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MATHEMATICAL PHYSICS

ON USING THIRD AND FOURTH KINDS CHEBYSHEV OPERATIONALMATRICES FOR SOLVING LANE-EMDEN TYPE EQUATIONS

E.H. DOHA1,a, W.M. ABD-ELHAMEED1,2,b, M.A. BASSUONY3

1Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, EgyptE-maila: [email protected]

2Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi ArabiaE-mailb: walee [email protected]

3Department of Mathematics, Faculty of Science, Fayoum University, Fayoum 63514, EgyptE-mail: [email protected]

Received July 27, 2014

This paper is concerned with deriving two new operational matrices of deriva-tives for Chebyshev polynomials of third and fourth kinds. As an important applicationof these introduced operational matrices, a certain class of linear and nonlinear Lane-Emden type singular initial value problems (IVPs) are treated. Two numerical algo-rithms are described in detail for solving such kinds of problems. The idea of obtainingour algorithms is essentially based on converting the differential equation with its initialconditions to a system of linear or nonlinear algebraic equations. Numerical examplesconcern some relevant physical problems are included to demonstrate the validity andapplicability of the proposed algorithms. In addition, some comparisons with someother methods are made.

Key words: Lane-Emden equations, operational matrices of differentiation, thirdand fourth kinds Chebyshev polynomials.

PACS: 02.60.Cb, 02.70.Hm, 02.70.Jn, 02.30.Mv, 02.30.Gp.

1. INTRODUCTION

The study of Lane-Emden type equations has attracted the attention of manymathematicians and physicists due to their great importance in many fields. It canmodel many phenomena in mathematical physics and astrophysics such as thermalexplosions [1], stellar structure [2] and isothermal gas spheres and thermionic cur-rents [3].

Lane-Emden type equations are singular IVPs. Their solutions, as well as othervarious linear and nonlinear singular initial value problems in quantum mechanicsand astrophysics, are numerically challenging because of the singularity behavior atthe origin. Adomian’s decomposition method (ADM) (see for instance, [4, 5]) isthe well-known systematic method used for solving these types of equations. Someof these algorithms are developed by Wazwaz [6–8]. Other approximate solutionsto Lane-Emden type equations are given by a large number of authors. Amongthem, approximations by homotopy perturbation method [9–11], variational itera-tion method [12], sinc-collocation method [13], optimal homotopy method [14], an

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282 E.H. Doha, W.M. Abd-Elhameed, M.A. Bassuony 2

implicit series solution [15] and perturbation techniques [16, 17]. Recently, someother approximate solutions of Lane-Emden equations are obtained using secondkind Chebyshev operational matrix algorithm [18], the square remainder minimiza-tion method [19], the improved Bessel collocation method [20] and Bessel orthogo-nal functions collocation method [21].

Chebyshev polynomials have become increasingly crucial in numerical analy-sis from both theoretical and practical points of view. It is well-known that thesepolynomials have strong links with Fourier and Laurent series, with minimality pro-perties in approximation theory and with discrete and continuous orthogonality infunction spaces. These links have led to important applications, especially in spec-tral methods for ordinary and partial differential equations. Most books and researchpapers which deal with Chebyshev polynomials, of course contain many results ofChebyshev polynomials of first and second kinds Tn(x) and Un(x) and their nu-merous applications uses in different applications, (see for instance, Boyd [22–25],Julien and Watson [26], Fox and Parker [27] and Mason [28]). The two other fam-ilies of polynomials, namely, Chebyshev polynomials of third kind Vn(x) and offourth kind Wn(x) may be constructed, which are related to Tn(x) and Un(x), buthave trigonometric definitions involving the half angle θ/2 (where x= cosθ ). Thesepolynomials are sometimes referred to as the airfoil polynomials (see for example,Fromme and Golberg, [29]), but Gautschi [30] rather appropriately named them thethird and fourth kinds Chebyshev polynomials. The main reason for the lack of useof such kinds of polynomials in various applications is that Vn(x) and Wn(x) aretwo special cases of the nonsymmetric Jacobi polynomials P (α,β)

n (x)(α 6= β), whichare more difficult to deal with than the symmetric cases. Recently, the numericaland theoretical studies of these kinds of polynomials have attracted the attention ofsome authors. For example, Doha and Abd-Elhameed in [31] have stated and provednew closed formulae for the coefficients of integrated expansions and integrals ofChebyshev polynomials of third and fourth kinds. Also, Doha et al. in [32, 33] havedeveloped new algorithms for solving high even/odd-order differential equations us-ing third and fourth kinds Chebyshev-Galerkin methods.

In this paper, we consider Lane-Emden type equation

u′′+λ

xu′+f(x,y) = g(x), 0< x < a, λ> 0, (1)

subject to the initial conditions

y(0) =A, y′(0) =B, (2)

whereA andB are known constants, f(x,y) is a continuous real valued function andg(x) ∈ C[0,1].

Briefly, our main objectives in this paper are to introduce third and fourth kinds

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3 Using 3rd & 4th kinds Chebyshev operational matrices for solving Lane-Emden type eqs. 283

Chebyshev operational matrices of derivatives and to develop two numerical algo-rithms for solving Lane-Emden type equations based on third and fourth Chebyshevoperational matrices.

2. SOME PROPERTIES OF THIRD AND FOURTH KINDS CHEBYSHEV POLYNOMIALSAND THEIR SHIFTED ONES

Third and fourth kinds Chebyshev polynomials Vn(x) and Wn(x) are definedby trigonometric functions, respectively, as (see, Mason and Handscomb [34])

Vn(cosθ) =cos(n+ 1

2)θ

cos θ2, and Wn(cosθ) =

sin(n+ 12)θ

sin θ2

,

or defined equivalently as two particular nonsymmetric special cases of Jacobi poly-nomials P (α,β)

n (x) as

Vn(x) =22n(2nn

) P (− 12, 12)

n (x), and Wn(x) =22n(2nn

) P ( 12,− 1

2)

n (x).

These polynomials are orthogonal on (−1,1) with respect to the weight functions√1+x

1−xand

√1−x1+x

, respectively, i.e.,∫ 1

−1

√1+x

1−xVn(x)Vm(x)dx=

∫ 1

−1

√1−x1+x

Wn(x)Wm(x)dx=

{π, n=m,

0, n 6=m

The Vn(x) and Wn(x) are related by the relation

Wn(x) = (−1)nVn(−x), (3)

and as a consequence of (3), it is sufficient to establish properties for Vn(x) only anddeduce directly the corresponding results for Wn(x).

The following structure formula is useful in the sequel

Vn(x) =1

2n(n+1)[nDVn+1(x)−DVn(x)− (n+1)DVn−1(x)] ,

n≥ 1, D ≡ d

dx, (4)

(see, Shen et al. [35], pp. 77, Theorem 3.23, with α = −β = −12 ). The interested

reader in Chebyshev polynomials is referred to the interesting book of Mason andHandscomb [34].

The shifted third and fourth kinds Chebyshev polynomials are defined on [a,b],respectively, as

V ∗n (x) = Vn

(2x−a− bb−a

), W ∗n(x) =Wn

(2x−a− bb−a

). (5)

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These polynomials are orthogonal on [a,b] with respect to the weight functions

w1(x) =√

x−ab−x and w2(x) =

√b−xx−a , respectively, i.e.,∫ b

aw1(x)V

∗n (x)V

∗m(x)dx=

∫ b

aw2(x)W

∗n(x)W

∗m(x)dx=

{(b−a)π2 , n=m,

0, n 6=m.(6)

3. THIRD AND FOURTH KINDS CHEBYSHEV OPERATIONAL MATRICES OFDERIVATIVES AND THEIR SHIFTED ONES

The main objective of this section is to state and prove new analytical formulaexpressing explicitly the first derivative of Chebyshev polynomials of third kind interms of their Chebyshev polynomials themselves. The corresponding analytical for-mula for Chebyshev polynomials of fourth kind will also be deduced. As a directconsequence, two new shifted algorithms based on third and fourth kinds Chebyshevoperational matrices of derivatives (SC3OMD and SC4OMD) are developed.Lemma 1. For all n≥ 1, one has

DVn(x) =n−1∑r=0

(n+r)odd

(n+ r+1)Vr(x)+n−2∑r=0

(n+r)even

(n− r)Vr(x). (7)

Proof. Let us denote

Sn(x) =

n∑r=0

ar Vr(x), (8)

and

In+1(x) =

∫Sn(x)dx=

n∑r=0

ar

∫Vr(x)dx, (9)

then integration of both sides of the structure formula (4) yields∫Vr(x)dx=

1

2r(r+1)[rVr+1(x)−Vr(x)− (r+1)Vr−1(x)] , r ≥ 1 (10)

which in turn implies that

In+1(x) = a0x+

n∑r=1

ar2r(r+1)

[rVr+1(x)−Vr(x)− (r+1)Vr−1(x)] . (11)

This equation may be written in the form

In+1(x) =n+1∑r=0

Ar Vr(x), (12)

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where

A0 =a0−a1

2,

Ar =(r+1)ar−1−ar− rar+1

2r(r+1), r = 1,2, . . . ,n,n+1, an+1 = an+2 = 0.

(13)

The difference equation (13) is solved to give

ar =n+1∑

k=r+1(k+r)odd

(k+ r+1)Ak+n+1∑

k=r+2(k+r) even

(k− r)Ak, r = 0,1, . . . ,n. (14)

Substitute of relation (14) into Eq. (8), yields

Sn(x) =n∑r=0

n+1∑

k=r+1(k+r)odd

(k+ r+1)Ak+n+1∑

k=r+2(k+r) even

(k− r)Ak

Vr(x). (15)

On the other hand, if we differentiate Eq. (12) with respect to x, then we get

Sn(x) =n+1∑r=0

AnDVr(x). (16)

Now, relations (15) and (16) immediately give

n∑r=0

n+1∑

k=r+1(k+r)odd

(k+ r+1)Ak+n+1∑

k=r+2(k+r) even

(k− r)Ak,

Vr(x) =n+1∑r=0

AnDVr(x). (17)

Expanding the left hand side of (17) and collecting similar terms, and after somerather lengthy manipulation, one can write

n+1∑r=0

ArBr(x) =n+1∑r=0

ArDVr(x),

where

Br(x) =r−1∑k=0

(r+k)odd

(r+k+1)Vk(x)+r−2∑k=0

(k+r)even

(k+ r)Vk(x),

which immediately yields

DVr(x) =

r−1∑k=0

(r+k)odd

(r+k+1)Vk(x)+

r−2∑k=0

(r+k)even

(r−k)Vk(x),

and this completes the proof of of Lemma 1.

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Remark 1. It is to be noted here that relation (7) may be written alternatively in thecompact form

DVn(x) =n−1∑r=0

en,r Vr(x), (18)

where

en,r =

{n+ r+1, (n+ r) odd,n− r, (n+ r) even.

(19)

Corollary 1. For shifted third Chebyshev polynomials, we have

DV ∗n (x) =2

b−a

n−1∑r=0

en,r V∗r (x), (20)

where en,r is defined as in (19).

3.1. CONSTRUCTION OF SC3OMD AND SC4OMD

Any function u(x) ∈ L2w1[a,b] can be expanded in terms of shifted third kind

Chebyshev polynomials in the form

u(x) =∞∑i=0

ciV∗i (x), (21)

where

ci =2

π(b−a)

∫ b

aw1(x)u(x)V

∗i (x)dx. (22)

In general, we approximate the series in Eq. (21) by taking the first (N +1) terms as

uN (x) =N∑i=0

ciV∗i (x) = CT Φ(x), (23)

whereCT = [c0, c1, . . . , cN ],Φ(x) = [V ∗0 (x),V

∗1 (x), . . . ,V

∗N (x)]

T . (24)The operational matrix of the first derivative of shifted third kind Chebyshev polyno-mials set Φ(x) may be defined by

dΦ(x)

dx= EΦ(x), (25)

where E = (eij)0≤i,j≤N is the square operational matrix of order (N +1) whosenonzero elements are given explicitly by

eij =2

b−a

{i+ j+1 i > j, (i+ j) odd,i− j, i > j, (i+ j) even.

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For example if N = 6, then the operational matrix E is given explicitly by

E =2

b−a

0 0 0 0 0 0 02 0 0 0 0 0 02 4 0 0 0 0 04 2 6 0 0 0 04 6 2 8 0 0 06 4 8 2 10 0 06 8 4 10 2 12 0

7×7

.

It can be easily shown that for any n ∈ Z+

dnΦ(x)

dxn= EnΦ(x), n= 0,1,2, . . . . (26)

3.2. CONVERGENCE ANALYSIS

In this section, we state and prove a theorem to ascertain that the shifted thirdkind Chebyshev expansion (21) of a function u(x) , with bounded second derivative,converges uniformly to f(x).

Theorem 1. A function u(x) ∈ L2w1[a,b], w1(x) =

√x−ab−x with |u′′(x)|6M, can be

expanded as an infinite sum of shifted third kind Chebyshev basis, and the series con-verges uniformly to u(x). Explicitly, the following inequality holds for the expansioncoefficients in (22), i.e.,

|cn|<M (b−a)2

n2, ∀n > 1. (27)

Proof. Returning to relation (22), we can write

cn =2

π(b−a)

∫ b

a

√x−ab−x

u(x)V ∗n (x)dx, (28)

and making use of the substitution 2x−a−bb−a = cosθ enables one to get

cn =2

π

∫ π

0u

(a+ b+(b−a)cosθ

2

)cos

(n+

1

2

)θ cos

(θ

2

)dθ,

which in turn and after integration by parts two times, gives

cn =(b−a)2

4π

∫ π

0u′′(a+ b+(b−a)cosθ

2

)γn(θ)dθ,

where

γn(θ)= sinθ

[1

n

(sin(n−1)θ

n−1− sin(n+1)θ

n+1

)+

1

n+1

(sin nθ

n− sin(n+2)θ

n+2

)].

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Now,

|cn|=∣∣∣∣(b−a)24π

∫ π

0u′′(a+ b+(b−a)cosθ

2

)γn(θ)dθ

∣∣∣∣6M (b−a)2

4π

∫ π

0|γn(θ)| dθ

6M (b−a)2

4

(1

n

[1

n−1+

1

n+1

]+

1

n+1

[1

n+

1

n+2

])=M (b−a)2 (2n2+2n−1)

2(n−1)n(n+1)(n+2).

Since n > 1, and for large n, then we get

|cn|<M (b−a)2

n2,

which completes the proof of the theorem.

Remark 2. It is worthy to mention here that inequality (27) holds also if uN (x) in(23) is expanded in terms of Wn(x).

4. APPLICATION OF SC3OMD AND SC4OMD TO SOLVE LANE-EMDEN TYPEEQUATIONS

Now, let us consider Lane-Emden equation (1) subject to the initial conditions(2). If we approximate u(x), f(x,y) and g(x) by the shifted third kind Chebyshevpolynomials as

uN (x) =

N∑i=0

ciV∗i (x) = CT Φ(x), (29)

f(x,y)≈ f(x,CT Φ(x)), (30)

g(x)≈N∑i=0

giV∗i (x) = GT Φ(x), (31)

then making use of SC3OMD, enables one to write the residual of Eq. (1) as

RN (x) = CTE2Φ(x)+λ

xCTEΦ(x)+f(x,CTΦ(x))−GTΦ(x). (32)

If we apply typical tau method, which is used in the sense of a particular form ofthe Petrov-Galerkin method (see, [36]), then Eq. (32) is reduced to (N −1) linear ornonlinear equations, namely

(RN (x),V∗i (x))w1 =

∫ b

aw1(x) RN (x)V

∗i (x)dx= 0, i= 0,1, . . . ,N −2. (33)

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Now, the initial conditions (2) lead to the two equations

u(0) = CTΦ(0) =A, u′(0) = CTEΦ(0) =B, (34)

and therefore Eqs. (33) together with Eqs. (34) generate a set of (N +1) linear ornonlinear equations. The linear system can be solved by using any suitable solver,while the nonlinear equations can be solved with the aid of Newton’s iteration methodfor the unknown components of vector C, and hence the approximate solution uN (x)can be obtained.

5. NUMERICAL RESULTS

In this section, the two developed algorithms presented in Section 4 are em-ployed to solve the linear and nonlinear Lane-Emden equations aiming to illustratethe efficiency and the applicability of them.Example 1. Consider the isothermal gas spheres equation which is modeled byDavis [37] in the form

u′′(x)+2

xu′(x)+eu(x) = 0; 0< x6 1, u(0) = 0, u′(0) = 0. (35)

The numerical solutions obtained using SC3OMD and SC4OMD are, respectively,

u6 =−2.77556×10−17x−0.166666x2−0.0000129048x3+0.00840262x4−0.000152895x5−0.000399899x6,

and

u6 =0.166667x2−2.48861×10−6x3+0.00835658x4−0.000074838x5−0.000444629x6.

Table 1 lists the maximum pointwise error |uR.K−uN | obtained by using SC3OMDand SC4OMD for various values of N , where uR.K is the approximate solution ob-tained by using a Runge-Kutta of order fourth. In Fig. 1, we compare our solutions(for N = 6) with SRMM (square remainder minimization method) in [19].

Table 1

Maximum pointwise error |uR.K−uN | using SC3OMD and SC4OMD for Example 1.

N SC3OMD SC4OMD6 1.3291×10−6 4.31398×10−6

8 1.13628×10−8 5.00007×10−8

10 7.78796×10−11 3.02559×10−10

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Fig. 1 – Comparison for Example 1

0.0 0.2 0.4 0.6 0.8 1.0

0

5. µ 10-8

1. µ 10-7

1.5 µ 10-7

x

Err

or

SRMM

SC4OMD

SC3OMD

Example 2. Consider Lane-Emden equation used by Richardson in his theory ofthermionic currents, which studies the density and electric force of an electron gasin the neighborhood of a hot body in thermal equilibrium (see, [19]). The equationand corresponding initial conditions are:

u′′(x)+2

xu′(x)+e−u(x) = 0; 0< x6 1, u(0) = 0, u′(0) = 0. (36)

The solutions obtained using SC3OMD and SC4OMD methods are, respectively,

u6 =−0.166668x2+0.0000169745x3−0.00842347x4+0.000194883x5

−0.000687951x6,

and

u6 =−6.93889×10−18−0.166667x2+2.99349×10−6x3−0.00836113x4

+0.0000883397x5−0.000626463x6.

Table 2 lists the maximum pointwise error |uR.K−uN | for various values ofN . Fig. 2gives a comparison between the absolute errors obtained by our methods and SRMMin [19].

Table 2

Maximum pointwise error |uR.K−uN | using SC3OMD and SC4OMD for Example 2.

N SC3OMD SC4OMD6 2.28512×10−6 6.50908×10−6

8 2.56751×10−8 9.76758×10−8

10 2.09807×10−10 7.85315×10−10

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Fig. 2 – Comparison for Example 2

0.0 0.2 0.4 0.6 0.8 1.0

0

5. µ 10-8

1. µ 10-7

1.5 µ 10-7

2. µ 10-7

2.5 µ 10-7

3. µ 10-7

3.5 µ 10-7

x

Err

or

SRMM

SC4OMD

SC3OMD

6. CONCLUDING REMARKS

In this paper, two new operational matrices of derivatives of Chebyshev polyno-mials of third and fourth kinds are derived. These operational matrices are employedfor obtaining some numerical solutions of Lane-Emden type equations. The mainadvantages of the developed algorithms are their simplicity and high accuracy whichwere achieved by using a few number of terms. The algorithms are applicable andthe obtained numerical results are convincing.

Acknowledgements. The authors would like to thank the anonymous referee for his carefullyreading the manuscript and for his valuable comments and suggestions.

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