]119904) 119890minus1205732]119904119905

(1205732

]119904 minus ]2 + 1198662)

119869] (120573]119904119903)

1198692

] (120573]119904) (50)


119865] (120573]119904) = 119874(1

12057332

]119904)

120573]119904 asymp 120587(119904 +1

4

)

1198692

] (120573]119904) asymp2

120587120573]119904


(51)


(Ω]119903 minus120597

120597119905

) 119906 (119903 119905)


119873

sum

119904=1

21205734

]119904119865] (120573]119904) 119890minus1205732]119904119905

(1205732

]119904 minus ]2 + 1198662)

sdot [

[

(1205732

]1199041199032

11986910158401015840

] (120573]119904119903) + 120573]1199041199031198691015840

] (120573]119904119903)

+ (1205732

]1199041199032

minus ]2) 119869] (120573]119904119903))

times (1205732

]1199041199032

1198692

] (120573]119904))minus1

]

]

) = 0

(52)



120597119906 (119903 119905)

120597119903

+ 119866119906 (119903 119905)]


[

[

119873

sum

119904=1

21205732

]119904119865] (120573]119904) 119890minus1205732]119904119905

(1205732

]119904 minus ]2 + 1198662) 1198692] (120573]119904)

times (120573]1199041198691015840

] (120573]119904119903) + 119866119869] (120573]119904119903))]

]

(53)





5(1205735119904) first


5119899(1205735119904) for 119899 = 100


119886(119903 119905) in the above


119906119886(119903 119905) as 119905 rarr 0




119895= 0 1100 150 125



119886(119903 119905) and 119906(119903 119905) at fixed







8 Conclusions




References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119892 (119903) cong

119899

sum

119894=0

119892 (119894ℎ) 120595119894(119903) (10)


119865] (119901) cong int1

0

(

119899

sum

119894=0

119892 (119894ℎ) 120595119894(119903)) 119869] (119901119903) 119889119903

=

119899

sum

119894=0

119892 (119894ℎ) int

1

0

120595119894(119903) 119869] (119901119903) 119889119903

(11)


119865] (119901) cong 119892 (0) intℎ

0

(

ℎ minus 119903

ℎ

) 119869] (119901119903) 119889119903

+

119899minus1

sum

119894=1

119892 (119894ℎ) [int

119894ℎ

(119894minus1)ℎ

(

119903 minus (119894 minus 1) ℎ

ℎ

) 119869] (119901119903) 119889119903

+int

(119894+1)ℎ

119894ℎ

((119894 + 1) ℎ minus 119903

ℎ

) 119869] (119901119903) 119889119903]

+ 119892 (119899ℎ) int

1

1minusℎ

(


ℎ

) 119869] (119901119903) 119889119903

(12)


int

119886

0

119869] (119905) 119889119905 = 2 lim119873rarrinfin

119873

sum

119899=0

119869]+2119899+1 (119886) Re (]) gt minus1 (13)

(see [18 p 333])

int

119886

0

1199051minus]119869] (119905) 119889119905 =

1

2]minus1 ]


(see [18 p 333])

int

119886

0

119905120583

119869] (119905) 119889119905

= 119886120583(] + 120583 + 1) 2

(] minus 120583 + 1) 2


119873

sum

119899=0

(] + 2119899 + 1) Γ (((] minus 120583 + 1) 2) + 119899)Γ (((] + 120583 + 3) 2) + 119899)

times 119869]+2119899+1 (119886) Re (] + 120583 + 1) gt 0

(15)

(see [19 p 480])

4 Error Analysis


119892119899(119903) =

119899

sum

119894=0

119892 (119894ℎ) 120595119894(119903) (16)




119865]119899 (119901) = int1

0

119892119899(119903) 119869] (119901119903) 119889119903

=

119899

sum

119894=0

119892 (119894ℎ) int

1

0

120595119894(119903) 119869] (119901119903) 119889119903

(17)




(i) |119892(119895ℎ) minus 119892119899(119895ℎ)| = 0 for 119895 = 0 1 2 119899

(ii) |119892(119903) minus 119892119899(119903)| le (12119899

2

)|11989210158401015840

(119895ℎ)| + 119874(11198993

) for 119895ℎ lt119903 lt (119895 + 1)ℎ 119895 = 0 1 2 119899 minus 1

(iii) 120576119899(119901) = |119865](119901) minus 119865]119899(119901)| le 1198722119899

2

+ 119874(11198993

) where|11989210158401015840

(119895ℎ)| le 119872



sum119899

119894=0119892(119894ℎ)120595

119894(119895ℎ) = 119892(119895ℎ)

So |119892(119895ℎ) minus 119892119899(119895ℎ)| = 0 for 119895 = 0 1 2 119899


119892119899(119903) =

119899

sum

119894=0

119892 (119894ℎ) 120595119894(119903)

= 119892 (119895ℎ) 120595119895(119903) + 119892 ((119895 + 1) ℎ) 120595

119895+1(119903)

= 119892 (119895ℎ) (

(119895 + 1) ℎ minus 119903

ℎ

)

+ 119892 (119895ℎ + ℎ) (

119903 minus 119895ℎ

ℎ

) (Using (3))

= 119892 (119895ℎ) minus 119895ℎ(

119892 (119895ℎ + ℎ) minus 119892 (119895ℎ)

ℎ

)

+ 119903(

119892 (119895ℎ + ℎ) minus 119892 (119895ℎ)

ℎ

)

= 119892 (119895ℎ) + (119903 minus 119895ℎ)(

119892 (119895ℎ + ℎ) minus 119892 (119895ℎ)

ℎ

)

(18)


119892119899(119903) cong 119892 (119895ℎ) + (119903 minus 119895ℎ) 119892

1015840

(119895ℎ) (19)By expanding 119892



119892 (119903) =

infin

sum

119896=0

(119903 minus 119895ℎ)119896

119896

119892(119896)

(119895ℎ) (20)



119892 (119903) minus 119892119899(119903) =

infin

sum

119896=2

(119903 minus 119895ℎ)119896

119896

119892(119896)

(119895ℎ)

=

(119903 minus 119895ℎ)2

2

11989210158401015840

(119895ℎ) + 119874 (119903 minus 119895ℎ)3

(21)


1003816100381610038161003816119892 (119903) minus 119892

119899(119903)1003816100381610038161003816le

1

21198992

1003816100381610038161003816100381611989210158401015840

(119895ℎ)

10038161003816100381610038161003816+ 119874(

1

1198993) (22)



120576119899(119901) =

10038161003816100381610038161003816119865] (119901) minus 119865]119899 (119901)

10038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

int

1

0

119892 (119903) 119869] (119901119903) 119889119903 minus int1

0

119892119899(119903) 119869] (119901119903) 119889119903

100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

119899minus1

sum

119895=0

int

(119895+1)ℎ

119895ℎ

(119892 (119903) minus 119892119899(119903)) 119869] (119901119903) 119889119903

10038161003816100381610038161003816100381610038161003816100381610038161003816

(23)

As |119869](119901119903)| le 1 we have

120576119899(119901) le

119899minus1

sum

119895=0

int

(119895+1)ℎ

119895ℎ

1003816100381610038161003816119892 (119903) minus 119892

119899(119903)1003816100381610038161003816119889119903

le

119899minus1

sum

119895=0

int

(119895+1)ℎ

119895ℎ

(

1

21198992

1003816100381610038161003816100381611989210158401015840

(119895ℎ)

10038161003816100381610038161003816+ 119874(

1

1198993))119889119903

(Follows from (22))

(24)


120576119899(119901) le

119872

21198992

119899minus1

sum

119895=0

int

(119895+1)ℎ

119895ℎ

119889119903

+ 119874(

1

1198993)

119899minus1

sum

119895=0

int

(119895+1)ℎ

119895ℎ

119889119903

=

119872

21198992+ 119874(

1

1198993) as 119899ℎ = 1

(25)



119894) = 119891(119903

119894) + 120572120579

119894 where 119903

119894= 119894ℎ 119894 = 0 1 2 119899



119894)| le 120572 0 le 119894 le 119899 If 119903119891120572(119903)


119865120572

]119899 (119901) =

119899

sum

119894=0

119892120572

(119894ℎ) int

1

0

120595119894(119903) 119869] (119901119903) 119889119903

=

119899

sum

119894=0

119894ℎ (119891 (119894ℎ) + 120572120579119894) int

1

0

120595119894(119903) 119869] (119901119903) 119889119903

(26)


10038161003816100381610038161003816119865120572

]119899 (119901) minus 119865]119899 (119901)10038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum

119894=0

119892120572

(119894ℎ) int

1

0

120595119894(119903) 119869] (119901119903) 119889119903

minus

119899

sum

119894=0

119892 (119894ℎ) int

1

0

120595119894(119903) 119869] (119901119903) 119889119903

1003816100381610038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum

119894=0

119894ℎ (120572120579119894) int

1

0

120595119894(119903) 119869] (119901119903) 119889119903

1003816100381610038161003816100381610038161003816100381610038161003816

le

119899

sum

119894=0

119894ℎ1003816100381610038161003816120572120579119894

1003816100381610038161003816int

1

0

1003816100381610038161003816120595119894(119903)1003816100381610038161003816

1003816100381610038161003816119869] (119901119903)

1003816100381610038161003816119889119903

le

119899

sum

119894=0

119894ℎ1003816100381610038161003816120572120579119894

1003816100381610038161003816int

1

0

1003816100381610038161003816120595119894(119903)1003816100381610038161003816119889119903 (As 100381610038161003816

1003816119869] (119901119903)

1003816100381610038161003816le 1)

(27)

Let 120579119872= max|120579


119894(119903) from (3) we get

10038161003816100381610038161003816119865120572

]119899 (119901) minus 119865]119899 (119901)10038161003816100381610038161003816

le 120579119872ℎ |120572|

times [

119899minus1

sum

119894=1

119894 (int

119894ℎ

(119894minus1)ℎ

(119903 minus (119894 minus 1) ℎ)

ℎ

119889119903

+int

(119894+1)ℎ

119894ℎ

((119894 + 1) ℎ minus 119903)

ℎ

119889119903)

+ 119899int

1

1minusℎ


ℎ

119889119903]

= 120579119872|120572| [ℎ

2

119899minus1

sum

119894=1

119894 + 119899

ℎ2

2

] = 120579119872|120572| (

1198992

ℎ2

2

)

le|120572|

2

(As 120579119872le 1 119899ℎ = 1)

(28)






119864119895(119901)2


10038171003817100381710038172


10038171003817100381710038171198641(119901)

10038171003817100381710038172


10038171003817100381710038171198642(119901)

10038171003817100381710038172




0= 0 120572

1= 0002 and





119895)



using the formula

10038171003817100381710038171003817119864119895(119901)

100381710038171003817100381710038172=radicsum119899

119894=0(119864119895(119901119894))

2

119899 + 1

(29)



1198651(119901) =

120587

1198692

1(1199012)

2119901


0 119901 = 0

(30)



1119899(119901) The



0(119901)

1198641(119901) and119864


1119899(119901)minus




Circ( 119903119886

) =

1 119903 le 119886

0 119903 gt 119886

(31)

0 5 10 15 20 25 30

0

002

004

006

008

01

012

014

016

018

p

Exac

t and

appr

oxim

ate F

HT

minus002


1119899(119901)





1198780(119901) = 119886

21198691(119886119901)

119886119901

(32)






1(119901) and 119864

2(119901) for 119899 = 100 Figure 6


0119899(119901)| for 120572 = 0002 and 120572 =

0005 with 119899 = 100



exact HT is

1198650(119901) = 2

1198692

1(1199012)

1199012

0 le 119901 lt infin (33)






0 5 10 15 20 25 30

0

1

2

3

p

minus5

minus4

minus3

minus2

minus1

times10minus4

E0(p)E1(p)

andE2(p)

E0(p)

E1(p)

E2(p)

Figure 2 1198640(119901) 119864

1(119901) 119864

2(119901) for 119899 = 100 Example 1

0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

p

times10minus4

For noise 120572 = 0005

For noise 120572 = 0002


1119899(119901)| for 120572 = 0002 and 120572 = 0005with

119899 = 100 Example 1



2(119901) for 119899 = 100


minus 1199032

)120583


defined by

119867(119903) =

1 119903 ge 0

0 119903 lt 0

(34)


119865] (119901) = 2]119886120583+]+1


120583 + 1 119869]+120583+1 (119886119901) (35)


0 5 10 15 20 25 30p

0

05

1

15

2

25

3

minus05

E0(p)

times10minus11


0 5 10 15 20 25 30p

0

2

minus14

minus12

minus10

minus8

minus6

minus4

minus2

times10minus4

E1(p)

andE2(p)

E1(p)

E2(p)

Figure 5 1198641(119901) and 119864

2(119901) for 119899 = 100 Example 2

0 5 10 15 20 25 30p

0

02

04

06

08

1

12times10minus3

For noise 120572 = 0005

For noise 120572 = 0002


0119899(119901)| for 120572 = 0002 and 120572 = 0005with

119899 = 100 Example 2


0

01

02

03

04

05

06

07

08

09

1

0 5 10 15 20 25 30p

times10minus5

E0(p)


0

2

0 5 10 15 20 25 30p

times10minus4

E1(p)

andE2(p)

E1(p)

E2(p)

minus12

minus10

minus8

minus6

minus4

minus2

Figure 8 1198641(119901) and 119864

2(119901) for 119899 = 100 in Example 3

0

001

002

003

004

005

006

007

008

Exac

t and

appr

oxim

ate F

HT

0 5 10 15 20 25 30p

minus002

minus001


11986532119899


0

1

2

0 5 10 15 20 25 30p

minus4

minus3

minus2

minus1

times10minus4

E0(p)

E1(p)

E2(p)

E0(p)E1(p)

andE2(p)

Figure 10 1198640(119901) 119864

1(119901) 119864

2(119901) for 119899 = 100 in Example 4

0

05

1

0 5 10 15 20 25 30p

Erro

r (w

ithou

t noi

se )E0(p)

minus2

minus15

minus1

minus05

times10minus5



(119901) andexact FHT 119865



1(119901) and

1198642(119901) is shown for 119899 = 100



119865] (119901) =119869]+1 (119901)

119901

(36)




0

05

0 5 10 15 20 25 30p

minus3

minus25

minus2

minus15

minus1

minus05

times10minus4

2E1(p)

andE2(p)

2E1(p)

E2(p)


2(119901) for 119899 = 100 in

Example 5

0 02 04 06 08 1

0

02

04

06

08

1

12

r

minus02

f(r)

ua(r0)

andf(r)

ua(r 0)


119905rarr0+119906119886(119903 119905)) (lowastblue)

0 01 02 03 04 05 06 07 08 09

0

1

2

3

4

r

minus3

minus2

minus1

times10minus3

E(r)


0 02 04 06 08 1

0

02

04

06

08

1

12

r

Solu

tion

for d

iffer

ent t

minus02

u0 for t = 0

u1 for t = 001

u2 for t = 002

u3 for t = 004


0 01 02 03 04 05 06 07 08 09 10

1

2

r

Abso

lute

erro

rtimes10minus4

For t = 0

For t = 001

For t = 002

For t = 004


different 119905

0

05

1

0

05

1

0

02

04

06

08

1

rt

minus02

u(rt)



0

05

1

0

05

10

02

04

06

08

1

rt

ua(rt)




2(119901) for 119899 = 100

are shown

7 Applications


1205972

119906 (119903 119905)

1205972119903

+

1

119903

120597119906 (119903 119905)

120597119903

minus

]2

1199032119906 (119903 119905)

=

120597119906 (119903 119905)

120597119905

(0 lt 119903 lt 1 0 lt 119905 lt infin)

(37)


(i) as 119905 rarr 0+ 119906(119903 119905) rarr 119891(119903) = 119903

][119867(119903) minus 119867(119903 minus 1)]



and 119866 gt 0



1205972

119903) +


Ω]119903119906 =120597119906

120597119905

(38)


int

1

0

119903Ω]119903119906119869] (120573]119904119903) 119889119903 = int1

0

119903 (

120597119906

120597119905

) 119869] (120573]119904119903) 119889119903 (39)


119866119869] (119911) + 1199111198691015840

] (119911) = 0 ] ge minus1

2

(40)


int

1

0

119903 (

1205972

119906

1205972119903

+

1

119903

120597119906

120597119903

minus

]2

1199032119906) 119869] (120573]119904119903) 119889119903

= int

1

0

(

1205972

119906

1205972119903

) 119903119869] (120573]119904119903) 119889119903

+ int

1

0

(

120597119906

120597119903

) 119869] (120573]119904119903) 119889119903 minus ]2 int1

0

(

119906

119903

) 119869] (120573]119904119903) 119889119903


120597119906 (119903 119905)

120597119903

119869] (120573]119904) minus 120573]119904 int1

0

119903

120597119906

120597119903

1198691015840

] (120573]119904119903) 119889119903

minus ]2 int1

0

119906

119903

119869] (120573]119904119903) 119889119903


1

0

120597119906

120597119903

1199031198691015840

] (120573]119904119903) 119889119903

minus ]2 int1

0

119906

119903

119869] (120573]119904119903) 119889119903 (As 119903 997888rarr 1minus

120597119906

120597119903

+ 119866119906 997888rarr 0)

(41)

Since 120573]119904 is a root of 119866119869](119911) + 1199111198691015840

](119911) = 0 we have 119866119869](120573]119904) =minus120573]119904119869

1015840

](120573]119904) and thus

int

1

0

119903 (

1205972

119906

1205972119903

+

1

119903

120597119906

120597119903

minus

]2

1199032119906) 119869] (120573]119904119903) 119889119903

= 120573]1199041198691015840

] (120573]119904) lim119903rarr1minus119906 (119903 119905)

minus 120573]1199041198691015840

] (120573]119904) lim119903rarr1minus119906 (119903 119905)

+ 120573]119904 int1

0

119906 (1198691015840

] (120573]119904119903) + 120573]11990411990311986910158401015840

] (120573]119904119903)) 119889119903

minus ]2 int1

0

119906

119903

119869] (120573]119904119903) 119889119903

(42)


2


int

1

0

119903 (

1205972

119906

1205972119903

+

1

119903

120597119906

120597119903

minus

]2

1199032119906) 119869] (120573]119904119903) 119889119903

= minus1205732

]119904 int1

0

119906 (119903 119905) 119903119869] (120573]119904119903) 119889119903

(43)



minus 1205732

]119904 int1

0

119906 (119903 119905) 119903119869] (120573]119904119903) 119889119903

=

120597

120597119905

(int

1

0

119903119906 (119903 119905) 119869] (120573]119904119903) 119889119903)

(44)

If FHT int10


120597119880 (120573]119904 119905)

120597119905

= minus1205732

]119904119880(120573]119904 119905) (45)


119880(120573]119904 119905) = 119860 (120573]119904) 119890minus1205732]119904119905 (46)


119860 (120573]119904) = 119865] (120573]119904) = int1

0

119891 (119903) 119903119869] (120573]119904119903) 119889119903 (47)


119880 (120573]119904 119905) = 119865] (120573]119904) 119890minus1205732]119904119905 (48)


119906 (119903 119905) = lim119873rarrinfin

119873

sum

119904=1

21205732

]119904119865] (120573]119904) 119890minus1205732]119904119905119869] (120573]119904119903)

(1205732

]119904 minus ]2) 1198692] (120573]119904) + 1205732]11990411986910158402] (120573]119904)


119873

sum

119904=1

21205732

]119904119865] (120573]119904) 119890minus1205732]119904119905119869] (120573]119904119903)

(1205732

]119904 minus ]2) 1198692] (120573]119904) + 11986621198692] (120573]119904)

(As 119866119869] (120573]119904) = minus120573]1199041198691015840

] (120573]119904))

(49)

So

119906 (119903 119905) = lim119873rarrinfin

119873

sum

119904=1

21205732

]119904119865] (120573]119904) 119890minus1205732]119904119905

(1205732

]119904 minus ]2 + 1198662)

119869] (120573]119904119903)

1198692

] (120573]119904) (50)


119865] (120573]119904) = 119874(1

12057332

]119904)

120573]119904 asymp 120587(119904 +1

4

)

1198692

] (120573]119904) asymp2

120587120573]119904


(51)


(Ω]119903 minus120597

120597119905

) 119906 (119903 119905)


119873

sum

119904=1

21205734

]119904119865] (120573]119904) 119890minus1205732]119904119905

(1205732

]119904 minus ]2 + 1198662)

sdot [

[

(1205732

]1199041199032

11986910158401015840

] (120573]119904119903) + 120573]1199041199031198691015840

] (120573]119904119903)

+ (1205732

]1199041199032

minus ]2) 119869] (120573]119904119903))

times (1205732

]1199041199032

1198692

] (120573]119904))minus1

]

]

) = 0

(52)



120597119906 (119903 119905)

120597119903

+ 119866119906 (119903 119905)]


[

[

119873

sum

119904=1

21205732

]119904119865] (120573]119904) 119890minus1205732]119904119905

(1205732

]119904 minus ]2 + 1198662) 1198692] (120573]119904)

times (120573]1199041198691015840

] (120573]119904119903) + 119866119869] (120573]119904119903))]

]

(53)





5(1205735119904) first


5119899(1205735119904) for 119899 = 100


119886(119903 119905) in the above


119906119886(119903 119905) as 119905 rarr 0




119895= 0 1100 150 125



119886(119903 119905) and 119906(119903 119905) at fixed







8 Conclusions




References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119892 (119903) minus 119892119899(119903) =

infin

sum

119896=2

(119903 minus 119895ℎ)119896

119896

119892(119896)

(119895ℎ)

=

(119903 minus 119895ℎ)2

2

11989210158401015840

(119895ℎ) + 119874 (119903 minus 119895ℎ)3

(21)


1003816100381610038161003816119892 (119903) minus 119892

119899(119903)1003816100381610038161003816le

1

21198992

1003816100381610038161003816100381611989210158401015840

(119895ℎ)

10038161003816100381610038161003816+ 119874(

1

1198993) (22)



120576119899(119901) =

10038161003816100381610038161003816119865] (119901) minus 119865]119899 (119901)

10038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

int

1

0

119892 (119903) 119869] (119901119903) 119889119903 minus int1

0

119892119899(119903) 119869] (119901119903) 119889119903

100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

119899minus1

sum

119895=0

int

(119895+1)ℎ

119895ℎ

(119892 (119903) minus 119892119899(119903)) 119869] (119901119903) 119889119903

10038161003816100381610038161003816100381610038161003816100381610038161003816

(23)

As |119869](119901119903)| le 1 we have

120576119899(119901) le

119899minus1

sum

119895=0

int

(119895+1)ℎ

119895ℎ

1003816100381610038161003816119892 (119903) minus 119892

119899(119903)1003816100381610038161003816119889119903

le

119899minus1

sum

119895=0

int

(119895+1)ℎ

119895ℎ

(

1

21198992

1003816100381610038161003816100381611989210158401015840

(119895ℎ)

10038161003816100381610038161003816+ 119874(

1

1198993))119889119903

(Follows from (22))

(24)


120576119899(119901) le

119872

21198992

119899minus1

sum

119895=0

int

(119895+1)ℎ

119895ℎ

119889119903

+ 119874(

1

1198993)

119899minus1

sum

119895=0

int

(119895+1)ℎ

119895ℎ

119889119903

=

119872

21198992+ 119874(

1

1198993) as 119899ℎ = 1

(25)



119894) = 119891(119903

119894) + 120572120579

119894 where 119903

119894= 119894ℎ 119894 = 0 1 2 119899



119894)| le 120572 0 le 119894 le 119899 If 119903119891120572(119903)


119865120572

]119899 (119901) =

119899

sum

119894=0

119892120572

(119894ℎ) int

1

0

120595119894(119903) 119869] (119901119903) 119889119903

=

119899

sum

119894=0

119894ℎ (119891 (119894ℎ) + 120572120579119894) int

1

0

120595119894(119903) 119869] (119901119903) 119889119903

(26)


10038161003816100381610038161003816119865120572

]119899 (119901) minus 119865]119899 (119901)10038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum

119894=0

119892120572

(119894ℎ) int

1

0

120595119894(119903) 119869] (119901119903) 119889119903

minus

119899

sum

119894=0

119892 (119894ℎ) int

1

0

120595119894(119903) 119869] (119901119903) 119889119903

1003816100381610038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum

119894=0

119894ℎ (120572120579119894) int

1

0

120595119894(119903) 119869] (119901119903) 119889119903

1003816100381610038161003816100381610038161003816100381610038161003816

le

119899

sum

119894=0

119894ℎ1003816100381610038161003816120572120579119894

1003816100381610038161003816int

1

0

1003816100381610038161003816120595119894(119903)1003816100381610038161003816

1003816100381610038161003816119869] (119901119903)

1003816100381610038161003816119889119903

le

119899

sum

119894=0

119894ℎ1003816100381610038161003816120572120579119894

1003816100381610038161003816int

1

0

1003816100381610038161003816120595119894(119903)1003816100381610038161003816119889119903 (As 100381610038161003816

1003816119869] (119901119903)

1003816100381610038161003816le 1)

(27)

Let 120579119872= max|120579


119894(119903) from (3) we get

10038161003816100381610038161003816119865120572

]119899 (119901) minus 119865]119899 (119901)10038161003816100381610038161003816

le 120579119872ℎ |120572|

times [

119899minus1

sum

119894=1

119894 (int

119894ℎ

(119894minus1)ℎ

(119903 minus (119894 minus 1) ℎ)

ℎ

119889119903

+int

(119894+1)ℎ

119894ℎ

((119894 + 1) ℎ minus 119903)

ℎ

119889119903)

+ 119899int

1

1minusℎ


ℎ

119889119903]

= 120579119872|120572| [ℎ

2

119899minus1

sum

119894=1

119894 + 119899

ℎ2

2

] = 120579119872|120572| (

1198992

ℎ2

2

)

le|120572|

2

(As 120579119872le 1 119899ℎ = 1)

(28)






119864119895(119901)2


10038171003817100381710038172


10038171003817100381710038171198641(119901)

10038171003817100381710038172


10038171003817100381710038171198642(119901)

10038171003817100381710038172




0= 0 120572

1= 0002 and





119895)



using the formula

10038171003817100381710038171003817119864119895(119901)

100381710038171003817100381710038172=radicsum119899

119894=0(119864119895(119901119894))

2

119899 + 1

(29)



1198651(119901) =

120587

1198692

1(1199012)

2119901


0 119901 = 0

(30)



1119899(119901) The



0(119901)

1198641(119901) and119864


1119899(119901)minus




Circ( 119903119886

) =

1 119903 le 119886

0 119903 gt 119886

(31)

0 5 10 15 20 25 30

0

002

004

006

008

01

012

014

016

018

p

Exac

t and

appr

oxim

ate F

HT

minus002


1119899(119901)





1198780(119901) = 119886

21198691(119886119901)

119886119901

(32)






1(119901) and 119864

2(119901) for 119899 = 100 Figure 6


0119899(119901)| for 120572 = 0002 and 120572 =

0005 with 119899 = 100



exact HT is

1198650(119901) = 2

1198692

1(1199012)

1199012

0 le 119901 lt infin (33)






0 5 10 15 20 25 30

0

1

2

3

p

minus5

minus4

minus3

minus2

minus1

times10minus4

E0(p)E1(p)

andE2(p)

E0(p)

E1(p)

E2(p)

Figure 2 1198640(119901) 119864

1(119901) 119864

2(119901) for 119899 = 100 Example 1

0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

p

times10minus4

For noise 120572 = 0005

For noise 120572 = 0002


1119899(119901)| for 120572 = 0002 and 120572 = 0005with

119899 = 100 Example 1



2(119901) for 119899 = 100


minus 1199032

)120583


defined by

119867(119903) =

1 119903 ge 0

0 119903 lt 0

(34)


119865] (119901) = 2]119886120583+]+1


120583 + 1 119869]+120583+1 (119886119901) (35)


0 5 10 15 20 25 30p

0

05

1

15

2

25

3

minus05

E0(p)

times10minus11


0 5 10 15 20 25 30p

0

2

minus14

minus12

minus10

minus8

minus6

minus4

minus2

times10minus4

E1(p)

andE2(p)

E1(p)

E2(p)

Figure 5 1198641(119901) and 119864

2(119901) for 119899 = 100 Example 2

0 5 10 15 20 25 30p

0

02

04

06

08

1

12times10minus3

For noise 120572 = 0005

For noise 120572 = 0002


0119899(119901)| for 120572 = 0002 and 120572 = 0005with

119899 = 100 Example 2


0

01

02

03

04

05

06

07

08

09

1

0 5 10 15 20 25 30p

times10minus5

E0(p)


0

2

0 5 10 15 20 25 30p

times10minus4

E1(p)

andE2(p)

E1(p)

E2(p)

minus12

minus10

minus8

minus6

minus4

minus2

Figure 8 1198641(119901) and 119864

2(119901) for 119899 = 100 in Example 3

0

001

002

003

004

005

006

007

008

Exac

t and

appr

oxim

ate F

HT

0 5 10 15 20 25 30p

minus002

minus001


11986532119899


0

1

2

0 5 10 15 20 25 30p

minus4

minus3

minus2

minus1

times10minus4

E0(p)

E1(p)

E2(p)

E0(p)E1(p)

andE2(p)

Figure 10 1198640(119901) 119864

1(119901) 119864

2(119901) for 119899 = 100 in Example 4

0

05

1

0 5 10 15 20 25 30p

Erro

r (w

ithou

t noi

se )E0(p)

minus2

minus15

minus1

minus05

times10minus5



(119901) andexact FHT 119865



1(119901) and

1198642(119901) is shown for 119899 = 100



119865] (119901) =119869]+1 (119901)

119901

(36)




0

05

0 5 10 15 20 25 30p

minus3

minus25

minus2

minus15

minus1

minus05

times10minus4

2E1(p)

andE2(p)

2E1(p)

E2(p)


2(119901) for 119899 = 100 in

Example 5

0 02 04 06 08 1

0

02

04

06

08

1

12

r

minus02

f(r)

ua(r0)

andf(r)

ua(r 0)


119905rarr0+119906119886(119903 119905)) (lowastblue)

0 01 02 03 04 05 06 07 08 09

0

1

2

3

4

r

minus3

minus2

minus1

times10minus3

E(r)


0 02 04 06 08 1

0

02

04

06

08

1

12

r

Solu

tion

for d

iffer

ent t

minus02

u0 for t = 0

u1 for t = 001

u2 for t = 002

u3 for t = 004


0 01 02 03 04 05 06 07 08 09 10

1

2

r

Abso

lute

erro

rtimes10minus4

For t = 0

For t = 001

For t = 002

For t = 004


different 119905

0

05

1

0

05

1

0

02

04

06

08

1

rt

minus02

u(rt)



0

05

1

0

05

10

02

04

06

08

1

rt

ua(rt)




2(119901) for 119899 = 100

are shown

7 Applications


1205972

119906 (119903 119905)

1205972119903

+

1

119903

120597119906 (119903 119905)

120597119903

minus

]2

1199032119906 (119903 119905)

=

120597119906 (119903 119905)

120597119905

(0 lt 119903 lt 1 0 lt 119905 lt infin)

(37)


(i) as 119905 rarr 0+ 119906(119903 119905) rarr 119891(119903) = 119903

][119867(119903) minus 119867(119903 minus 1)]



and 119866 gt 0



1205972

119903) +


Ω]119903119906 =120597119906

120597119905

(38)


int

1

0

119903Ω]119903119906119869] (120573]119904119903) 119889119903 = int1

0

119903 (

120597119906

120597119905

) 119869] (120573]119904119903) 119889119903 (39)


119866119869] (119911) + 1199111198691015840

] (119911) = 0 ] ge minus1

2

(40)


int

1

0

119903 (

1205972

119906

1205972119903

+

1

119903

120597119906

120597119903

minus

]2

1199032119906) 119869] (120573]119904119903) 119889119903

= int

1

0

(

1205972

119906

1205972119903

) 119903119869] (120573]119904119903) 119889119903

+ int

1

0

(

120597119906

120597119903

) 119869] (120573]119904119903) 119889119903 minus ]2 int1

0

(

119906

119903

) 119869] (120573]119904119903) 119889119903


120597119906 (119903 119905)

120597119903

119869] (120573]119904) minus 120573]119904 int1

0

119903

120597119906

120597119903

1198691015840

] (120573]119904119903) 119889119903

minus ]2 int1

0

119906

119903

119869] (120573]119904119903) 119889119903


1

0

120597119906

120597119903

1199031198691015840

] (120573]119904119903) 119889119903

minus ]2 int1

0

119906

119903

119869] (120573]119904119903) 119889119903 (As 119903 997888rarr 1minus

120597119906

120597119903

+ 119866119906 997888rarr 0)

(41)

Since 120573]119904 is a root of 119866119869](119911) + 1199111198691015840

](119911) = 0 we have 119866119869](120573]119904) =minus120573]119904119869

1015840

](120573]119904) and thus

int

1

0

119903 (

1205972

119906

1205972119903

+

1

119903

120597119906

120597119903

minus

]2

1199032119906) 119869] (120573]119904119903) 119889119903

= 120573]1199041198691015840

] (120573]119904) lim119903rarr1minus119906 (119903 119905)

minus 120573]1199041198691015840

] (120573]119904) lim119903rarr1minus119906 (119903 119905)

+ 120573]119904 int1

0

119906 (1198691015840

] (120573]119904119903) + 120573]11990411990311986910158401015840

] (120573]119904119903)) 119889119903

minus ]2 int1

0

119906

119903

119869] (120573]119904119903) 119889119903

(42)


2


int

1

0

119903 (

1205972

119906

1205972119903

+

1

119903

120597119906

120597119903

minus

]2

1199032119906) 119869] (120573]119904119903) 119889119903

= minus1205732

]119904 int1

0

119906 (119903 119905) 119903119869] (120573]119904119903) 119889119903

(43)



minus 1205732

]119904 int1

0

119906 (119903 119905) 119903119869] (120573]119904119903) 119889119903

=

120597

120597119905

(int

1

0

119903119906 (119903 119905) 119869] (120573]119904119903) 119889119903)

(44)

If FHT int10


120597119880 (120573]119904 119905)

120597119905

= minus1205732

]119904119880(120573]119904 119905) (45)


119880(120573]119904 119905) = 119860 (120573]119904) 119890minus1205732]119904119905 (46)


119860 (120573]119904) = 119865] (120573]119904) = int1

0

119891 (119903) 119903119869] (120573]119904119903) 119889119903 (47)


119880 (120573]119904 119905) = 119865] (120573]119904) 119890minus1205732]119904119905 (48)


119906 (119903 119905) = lim119873rarrinfin

119873

sum

119904=1

21205732

]119904119865] (120573]119904) 119890minus1205732]119904119905119869] (120573]119904119903)

(1205732

]119904 minus ]2) 1198692] (120573]119904) + 1205732]11990411986910158402] (120573]119904)


119873

sum

119904=1

21205732

]119904119865] (120573]119904) 119890minus1205732]119904119905119869] (120573]119904119903)

(1205732

]119904 minus ]2) 1198692] (120573]119904) + 11986621198692] (120573]119904)

(As 119866119869] (120573]119904) = minus120573]1199041198691015840

] (120573]119904))

(49)

So

119906 (119903 119905) = lim119873rarrinfin

119873

sum

119904=1

21205732

]119904119865] (120573]119904) 119890minus1205732]119904119905

(1205732

]119904 minus ]2 + 1198662)

119869] (120573]119904119903)

1198692

] (120573]119904) (50)


119865] (120573]119904) = 119874(1

12057332

]119904)

120573]119904 asymp 120587(119904 +1

4

)

1198692

] (120573]119904) asymp2

120587120573]119904


(51)


(Ω]119903 minus120597

120597119905

) 119906 (119903 119905)


119873

sum

119904=1

21205734

]119904119865] (120573]119904) 119890minus1205732]119904119905

(1205732

]119904 minus ]2 + 1198662)

sdot [

[

(1205732

]1199041199032

11986910158401015840

] (120573]119904119903) + 120573]1199041199031198691015840

] (120573]119904119903)

+ (1205732

]1199041199032

minus ]2) 119869] (120573]119904119903))

times (1205732

]1199041199032

1198692

] (120573]119904))minus1

]

]

) = 0

(52)



120597119906 (119903 119905)

120597119903

+ 119866119906 (119903 119905)]


[

[

119873

sum

119904=1

21205732

]119904119865] (120573]119904) 119890minus1205732]119904119905

(1205732

]119904 minus ]2 + 1198662) 1198692] (120573]119904)

times (120573]1199041198691015840

] (120573]119904119903) + 119866119869] (120573]119904119903))]

]

(53)





5(1205735119904) first


5119899(1205735119904) for 119899 = 100


119886(119903 119905) in the above


119906119886(119903 119905) as 119905 rarr 0




119895= 0 1100 150 125



119886(119903 119905) and 119906(119903 119905) at fixed







8 Conclusions




References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119864119895(119901)2


10038171003817100381710038172


10038171003817100381710038171198641(119901)

10038171003817100381710038172


10038171003817100381710038171198642(119901)

10038171003817100381710038172




0= 0 120572

1= 0002 and





119895)



using the formula

10038171003817100381710038171003817119864119895(119901)

100381710038171003817100381710038172=radicsum119899

119894=0(119864119895(119901119894))

2

119899 + 1

(29)



1198651(119901) =

120587

1198692

1(1199012)

2119901


0 119901 = 0

(30)



1119899(119901) The



0(119901)

1198641(119901) and119864


1119899(119901)minus




Circ( 119903119886

) =

1 119903 le 119886

0 119903 gt 119886

(31)

0 5 10 15 20 25 30

0

002

004

006

008

01

012

014

016

018

p

Exac

t and

appr

oxim

ate F

HT

minus002


1119899(119901)





1198780(119901) = 119886

21198691(119886119901)

119886119901

(32)






1(119901) and 119864

2(119901) for 119899 = 100 Figure 6


0119899(119901)| for 120572 = 0002 and 120572 =

0005 with 119899 = 100



exact HT is

1198650(119901) = 2

1198692

1(1199012)

1199012

0 le 119901 lt infin (33)






0 5 10 15 20 25 30

0

1

2

3

p

minus5

minus4

minus3

minus2

minus1

times10minus4

E0(p)E1(p)

andE2(p)

E0(p)

E1(p)

E2(p)

Figure 2 1198640(119901) 119864

1(119901) 119864

2(119901) for 119899 = 100 Example 1

0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

p

times10minus4

For noise 120572 = 0005

For noise 120572 = 0002


1119899(119901)| for 120572 = 0002 and 120572 = 0005with

119899 = 100 Example 1



2(119901) for 119899 = 100


minus 1199032

)120583


defined by

119867(119903) =

1 119903 ge 0

0 119903 lt 0

(34)


119865] (119901) = 2]119886120583+]+1


120583 + 1 119869]+120583+1 (119886119901) (35)


0 5 10 15 20 25 30p

0

05

1

15

2

25

3

minus05

E0(p)

times10minus11


0 5 10 15 20 25 30p

0

2

minus14

minus12

minus10

minus8

minus6

minus4

minus2

times10minus4

E1(p)

andE2(p)

E1(p)

E2(p)

Figure 5 1198641(119901) and 119864

2(119901) for 119899 = 100 Example 2

0 5 10 15 20 25 30p

0

02

04

06

08

1

12times10minus3

For noise 120572 = 0005

For noise 120572 = 0002


0119899(119901)| for 120572 = 0002 and 120572 = 0005with

119899 = 100 Example 2


0

01

02

03

04

05

06

07

08

09

1

0 5 10 15 20 25 30p

times10minus5

E0(p)


0

2

0 5 10 15 20 25 30p

times10minus4

E1(p)

andE2(p)

E1(p)

E2(p)

minus12

minus10

minus8

minus6

minus4

minus2

Figure 8 1198641(119901) and 119864

2(119901) for 119899 = 100 in Example 3

0

001

002

003

004

005

006

007

008

Exac

t and

appr

oxim

ate F

HT

0 5 10 15 20 25 30p

minus002

minus001


11986532119899


0

1

2

0 5 10 15 20 25 30p

minus4

minus3

minus2

minus1

times10minus4

E0(p)

E1(p)

E2(p)

E0(p)E1(p)

andE2(p)

Figure 10 1198640(119901) 119864

1(119901) 119864

2(119901) for 119899 = 100 in Example 4

0

05

1

0 5 10 15 20 25 30p

Erro

r (w

ithou

t noi

se )E0(p)

minus2

minus15

minus1

minus05

times10minus5



(119901) andexact FHT 119865



1(119901) and

1198642(119901) is shown for 119899 = 100



119865] (119901) =119869]+1 (119901)

119901

(36)




0

05

0 5 10 15 20 25 30p

minus3

minus25

minus2

minus15

minus1

minus05

times10minus4

2E1(p)

andE2(p)

2E1(p)

E2(p)


2(119901) for 119899 = 100 in

Example 5

0 02 04 06 08 1

0

02

04

06

08

1

12

r

minus02

f(r)

ua(r0)

andf(r)

ua(r 0)


119905rarr0+119906119886(119903 119905)) (lowastblue)

0 01 02 03 04 05 06 07 08 09

0

1

2

3

4

r

minus3

minus2

minus1

times10minus3

E(r)


0 02 04 06 08 1

0

02

04

06

08

1

12

r

Solu

tion

for d

iffer

ent t

minus02

u0 for t = 0

u1 for t = 001

u2 for t = 002

u3 for t = 004


0 01 02 03 04 05 06 07 08 09 10

1

2

r

Abso

lute

erro

rtimes10minus4

For t = 0

For t = 001

For t = 002

For t = 004


different 119905

0

05

1

0

05

1

0

02

04

06

08

1

rt

minus02

u(rt)



0

05

1

0

05

10

02

04

06

08

1

rt

ua(rt)




2(119901) for 119899 = 100

are shown

7 Applications


1205972

119906 (119903 119905)

1205972119903

+

1

119903

120597119906 (119903 119905)

120597119903

minus

]2

1199032119906 (119903 119905)

=

120597119906 (119903 119905)

120597119905

(0 lt 119903 lt 1 0 lt 119905 lt infin)

(37)


(i) as 119905 rarr 0+ 119906(119903 119905) rarr 119891(119903) = 119903

][119867(119903) minus 119867(119903 minus 1)]



and 119866 gt 0



1205972

119903) +


Ω]119903119906 =120597119906

120597119905

(38)


int

1

0

119903Ω]119903119906119869] (120573]119904119903) 119889119903 = int1

0

119903 (

120597119906

120597119905

) 119869] (120573]119904119903) 119889119903 (39)


119866119869] (119911) + 1199111198691015840

] (119911) = 0 ] ge minus1

2

(40)


int

1

0

119903 (

1205972

119906

1205972119903

+

1

119903

120597119906

120597119903

minus

]2

1199032119906) 119869] (120573]119904119903) 119889119903

= int

1

0

(

1205972

119906

1205972119903

) 119903119869] (120573]119904119903) 119889119903

+ int

1

0

(

120597119906

120597119903

) 119869] (120573]119904119903) 119889119903 minus ]2 int1

0

(

119906

119903

) 119869] (120573]119904119903) 119889119903


120597119906 (119903 119905)

120597119903

119869] (120573]119904) minus 120573]119904 int1

0

119903

120597119906

120597119903

1198691015840

] (120573]119904119903) 119889119903

minus ]2 int1

0

119906

119903

119869] (120573]119904119903) 119889119903


1

0

120597119906

120597119903

1199031198691015840

] (120573]119904119903) 119889119903

minus ]2 int1

0

119906

119903

119869] (120573]119904119903) 119889119903 (As 119903 997888rarr 1minus

120597119906

120597119903

+ 119866119906 997888rarr 0)

(41)

Since 120573]119904 is a root of 119866119869](119911) + 1199111198691015840

](119911) = 0 we have 119866119869](120573]119904) =minus120573]119904119869

1015840

](120573]119904) and thus

int

1

0

119903 (

1205972

119906

1205972119903

+

1

119903

120597119906

120597119903

minus

]2

1199032119906) 119869] (120573]119904119903) 119889119903

= 120573]1199041198691015840

] (120573]119904) lim119903rarr1minus119906 (119903 119905)

minus 120573]1199041198691015840

] (120573]119904) lim119903rarr1minus119906 (119903 119905)

+ 120573]119904 int1

0

119906 (1198691015840

] (120573]119904119903) + 120573]11990411990311986910158401015840

] (120573]119904119903)) 119889119903

minus ]2 int1

0

119906

119903

119869] (120573]119904119903) 119889119903

(42)


2


int

1

0

119903 (

1205972

119906

1205972119903

+

1

119903

120597119906

120597119903

minus

]2

1199032119906) 119869] (120573]119904119903) 119889119903

= minus1205732

]119904 int1

0

119906 (119903 119905) 119903119869] (120573]119904119903) 119889119903

(43)



minus 1205732

]119904 int1

0

119906 (119903 119905) 119903119869] (120573]119904119903) 119889119903

=

120597

120597119905

(int

1

0

119903119906 (119903 119905) 119869] (120573]119904119903) 119889119903)

(44)

If FHT int10


120597119880 (120573]119904 119905)

120597119905

= minus1205732

]119904119880(120573]119904 119905) (45)


119880(120573]119904 119905) = 119860 (120573]119904) 119890minus1205732]119904119905 (46)


119860 (120573]119904) = 119865] (120573]119904) = int1

0

119891 (119903) 119903119869] (120573]119904119903) 119889119903 (47)


119880 (120573]119904 119905) = 119865] (120573]119904) 119890minus1205732]119904119905 (48)


119906 (119903 119905) = lim119873rarrinfin

119873

sum

119904=1

21205732

]119904119865] (120573]119904) 119890minus1205732]119904119905119869] (120573]119904119903)

(1205732

]119904 minus ]2) 1198692] (120573]119904) + 1205732]11990411986910158402] (120573]119904)


119873

sum

119904=1

21205732

]119904119865] (120573]119904) 119890minus1205732]119904119905119869] (120573]119904119903)

(1205732

]119904 minus ]2) 1198692] (120573]119904) + 11986621198692] (120573]119904)

(As 119866119869] (120573]119904) = minus120573]1199041198691015840

] (120573]119904))

(49)

So

119906 (119903 119905) = lim119873rarrinfin

119873

sum

119904=1

21205732

]119904119865] (120573]119904) 119890minus1205732]119904119905

(1205732

]119904 minus ]2 + 1198662)

119869] (120573]119904119903)

1198692

] (120573]119904) (50)


119865] (120573]119904) = 119874(1

12057332

]119904)

120573]119904 asymp 120587(119904 +1

4

)

1198692

] (120573]119904) asymp2

120587120573]119904


(51)


(Ω]119903 minus120597

120597119905

) 119906 (119903 119905)


119873

sum

119904=1

21205734

]119904119865] (120573]119904) 119890minus1205732]119904119905

(1205732

]119904 minus ]2 + 1198662)

sdot [

[

(1205732

]1199041199032

11986910158401015840

] (120573]119904119903) + 120573]1199041199031198691015840

] (120573]119904119903)

+ (1205732

]1199041199032

minus ]2) 119869] (120573]119904119903))

times (1205732

]1199041199032

1198692

] (120573]119904))minus1

]

]

) = 0

(52)



120597119906 (119903 119905)

120597119903

+ 119866119906 (119903 119905)]


[

[

119873

sum

119904=1

21205732

]119904119865] (120573]119904) 119890minus1205732]119904119905

(1205732

]119904 minus ]2 + 1198662) 1198692] (120573]119904)

times (120573]1199041198691015840

] (120573]119904119903) + 119866119869] (120573]119904119903))]

]

(53)





5(1205735119904) first


5119899(1205735119904) for 119899 = 100


119886(119903 119905) in the above


119906119886(119903 119905) as 119905 rarr 0




119895= 0 1100 150 125



119886(119903 119905) and 119906(119903 119905) at fixed







8 Conclusions




References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 5 10 15 20 25 30

0

1

2

3

p

minus5

minus4

minus3

minus2

minus1

times10minus4

E0(p)E1(p)

andE2(p)

E0(p)

E1(p)

E2(p)

Figure 2 1198640(119901) 119864

1(119901) 119864

2(119901) for 119899 = 100 Example 1

0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

p

times10minus4

For noise 120572 = 0005

For noise 120572 = 0002


1119899(119901)| for 120572 = 0002 and 120572 = 0005with

119899 = 100 Example 1



2(119901) for 119899 = 100


minus 1199032

)120583


defined by

119867(119903) =

1 119903 ge 0

0 119903 lt 0

(34)


119865] (119901) = 2]119886120583+]+1


120583 + 1 119869]+120583+1 (119886119901) (35)


0 5 10 15 20 25 30p

0

05

1

15

2

25

3

minus05

E0(p)

times10minus11


0 5 10 15 20 25 30p

0

2

minus14

minus12

minus10

minus8

minus6

minus4

minus2

times10minus4

E1(p)

andE2(p)

E1(p)

E2(p)

Figure 5 1198641(119901) and 119864

2(119901) for 119899 = 100 Example 2

0 5 10 15 20 25 30p

0

02

04

06

08

1

12times10minus3

For noise 120572 = 0005

For noise 120572 = 0002


0119899(119901)| for 120572 = 0002 and 120572 = 0005with

119899 = 100 Example 2


0

01

02

03

04

05

06

07

08

09

1

0 5 10 15 20 25 30p

times10minus5

E0(p)


0

2

0 5 10 15 20 25 30p

times10minus4

E1(p)

andE2(p)

E1(p)

E2(p)

minus12

minus10

minus8

minus6

minus4

minus2

Figure 8 1198641(119901) and 119864

2(119901) for 119899 = 100 in Example 3

0

001

002

003

004

005

006

007

008

Exac

t and

appr

oxim

ate F

HT

0 5 10 15 20 25 30p

minus002

minus001


11986532119899


0

1

2

0 5 10 15 20 25 30p

minus4

minus3

minus2

minus1

times10minus4

E0(p)

E1(p)

E2(p)

E0(p)E1(p)

andE2(p)

Figure 10 1198640(119901) 119864

1(119901) 119864

2(119901) for 119899 = 100 in Example 4

0

05

1

0 5 10 15 20 25 30p

Erro

r (w

ithou

t noi

se )E0(p)

minus2

minus15

minus1

minus05

times10minus5



(119901) andexact FHT 119865



1(119901) and

1198642(119901) is shown for 119899 = 100



119865] (119901) =119869]+1 (119901)

119901

(36)




0

05

0 5 10 15 20 25 30p

minus3

minus25

minus2

minus15

minus1

minus05

times10minus4

2E1(p)

andE2(p)

2E1(p)

E2(p)


2(119901) for 119899 = 100 in

Example 5

0 02 04 06 08 1

0

02

04

06

08

1

12

r

minus02

f(r)

ua(r0)

andf(r)

ua(r 0)


119905rarr0+119906119886(119903 119905)) (lowastblue)

0 01 02 03 04 05 06 07 08 09

0

1

2

3

4

r

minus3

minus2

minus1

times10minus3

E(r)


0 02 04 06 08 1

0

02

04

06

08

1

12

r

Solu

tion

for d

iffer

ent t

minus02

u0 for t = 0

u1 for t = 001

u2 for t = 002

u3 for t = 004


0 01 02 03 04 05 06 07 08 09 10

1

2

r

Abso

lute

erro

rtimes10minus4

For t = 0

For t = 001

For t = 002

For t = 004


different 119905

0

05

1

0

05

1

0

02

04

06

08

1

rt

minus02

u(rt)



0

05

1

0

05

10

02

04

06

08

1

rt

ua(rt)




2(119901) for 119899 = 100

are shown

7 Applications


1205972

119906 (119903 119905)

1205972119903

+

1

119903

120597119906 (119903 119905)

120597119903

minus

]2

1199032119906 (119903 119905)

=

120597119906 (119903 119905)

120597119905

(0 lt 119903 lt 1 0 lt 119905 lt infin)

(37)


(i) as 119905 rarr 0+ 119906(119903 119905) rarr 119891(119903) = 119903

][119867(119903) minus 119867(119903 minus 1)]



and 119866 gt 0



1205972

119903) +


Ω]119903119906 =120597119906

120597119905

(38)


int

1

0

119903Ω]119903119906119869] (120573]119904119903) 119889119903 = int1

0

119903 (

120597119906

120597119905

) 119869] (120573]119904119903) 119889119903 (39)


119866119869] (119911) + 1199111198691015840

] (119911) = 0 ] ge minus1

2

(40)


int

1

0

119903 (

1205972

119906

1205972119903

+

1

119903

120597119906

120597119903

minus

]2

1199032119906) 119869] (120573]119904119903) 119889119903

= int

1

0

(

1205972

119906

1205972119903

) 119903119869] (120573]119904119903) 119889119903

+ int

1

0

(

120597119906

120597119903

) 119869] (120573]119904119903) 119889119903 minus ]2 int1

0

(

119906

119903

) 119869] (120573]119904119903) 119889119903


120597119906 (119903 119905)

120597119903

119869] (120573]119904) minus 120573]119904 int1

0

119903

120597119906

120597119903

1198691015840

] (120573]119904119903) 119889119903

minus ]2 int1

0

119906

119903

119869] (120573]119904119903) 119889119903


1

0

120597119906

120597119903

1199031198691015840

] (120573]119904119903) 119889119903

minus ]2 int1

0

119906

119903

119869] (120573]119904119903) 119889119903 (As 119903 997888rarr 1minus

120597119906

120597119903

+ 119866119906 997888rarr 0)

(41)

Since 120573]119904 is a root of 119866119869](119911) + 1199111198691015840

](119911) = 0 we have 119866119869](120573]119904) =minus120573]119904119869

1015840

](120573]119904) and thus

int

1

0

119903 (

1205972

119906

1205972119903

+

1

119903

120597119906

120597119903

minus

]2

1199032119906) 119869] (120573]119904119903) 119889119903

= 120573]1199041198691015840

] (120573]119904) lim119903rarr1minus119906 (119903 119905)

minus 120573]1199041198691015840

] (120573]119904) lim119903rarr1minus119906 (119903 119905)

+ 120573]119904 int1

0

119906 (1198691015840

] (120573]119904119903) + 120573]11990411990311986910158401015840

] (120573]119904119903)) 119889119903

minus ]2 int1

0

119906

119903

119869] (120573]119904119903) 119889119903

(42)


2


int

1

0

119903 (

1205972

119906

1205972119903

+

1

119903

120597119906

120597119903

minus

]2

1199032119906) 119869] (120573]119904119903) 119889119903

= minus1205732

]119904 int1

0

119906 (119903 119905) 119903119869] (120573]119904119903) 119889119903

(43)



minus 1205732

]119904 int1

0

119906 (119903 119905) 119903119869] (120573]119904119903) 119889119903

=

120597

120597119905

(int

1

0

119903119906 (119903 119905) 119869] (120573]119904119903) 119889119903)

(44)

If FHT int10


120597119880 (120573]119904 119905)

120597119905

= minus1205732

]119904119880(120573]119904 119905) (45)


119880(120573]119904 119905) = 119860 (120573]119904) 119890minus1205732]119904119905 (46)


119860 (120573]119904) = 119865] (120573]119904) = int1

0

119891 (119903) 119903119869] (120573]119904119903) 119889119903 (47)


119880 (120573]119904 119905) = 119865] (120573]119904) 119890minus1205732]119904119905 (48)


119906 (119903 119905) = lim119873rarrinfin

119873

sum

119904=1

21205732

]119904119865] (120573]119904) 119890minus1205732]119904119905119869] (120573]119904119903)

(1205732

]119904 minus ]2) 1198692] (120573]119904) + 1205732]11990411986910158402] (120573]119904)


119873

sum

119904=1

21205732

]119904119865] (120573]119904) 119890minus1205732]119904119905119869] (120573]119904119903)

(1205732

]119904 minus ]2) 1198692] (120573]119904) + 11986621198692] (120573]119904)

(As 119866119869] (120573]119904) = minus120573]1199041198691015840

] (120573]119904))

(49)

So

119906 (119903 119905) = lim119873rarrinfin

119873

sum

119904=1

21205732

]119904119865] (120573]119904) 119890minus1205732]119904119905

(1205732

]119904 minus ]2 + 1198662)

119869] (120573]119904119903)

1198692

] (120573]119904) (50)


119865] (120573]119904) = 119874(1

12057332

]119904)

120573]119904 asymp 120587(119904 +1

4

)

1198692

] (120573]119904) asymp2

120587120573]119904


(51)


(Ω]119903 minus120597

120597119905

) 119906 (119903 119905)


119873

sum

119904=1

21205734

]119904119865] (120573]119904) 119890minus1205732]119904119905

(1205732

]119904 minus ]2 + 1198662)

sdot [

[

(1205732

]1199041199032

11986910158401015840

] (120573]119904119903) + 120573]1199041199031198691015840

] (120573]119904119903)

+ (1205732

]1199041199032

minus ]2) 119869] (120573]119904119903))

times (1205732

]1199041199032

1198692

] (120573]119904))minus1

]

]

) = 0

(52)



120597119906 (119903 119905)

120597119903

+ 119866119906 (119903 119905)]


[

[

119873

sum

119904=1

21205732

]119904119865] (120573]119904) 119890minus1205732]119904119905

(1205732

]119904 minus ]2 + 1198662) 1198692] (120573]119904)

times (120573]1199041198691015840

] (120573]119904119903) + 119866119869] (120573]119904119903))]

]

(53)





5(1205735119904) first


5119899(1205735119904) for 119899 = 100


119886(119903 119905) in the above


119906119886(119903 119905) as 119905 rarr 0




119895= 0 1100 150 125



119886(119903 119905) and 119906(119903 119905) at fixed







8 Conclusions




References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

01

02

03

04

05

06

07

08

09

1

0 5 10 15 20 25 30p

times10minus5

E0(p)


0

2

0 5 10 15 20 25 30p

times10minus4

E1(p)

andE2(p)

E1(p)

E2(p)

minus12

minus10

minus8

minus6

minus4

minus2

Figure 8 1198641(119901) and 119864

2(119901) for 119899 = 100 in Example 3

0

001

002

003

004

005

006

007

008

Exac

t and

appr

oxim

ate F

HT

0 5 10 15 20 25 30p

minus002

minus001


11986532119899


0

1

2

0 5 10 15 20 25 30p

minus4

minus3

minus2

minus1

times10minus4

E0(p)

E1(p)

E2(p)

E0(p)E1(p)

andE2(p)

Figure 10 1198640(119901) 119864

1(119901) 119864

2(119901) for 119899 = 100 in Example 4

0

05

1

0 5 10 15 20 25 30p

Erro

r (w

ithou

t noi

se )E0(p)

minus2

minus15

minus1

minus05

times10minus5



(119901) andexact FHT 119865



1(119901) and

1198642(119901) is shown for 119899 = 100



119865] (119901) =119869]+1 (119901)

119901

(36)




0

05

0 5 10 15 20 25 30p

minus3

minus25

minus2

minus15

minus1

minus05

times10minus4

2E1(p)

andE2(p)

2E1(p)

E2(p)


2(119901) for 119899 = 100 in

Example 5

0 02 04 06 08 1

0

02

04

06

08

1

12

r

minus02

f(r)

ua(r0)

andf(r)

ua(r 0)


119905rarr0+119906119886(119903 119905)) (lowastblue)

0 01 02 03 04 05 06 07 08 09

0

1

2

3

4

r

minus3

minus2

minus1

times10minus3

E(r)


0 02 04 06 08 1

0

02

04

06

08

1

12

r

Solu

tion

for d

iffer

ent t

minus02

u0 for t = 0

u1 for t = 001

u2 for t = 002

u3 for t = 004


0 01 02 03 04 05 06 07 08 09 10

1

2

r

Abso

lute

erro

rtimes10minus4

For t = 0

For t = 001

For t = 002

For t = 004


different 119905

0

05

1

0

05

1

0

02

04

06

08

1

rt

minus02

u(rt)



0

05

1

0

05

10

02

04

06

08

1

rt

ua(rt)




2(119901) for 119899 = 100

are shown

7 Applications


1205972

119906 (119903 119905)

1205972119903

+

1

119903

120597119906 (119903 119905)

120597119903

minus

]2

1199032119906 (119903 119905)

=

120597119906 (119903 119905)

120597119905

(0 lt 119903 lt 1 0 lt 119905 lt infin)

(37)


(i) as 119905 rarr 0+ 119906(119903 119905) rarr 119891(119903) = 119903

][119867(119903) minus 119867(119903 minus 1)]



and 119866 gt 0



1205972

119903) +


Ω]119903119906 =120597119906

120597119905

(38)


int

1

0

119903Ω]119903119906119869] (120573]119904119903) 119889119903 = int1

0

119903 (

120597119906

120597119905

) 119869] (120573]119904119903) 119889119903 (39)


119866119869] (119911) + 1199111198691015840

] (119911) = 0 ] ge minus1

2

(40)


int

1

0

119903 (

1205972

119906

1205972119903

+

1

119903

120597119906

120597119903

minus

]2

1199032119906) 119869] (120573]119904119903) 119889119903

= int

1

0

(

1205972

119906

1205972119903

) 119903119869] (120573]119904119903) 119889119903

+ int

1

0

(

120597119906

120597119903

) 119869] (120573]119904119903) 119889119903 minus ]2 int1

0

(

119906

119903

) 119869] (120573]119904119903) 119889119903


120597119906 (119903 119905)

120597119903

119869] (120573]119904) minus 120573]119904 int1

0

119903

120597119906

120597119903

1198691015840

] (120573]119904119903) 119889119903

minus ]2 int1

0

119906

119903

119869] (120573]119904119903) 119889119903


1

0

120597119906

120597119903

1199031198691015840

] (120573]119904119903) 119889119903

minus ]2 int1

0

119906

119903

119869] (120573]119904119903) 119889119903 (As 119903 997888rarr 1minus

120597119906

120597119903

+ 119866119906 997888rarr 0)

(41)

Since 120573]119904 is a root of 119866119869](119911) + 1199111198691015840

](119911) = 0 we have 119866119869](120573]119904) =minus120573]119904119869

1015840

](120573]119904) and thus

int

1

0

119903 (

1205972

119906

1205972119903

+

1

119903

120597119906

120597119903

minus

]2

1199032119906) 119869] (120573]119904119903) 119889119903

= 120573]1199041198691015840

] (120573]119904) lim119903rarr1minus119906 (119903 119905)

minus 120573]1199041198691015840

] (120573]119904) lim119903rarr1minus119906 (119903 119905)

+ 120573]119904 int1

0

119906 (1198691015840

] (120573]119904119903) + 120573]11990411990311986910158401015840

] (120573]119904119903)) 119889119903

minus ]2 int1

0

119906

119903

119869] (120573]119904119903) 119889119903

(42)


2


int

1

0

119903 (

1205972

119906

1205972119903

+

1

119903

120597119906

120597119903

minus

]2

1199032119906) 119869] (120573]119904119903) 119889119903

= minus1205732

]119904 int1

0

119906 (119903 119905) 119903119869] (120573]119904119903) 119889119903

(43)



minus 1205732

]119904 int1

0

119906 (119903 119905) 119903119869] (120573]119904119903) 119889119903

=

120597

120597119905

(int

1

0

119903119906 (119903 119905) 119869] (120573]119904119903) 119889119903)

(44)

If FHT int10


120597119880 (120573]119904 119905)

120597119905

= minus1205732

]119904119880(120573]119904 119905) (45)


119880(120573]119904 119905) = 119860 (120573]119904) 119890minus1205732]119904119905 (46)


119860 (120573]119904) = 119865] (120573]119904) = int1

0

119891 (119903) 119903119869] (120573]119904119903) 119889119903 (47)


119880 (120573]119904 119905) = 119865] (120573]119904) 119890minus1205732]119904119905 (48)


119906 (119903 119905) = lim119873rarrinfin

119873

sum

119904=1

21205732

]119904119865] (120573]119904) 119890minus1205732]119904119905119869] (120573]119904119903)

(1205732

]119904 minus ]2) 1198692] (120573]119904) + 1205732]11990411986910158402] (120573]119904)


119873

sum

119904=1

21205732

]119904119865] (120573]119904) 119890minus1205732]119904119905119869] (120573]119904119903)

(1205732

]119904 minus ]2) 1198692] (120573]119904) + 11986621198692] (120573]119904)

(As 119866119869] (120573]119904) = minus120573]1199041198691015840

] (120573]119904))

(49)

So

119906 (119903 119905) = lim119873rarrinfin

119873

sum

119904=1

21205732

]119904119865] (120573]119904) 119890minus1205732]119904119905

(1205732

]119904 minus ]2 + 1198662)

119869] (120573]119904119903)

1198692

] (120573]119904) (50)


119865] (120573]119904) = 119874(1

12057332

]119904)

120573]119904 asymp 120587(119904 +1

4

)

1198692

] (120573]119904) asymp2

120587120573]119904


(51)


(Ω]119903 minus120597

120597119905

) 119906 (119903 119905)


119873

sum

119904=1

21205734

]119904119865] (120573]119904) 119890minus1205732]119904119905

(1205732

]119904 minus ]2 + 1198662)

sdot [

[

(1205732

]1199041199032

11986910158401015840

] (120573]119904119903) + 120573]1199041199031198691015840

] (120573]119904119903)

+ (1205732

]1199041199032

minus ]2) 119869] (120573]119904119903))

times (1205732

]1199041199032

1198692

] (120573]119904))minus1

]

]

) = 0

(52)



120597119906 (119903 119905)

120597119903

+ 119866119906 (119903 119905)]


[

[

119873

sum

119904=1

21205732

]119904119865] (120573]119904) 119890minus1205732]119904119905

(1205732

]119904 minus ]2 + 1198662) 1198692] (120573]119904)

times (120573]1199041198691015840

] (120573]119904119903) + 119866119869] (120573]119904119903))]

]

(53)





5(1205735119904) first


5119899(1205735119904) for 119899 = 100


119886(119903 119905) in the above


119906119886(119903 119905) as 119905 rarr 0




119895= 0 1100 150 125



119886(119903 119905) and 119906(119903 119905) at fixed







8 Conclusions




References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

05

0 5 10 15 20 25 30p

minus3

minus25

minus2

minus15

minus1

minus05

times10minus4

2E1(p)

andE2(p)

2E1(p)

E2(p)


2(119901) for 119899 = 100 in

Example 5

0 02 04 06 08 1

0

02

04

06

08

1

12

r

minus02

f(r)

ua(r0)

andf(r)

ua(r 0)


119905rarr0+119906119886(119903 119905)) (lowastblue)

0 01 02 03 04 05 06 07 08 09

0

1

2

3

4

r

minus3

minus2

minus1

times10minus3

E(r)


0 02 04 06 08 1

0

02

04

06

08

1

12

r

Solu

tion

for d

iffer

ent t

minus02

u0 for t = 0

u1 for t = 001

u2 for t = 002

u3 for t = 004


0 01 02 03 04 05 06 07 08 09 10

1

2

r

Abso

lute

erro

rtimes10minus4

For t = 0

For t = 001

For t = 002

For t = 004


different 119905

0

05

1

0

05

1

0

02

04

06

08

1

rt

minus02

u(rt)



0

05

1

0

05

10

02

04

06

08

1

rt

ua(rt)




2(119901) for 119899 = 100

are shown

7 Applications


1205972

119906 (119903 119905)

1205972119903

+

1

119903

120597119906 (119903 119905)

120597119903

minus

]2

1199032119906 (119903 119905)

=

120597119906 (119903 119905)

120597119905

(0 lt 119903 lt 1 0 lt 119905 lt infin)

(37)


(i) as 119905 rarr 0+ 119906(119903 119905) rarr 119891(119903) = 119903

][119867(119903) minus 119867(119903 minus 1)]



and 119866 gt 0



1205972

119903) +


Ω]119903119906 =120597119906

120597119905

(38)


int

1

0

119903Ω]119903119906119869] (120573]119904119903) 119889119903 = int1

0

119903 (

120597119906

120597119905

) 119869] (120573]119904119903) 119889119903 (39)


119866119869] (119911) + 1199111198691015840

] (119911) = 0 ] ge minus1

2

(40)


int

1

0

119903 (

1205972

119906

1205972119903

+

1

119903

120597119906

120597119903

minus

]2

1199032119906) 119869] (120573]119904119903) 119889119903

= int

1

0

(

1205972

119906

1205972119903

) 119903119869] (120573]119904119903) 119889119903

+ int

1

0

(

120597119906

120597119903

) 119869] (120573]119904119903) 119889119903 minus ]2 int1

0

(

119906

119903

) 119869] (120573]119904119903) 119889119903


120597119906 (119903 119905)

120597119903

119869] (120573]119904) minus 120573]119904 int1

0

119903

120597119906

120597119903

1198691015840

] (120573]119904119903) 119889119903

minus ]2 int1

0

119906

119903

119869] (120573]119904119903) 119889119903


1

0

120597119906

120597119903

1199031198691015840

] (120573]119904119903) 119889119903

minus ]2 int1

0

119906

119903

119869] (120573]119904119903) 119889119903 (As 119903 997888rarr 1minus

120597119906

120597119903

+ 119866119906 997888rarr 0)

(41)

Since 120573]119904 is a root of 119866119869](119911) + 1199111198691015840

](119911) = 0 we have 119866119869](120573]119904) =minus120573]119904119869

1015840

](120573]119904) and thus

int

1

0

119903 (

1205972

119906

1205972119903

+

1

119903

120597119906

120597119903

minus

]2

1199032119906) 119869] (120573]119904119903) 119889119903

= 120573]1199041198691015840

] (120573]119904) lim119903rarr1minus119906 (119903 119905)

minus 120573]1199041198691015840

] (120573]119904) lim119903rarr1minus119906 (119903 119905)

+ 120573]119904 int1

0

119906 (1198691015840

] (120573]119904119903) + 120573]11990411990311986910158401015840

] (120573]119904119903)) 119889119903

minus ]2 int1

0

119906

119903

119869] (120573]119904119903) 119889119903

(42)


2


int

1

0

119903 (

1205972

119906

1205972119903

+

1

119903

120597119906

120597119903

minus

]2

1199032119906) 119869] (120573]119904119903) 119889119903

= minus1205732

]119904 int1

0

119906 (119903 119905) 119903119869] (120573]119904119903) 119889119903

(43)



minus 1205732

]119904 int1

0

119906 (119903 119905) 119903119869] (120573]119904119903) 119889119903

=

120597

120597119905

(int

1

0

119903119906 (119903 119905) 119869] (120573]119904119903) 119889119903)

(44)

If FHT int10


120597119880 (120573]119904 119905)

120597119905

= minus1205732

]119904119880(120573]119904 119905) (45)


119880(120573]119904 119905) = 119860 (120573]119904) 119890minus1205732]119904119905 (46)


119860 (120573]119904) = 119865] (120573]119904) = int1

0

119891 (119903) 119903119869] (120573]119904119903) 119889119903 (47)


119880 (120573]119904 119905) = 119865] (120573]119904) 119890minus1205732]119904119905 (48)


119906 (119903 119905) = lim119873rarrinfin

119873

sum

119904=1

21205732

]119904119865] (120573]119904) 119890minus1205732]119904119905119869] (120573]119904119903)

(1205732

]119904 minus ]2) 1198692] (120573]119904) + 1205732]11990411986910158402] (120573]119904)


119873

sum

119904=1

21205732

]119904119865] (120573]119904) 119890minus1205732]119904119905119869] (120573]119904119903)

(1205732

]119904 minus ]2) 1198692] (120573]119904) + 11986621198692] (120573]119904)

(As 119866119869] (120573]119904) = minus120573]1199041198691015840

] (120573]119904))

(49)

So

119906 (119903 119905) = lim119873rarrinfin

119873

sum

119904=1

21205732

]119904119865] (120573]119904) 119890minus1205732]119904119905

(1205732

]119904 minus ]2 + 1198662)

119869] (120573]119904119903)

1198692

] (120573]119904) (50)


119865] (120573]119904) = 119874(1

12057332

]119904)

120573]119904 asymp 120587(119904 +1

4

)

1198692

] (120573]119904) asymp2

120587120573]119904


(51)


(Ω]119903 minus120597

120597119905

) 119906 (119903 119905)


119873

sum

119904=1

21205734

]119904119865] (120573]119904) 119890minus1205732]119904119905

(1205732

]119904 minus ]2 + 1198662)

sdot [

[

(1205732

]1199041199032

11986910158401015840

] (120573]119904119903) + 120573]1199041199031198691015840

] (120573]119904119903)

+ (1205732

]1199041199032

minus ]2) 119869] (120573]119904119903))

times (1205732

]1199041199032

1198692

] (120573]119904))minus1

]

]

) = 0

(52)



120597119906 (119903 119905)

120597119903

+ 119866119906 (119903 119905)]


[

[

119873

sum

119904=1

21205732

]119904119865] (120573]119904) 119890minus1205732]119904119905

(1205732

]119904 minus ]2 + 1198662) 1198692] (120573]119904)

times (120573]1199041198691015840

] (120573]119904119903) + 119866119869] (120573]119904119903))]

]

(53)





5(1205735119904) first


5119899(1205735119904) for 119899 = 100


119886(119903 119905) in the above


119906119886(119903 119905) as 119905 rarr 0




119895= 0 1100 150 125



119886(119903 119905) and 119906(119903 119905) at fixed







8 Conclusions




References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

05

1

0

05

10

02

04

06

08

1

rt

ua(rt)




2(119901) for 119899 = 100

are shown

7 Applications


1205972

119906 (119903 119905)

1205972119903

+

1

119903

120597119906 (119903 119905)

120597119903

minus

]2

1199032119906 (119903 119905)

=

120597119906 (119903 119905)

120597119905

(0 lt 119903 lt 1 0 lt 119905 lt infin)

(37)


(i) as 119905 rarr 0+ 119906(119903 119905) rarr 119891(119903) = 119903

][119867(119903) minus 119867(119903 minus 1)]



and 119866 gt 0



1205972

119903) +


Ω]119903119906 =120597119906

120597119905

(38)


int

1

0

119903Ω]119903119906119869] (120573]119904119903) 119889119903 = int1

0

119903 (

120597119906

120597119905

) 119869] (120573]119904119903) 119889119903 (39)


119866119869] (119911) + 1199111198691015840

] (119911) = 0 ] ge minus1

2

(40)


int

1

0

119903 (

1205972

119906

1205972119903

+

1

119903

120597119906

120597119903

minus

]2

1199032119906) 119869] (120573]119904119903) 119889119903

= int

1

0

(

1205972

119906

1205972119903

) 119903119869] (120573]119904119903) 119889119903

+ int

1

0

(

120597119906

120597119903

) 119869] (120573]119904119903) 119889119903 minus ]2 int1

0

(

119906

119903

) 119869] (120573]119904119903) 119889119903


120597119906 (119903 119905)

120597119903

119869] (120573]119904) minus 120573]119904 int1

0

119903

120597119906

120597119903

1198691015840

] (120573]119904119903) 119889119903

minus ]2 int1

0

119906

119903

119869] (120573]119904119903) 119889119903


1

0

120597119906

120597119903

1199031198691015840

] (120573]119904119903) 119889119903

minus ]2 int1

0

119906

119903

119869] (120573]119904119903) 119889119903 (As 119903 997888rarr 1minus

120597119906

120597119903

+ 119866119906 997888rarr 0)

(41)

Since 120573]119904 is a root of 119866119869](119911) + 1199111198691015840

](119911) = 0 we have 119866119869](120573]119904) =minus120573]119904119869

1015840

](120573]119904) and thus

int

1

0

119903 (

1205972

119906

1205972119903

+

1

119903

120597119906

120597119903

minus

]2

1199032119906) 119869] (120573]119904119903) 119889119903

= 120573]1199041198691015840

] (120573]119904) lim119903rarr1minus119906 (119903 119905)

minus 120573]1199041198691015840

] (120573]119904) lim119903rarr1minus119906 (119903 119905)

+ 120573]119904 int1

0

119906 (1198691015840

] (120573]119904119903) + 120573]11990411990311986910158401015840

] (120573]119904119903)) 119889119903

minus ]2 int1

0

119906

119903

119869] (120573]119904119903) 119889119903

(42)


2


int

1

0

119903 (

1205972

119906

1205972119903

+

1

119903

120597119906

120597119903

minus

]2

1199032119906) 119869] (120573]119904119903) 119889119903

= minus1205732

]119904 int1

0

119906 (119903 119905) 119903119869] (120573]119904119903) 119889119903

(43)



minus 1205732

]119904 int1

0

119906 (119903 119905) 119903119869] (120573]119904119903) 119889119903

=

120597

120597119905

(int

1

0

119903119906 (119903 119905) 119869] (120573]119904119903) 119889119903)

(44)

If FHT int10


120597119880 (120573]119904 119905)

120597119905

= minus1205732

]119904119880(120573]119904 119905) (45)


119880(120573]119904 119905) = 119860 (120573]119904) 119890minus1205732]119904119905 (46)


119860 (120573]119904) = 119865] (120573]119904) = int1

0

119891 (119903) 119903119869] (120573]119904119903) 119889119903 (47)


119880 (120573]119904 119905) = 119865] (120573]119904) 119890minus1205732]119904119905 (48)


119906 (119903 119905) = lim119873rarrinfin

119873

sum

119904=1

21205732

]119904119865] (120573]119904) 119890minus1205732]119904119905119869] (120573]119904119903)

(1205732

]119904 minus ]2) 1198692] (120573]119904) + 1205732]11990411986910158402] (120573]119904)


119873

sum

119904=1

21205732

]119904119865] (120573]119904) 119890minus1205732]119904119905119869] (120573]119904119903)

(1205732

]119904 minus ]2) 1198692] (120573]119904) + 11986621198692] (120573]119904)

(As 119866119869] (120573]119904) = minus120573]1199041198691015840

] (120573]119904))

(49)

So

119906 (119903 119905) = lim119873rarrinfin

119873

sum

119904=1

21205732

]119904119865] (120573]119904) 119890minus1205732]119904119905

(1205732

]119904 minus ]2 + 1198662)

119869] (120573]119904119903)

1198692

] (120573]119904) (50)


119865] (120573]119904) = 119874(1

12057332

]119904)

120573]119904 asymp 120587(119904 +1

4

)

1198692

] (120573]119904) asymp2

120587120573]119904


(51)


(Ω]119903 minus120597

120597119905

) 119906 (119903 119905)


119873

sum

119904=1

21205734

]119904119865] (120573]119904) 119890minus1205732]119904119905

(1205732

]119904 minus ]2 + 1198662)

sdot [

[

(1205732

]1199041199032

11986910158401015840

] (120573]119904119903) + 120573]1199041199031198691015840

] (120573]119904119903)

+ (1205732

]1199041199032

minus ]2) 119869] (120573]119904119903))

times (1205732

]1199041199032

1198692

] (120573]119904))minus1

]

]

) = 0

(52)



120597119906 (119903 119905)

120597119903

+ 119866119906 (119903 119905)]


[

[

119873

sum

119904=1

21205732

]119904119865] (120573]119904) 119890minus1205732]119904119905

(1205732

]119904 minus ]2 + 1198662) 1198692] (120573]119904)

times (120573]1199041198691015840

] (120573]119904119903) + 119866119869] (120573]119904119903))]

]

(53)





5(1205735119904) first


5119899(1205735119904) for 119899 = 100


119886(119903 119905) in the above


119906119886(119903 119905) as 119905 rarr 0




119895= 0 1100 150 125



119886(119903 119905) and 119906(119903 119905) at fixed







8 Conclusions




References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











minus 1205732

]119904 int1

0

119906 (119903 119905) 119903119869] (120573]119904119903) 119889119903

=

120597

120597119905

(int

1

0

119903119906 (119903 119905) 119869] (120573]119904119903) 119889119903)

(44)

If FHT int10


120597119880 (120573]119904 119905)

120597119905

= minus1205732

]119904119880(120573]119904 119905) (45)


119880(120573]119904 119905) = 119860 (120573]119904) 119890minus1205732]119904119905 (46)


119860 (120573]119904) = 119865] (120573]119904) = int1

0

119891 (119903) 119903119869] (120573]119904119903) 119889119903 (47)


119880 (120573]119904 119905) = 119865] (120573]119904) 119890minus1205732]119904119905 (48)


119906 (119903 119905) = lim119873rarrinfin

119873

sum

119904=1

21205732

]119904119865] (120573]119904) 119890minus1205732]119904119905119869] (120573]119904119903)

(1205732

]119904 minus ]2) 1198692] (120573]119904) + 1205732]11990411986910158402] (120573]119904)


119873

sum

119904=1

21205732

]119904119865] (120573]119904) 119890minus1205732]119904119905119869] (120573]119904119903)

(1205732

]119904 minus ]2) 1198692] (120573]119904) + 11986621198692] (120573]119904)

(As 119866119869] (120573]119904) = minus120573]1199041198691015840

] (120573]119904))

(49)

So

119906 (119903 119905) = lim119873rarrinfin

119873

sum

119904=1

21205732

]119904119865] (120573]119904) 119890minus1205732]119904119905

(1205732

]119904 minus ]2 + 1198662)

119869] (120573]119904119903)

1198692

] (120573]119904) (50)


119865] (120573]119904) = 119874(1

12057332

]119904)

120573]119904 asymp 120587(119904 +1

4

)

1198692

] (120573]119904) asymp2

120587120573]119904


(51)


(Ω]119903 minus120597

120597119905

) 119906 (119903 119905)


119873

sum

119904=1

21205734

]119904119865] (120573]119904) 119890minus1205732]119904119905

(1205732

]119904 minus ]2 + 1198662)

sdot [

[

(1205732

]1199041199032

11986910158401015840

] (120573]119904119903) + 120573]1199041199031198691015840

] (120573]119904119903)

+ (1205732

]1199041199032

minus ]2) 119869] (120573]119904119903))

times (1205732

]1199041199032

1198692

] (120573]119904))minus1

]

]

) = 0

(52)



120597119906 (119903 119905)

120597119903

+ 119866119906 (119903 119905)]


[

[

119873

sum

119904=1

21205732

]119904119865] (120573]119904) 119890minus1205732]119904119905

(1205732

]119904 minus ]2 + 1198662) 1198692] (120573]119904)

times (120573]1199041198691015840

] (120573]119904119903) + 119866119869] (120573]119904119903))]

]

(53)





5(1205735119904) first


5119899(1205735119904) for 119899 = 100


119886(119903 119905) in the above


119906119886(119903 119905) as 119905 rarr 0




119895= 0 1100 150 125



119886(119903 119905) and 119906(119903 119905) at fixed







8 Conclusions




References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













8 Conclusions




References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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