an extension of the finite hankel transforms*

Znt. f. Engng Sci. Vol. 3, pp. 539-559. Pergamon Press 1965. Printed in Great Britain

AN EXTENSION OF THE FINITE HANKEL TRANSFORM AN-D APPLICATIONS*

G. CINELLI

Argonne National laboratory, Argonne, Illinois

(Communicated by I. N. SNEDDON)

Abstract-New finite Hankel transforms with kernels of the form [X,(&x) Yfi(&r)--J,(&r) Y,(&x)] and the corresponding infinite series are introduced, which bring the solution of Bessel’s equation with asymmetric endpoint conditions within the realm of integral transform theory. A general solution for the transient temperatures produced in a finite, hollow cylinder by an asymmetrical internal heat source, when radiation takes place on all four surfaces, is accomplished. It is shown how this solution contains Carslaw and Jaeger’s results [l] on hollow cylinders as a special case.

1. INTRODUCTION

THE solution of boundary vaIue problems in a finite domain requires the boundary~onditions to be specified on every surface of the region. Each spatial direction in a finite region has two surfaces at which boundary conditions must be known. If the boundary conditions are the same for both surfaces they are said to be ‘ symmetric’. If not, they are then called ‘ asymmetric’. Types of boundary conditions that usually arise in physical problems are three, namely; Dirichlet (f=O), Neumann (&f=O), and Cauchy (A&,f+3f==O). For two surfaces the number of possible combinations of these types are nine, three symmet~c and six asymmetric.

The use of finite integral transforms on the space variables considerably simplifies the solution of boundary value problems. In particular, cylindrical geometry problems are usually resolved by means of the finite Hankel transforms introduced by Sneddon [2]. However, his work applies only to those problems in which the boundary conditions are symmetric and of the Dirichlet type. Kaplan and Sonneman [3] have treated two of the asymmetric cases for Bessel kernels of order zero. It is the purpose of this paper to extend the finite Hankel transform method to include all symmetric and asymmetric cases using Bessel kernels of arbitrary order. This work thus enables the technique to be applied to a wider class of problems.

The paper is divided into three parts. The first part illustrates the method and formulas used in establishing the results. The second portion lists the new transforms, corresponding infinite series, and operational properties applicable to Cauchy boundary conditions. Then a general solution for the transient temperatures produced in a finite, hollow cylinder by an arbitrary asymmetrical internal heat source with radiation occurring on all four surfaces is accomplished. It is shown how Carslaw and Jaeger’s work [ 11 on hollow cylinders appears as a special case of the general solution.

* Work performed under the auspices of the U.S. Atomic Energy Commistiion.

539

540 G. CINELLI

2. ESTABLISHMENT OF TRANSFORMS

The basic technique used in establishing the new transforms can be found in [3-61. In essence, it consists of solving the Sturm-Liouville problem for the particular boundary value problem in question and defining the definite integral that arises during the norma- fixation process as the finite transform. To illustrate the method along with the necessary formulas to achieve the new results, the expressions given by Sneddon [23 for the case of Dirichlet conditions on both surfaces will be derived.

The Sturm-Liouville problem for this case is:

f=O asx<b

Boundary conditions :

fW=.f(b)=O

The general solution to equation (I) is

ftx)=AJ,fgiX)+BY~(5iX) where

J1,(5iX), Y,(tix) are Bessel functions of the first and second kind, and of order p. Using the boundary conditions in equation (2), a nontrivial solution is

.f(xl=AIJp(Six) y~(~~~)-~~(~~~) yfi(tjxll

where ti is a positive root of the equation

J&(5$) YJria) -Jp(<ia) YM(&b) ~0.

(1)

(2)

(3)

(4)

(5)

Assuming thatf(x) satisfies Dirichlet conditions in the range a<xlb it can be expanded as an infinite series of the typ:: in equation (4). Doing this and normalizing gives the value of the constant A as

f;i(SJ= ff[f (xl] = ~*f(~)~~,(~~~)y,(ii.)-~,(Cio)Y~(~~x)ld~ a

in which equation (7) defines the finite Hankel transform off(x). Expanding the denominator of equation (6) gives

Using the indefinite integral*

(6)

(7)

(8)

(9

* Appendix 1.

An extension of the finite Hankel transform and applications 541

where W,(5iX), W,(riX) can be any Bessel function of the first, second, or third kind and of order p.

In equation (8), putting in the limits and simplifying gives,

s b

x[~,(Six)Y~(i;i~)-J,(~i~)Y~(~ix)l’dX=~ [J’,(rib>Y,(Sia)-J,(5ia)Y;(5ib)12 a

+

Using the Wronskian relationship

J:(z)YJz)-J,(z)Y;(z)=L (11) 712

and equation (5), it can be shown that

Placing equations (11) and (12) into equation (10) gives as a final result

s b

a XCJ,(TiX)Y,(CiU)-J,(Sia)Y,(S,X)]’dX=~’:2 “r’(‘~~(~~~Cb)

I s i

Putting equations (6) and (13) into equation (4) gives the desired series expansion as

(13)

(14)

where S(ri) is defined in equation (7) Ti is a positive root of equation (5).

Returning to equations (2) and (3) it follows that there exists another solution similar to equation (4) which is

.Ax) = 41Jjt(Six) Yfl(5ib) -Jp(tib) yp(tix)l (15)

and where ti is still a root of equation (5). In this case the finite Hankel transform is defined as

3ACi)=HV(X)I= :X~(X)V,(C*)Y~(~~b)-J,(Cb)Y,(S,X)]dX s

(16)

Since equation (16) is very similar to equation (7), the same process as before is used to find the series expansion. Thus, equation (9) is used for all cases and plays a central role in obtaining the results.

542 G. CINELLI

3. TABULATION OF RESULTS

The new finite Hankel transforms and the corresponding operational properties are given in this section for the general case. By choosing appropriate values for the constants (h, k) in the boundary conditions, all other eight cases can be obtained. This has been accomplished and the results are listed in Appendix 3.

The constants can be positive, negative, or zero. A bar over the lower case letter indicates the finite Hankel transform. A prime on a lower case or capital letter indicates differentiation. Finally, the letter H stands for a linear operator on the function f(s).

i

r = a, Cauchy Boundary Conditions

lr = b, Cauchy A

f(s’i>=H[f(x)]= J :x~(x){J,(riX)[riYI(Tia)+ hyp(5ia>l

- Y,(Six)[SiJ~(gia>+hJ,(~ia>l)dx (17)

(18)

where gi is a positive root of

[tiYJtia) + h Yp(tia)][5J’p(5ib> + kJp(5ib)I

-[TiY1(Sib)+kY,,(5ib)][5iJ:(5ia)+hJ,(~ia)I=O (‘9)

B

f<tJ=H[f(x)]= bxf(x)(J,(rix)[5iY;(5ib)+kY,,(4,b)I s a

- Yp(tix)[tJ’p(tib) + kJp(tib)]}dx (21)

~~[~iJ~l(<iU) + hJ,(i”i” )]2J(4i)

{(k2++-($,)‘]) C<iJ;(<ia) + hJ,,(gia >I’ - (h2 + Cf [ I-( $J[eJ;tcib, + kJJrib)l’)


2 [5iJL(5ib) + kJfi(5ib)I - n [5iJh(5ia) + hJpttiu>] V’(4 + W(a)1

-Hiti) (23)

It is emphasized that the results shown here are formal in nature. The reason for this is that there are no readily available existence theorems for solutions of the class of boundary value problems considered here. Hence, questions as to the existence and uniqueness of solutions obtained by this technique must be established in each particular case. The operational method does not solve problems which cannot be solved by classical techniques. Its advantage lies, as in the case of the Laplace transform, in its direct, concise, and logical manner of finding the solution to a given problem.

4. TRANSIENT TEMPERATURES IN A FINITE HOLLOW CYLINDER DUE TO ASYMMETRICAL INTERNAL HEAT GENERATION

It is the goal of this section to determine the transient temperature distribution produced in a finite, hollow cylinder by an arbitrary asymmetrical internal heat source when radiation is taking place on all four surfaces. Problems of this type arise whenever ionizing particles (e.g. neutrons) are present as in a nuclear structure or space vehicle. Special cases of this problem have been solved in the case of nuclear reactors [7, 81.

The only assumption made in the analysis is that the properties of the cylinder material are independent of temperature. In addition, the work given in Carslaw and Jaeger’s book [l] on hollow cylinders is shown to be a special case of the general solution. Thus, as far as it can be determined, this problem has not previously been solved.

The boundary value problem under consideration is:

aT ( a2T 1aT 1 a2T a2T z=K p+; r+&p++aZZ

> +e, d,z, Q

a<r<b,

Initial conditions :

Boundary conditions : T(r, 8, Z, 0) = F(r, 8, z)

i;T x$y +hlT=A,(O, z, t)

aT ;r +h,T=A,@, Z, t)

dT 5 +h,T=A,(r, 0, t)

dT z +k+T=Ur, 0, 0

r=a, t>O

r=b, t>O

z=o, t>o

z=l, t>o

(24)

(25)

(26)

(27)

(28)

(2%

(30)

544 G.CINELLI

where K = k/p is a constant k = thermal conductivity of the material p = material density c = specific heat of the material 4 = internal heat source F= initial temperature distribution in the cylinder T= temperature distribution in the cylinder A i, AZ, A 3, A, = temperature distributions of the surrounding media

hi, h,, 4, h4 =constant coefficients whose value can be positive, negative, or zero.

As a preliminary to the solution, it is necessary to recall that any functionf(x) in the interval (q X) can be writtten as the sum of an even and odd function by means of the identity

S~~>=3~f~~>+f~-~~l+3Lf~~~-~-~~l (31)

Let F(x) be any function in (0, 7c), then an even (odd) functionf(x) exists in (- 71, n) which is identical with F(x) in (0, x). If the even (odd) functionf(x) is represented by its Fourier series in (-z, x), so is F(x) in (0, rr). Thus, use of the finite sine and cosine transforms will resolve the angular variable. Because of this, the solution of the problem is accomplished in two steps, one involving the sine transform and the other the cosine transform. The complete solution is then obtained by adding the two separate solutions.

The following finite transform formula for the axial variable are derived from results given in [3],

T=T(r, 8, A,, t)= {l T(r, 8, z, r)[sin+- $cosl.z]dr (32)

-

T(r, 0, z,O=2;: [ sin l,z - 2 2 cos l,z

n h, 1 where 1, is a positive root of

+[&tanIJ-h,]+[I.+h,tanLJ]=O 3

and

N,=l I+% - isin 1 %+I cosl,l+$sinJJ [ h:l A” ’ ii3 I[ 3 1

(33)

(34)

(35)

From Sneddon [2] and case A the following transforms are defined,

r= r(r, m, [,, t)= s

1 T(r, 8, z, t)sinmed0 m=1,2,... (37)


Using equations (29), (30), (32) and (34) the sine transform of equation (24) with respect to axial variable is

+kA,(r, 0, t)+ 3 [

” sin&l- +s1.,1

3 1 A,(r, 8, t) (39)

where

Applying equations (20), (27), (28), (37), and (38) to the angular and respectively, in equation (39) gives

dFS 2 [tiJX5ia) + hlJ*(5i”)] - 2_

dt=K n [4iri(l;ib)+h2J,(4ib),~z(m, L 0- iAl(m, &I, r)

(40)

radial variables,

-l

-K(t;f+{,2)F~+~(~iy in, A,, t)++ 3(5i, m, t)+ sin&l-ffkOSA,l A=,(~i, m, t)(41) 3 1 3 J

where ti is a positive root of

[<iK(<ia) + h, L<eia>I[SJXtib) + h2Jm(Sib)] = [5iG(tib)+ h2Ym(Sib)][SiJ~(~iu)+hlJ,(5iu>]

br%-, 03 43 t){J,(5ir)[5iY~(riU)+h, Y,<tIiU)]

- Ym(Sir)[SiJ6(CU) + hl Y,(@)])sin mOdOdr

sin medI3dz

sin m0dtIdz

= A3(L m, 0= b IA&, 0, t)(J,(Cir)[riYA(Sia) + hl Y,(Sia)]

- Y,(<ir)[~J~(~ia)+ h,J,(&z)])sin mededr

(42)

(43)

(44)

(45)

(46)

546 G. CINELLI

A=,(ti, m, t)’

- Ym(eir)[SiJ1,(Sia) + hi Y,,,(<ia)]}sin nzOd&Jdr (47)

Rearranging equation (41) into a more appropriate form gives

Using the Laplace transform, convolution integral, and initial condition (26) it follows that the solution of equation (48) is

~~(~i, m, An, t)=exp( -lC(cz+A,2)t) i-(<. [ I) m, i)+jbcxp[x(5r+li)rl

(49)

Using the inversion series given by equations (18) (33) and in Sneddon [2] the solution for the temperature is

* {Jm<tir>[tiUSia) + hl ym(tia>l - YJC5ir)[tiJX5P)+ hlJm(5iu)l} (51)

The solution of equation (24) for the even function proceeds in exactly the same manner as for the odd function. The only difference occurs in the inversion series for cos me which is

(52)

An extension of the finite Ha&e1 transform and applications 541

Therefore, the solution for the case of the even function is

1 &I .- cos me N” [

sin 1,~ - ~0s &z 3 1

where

?=(t, m, A,,,, t) is defined by equation (37) when sin m6 is replaced by cos me.

The solution for ?‘c is given by equation (49) where the cosine transform has been used on the angular variable.

The complete solution is obtained by adding equations (51) and (53) according to equation (31). Doing this gives the general solution to equation (24) as

*[fs(i;i, m, An, t)sin rnti+!i”&<, m, A,, ?)cos mfl]

2

1

C&, 6) * cos&z- &in&z - h, F,(6)

(54)

(55)

Cmtry ri)={Jmtgir)[giY,triu)+ hl K(&a>l- KXCr)CbUSr~)+ hJm(b)l) where

(56)

?“(Lj,, m, An, t), ~~;<r, m, A,, t) are given by equation (49) N,, is equation (35) rl is a positive root of equation (42) Iz, is a positive root of equation (34).

The problem considered by Carslaw and surrounded by media at constant temperatures.

Jaeger [l] is an inkite hollow cylinder Their formation of the problem is :

a<r<b (57)

548 G. CINELIA

Initial conditions: T(r, 0) = 0

Boundary conditions :

klg -k,T=k, r=a, t>O

kia$:+ k;T= k; r=b, t>O

where kl, k2, k;, k;, k3, k; are constants and may be positive, negative, or zero.

For this problem the genera1 solution in equation (54) reduces to

i5@

(59)

(60)

(62)

~(ri, t)= - %exp[ --z&t]

where 5i is a root of

Comparing equations (27), (28) and (59), the relationships between the constants are

(65)

Integrating equation (63) and putting the values of the constants in equation (65) into equations (60), (61), and (62) gives

. ljJ,(<,b)- ~~,(5ib) 1 [ - ~ 11 (66) 1 1


Fo(C,)=[ (>’ + 5:][ tiJ1(5ia) + 2JO(tia)]

-[ (2>’ + Cf ][ gJl(<ib)- 2dtib)J (67)

Cob-, ti>= Jo(b) i [ Ciyl(5ia) + p yO(tia) - G(tir)

1 1 [

CJl(tiu) + 2JO(tiu) 1 1

(68)

Multiplying out the constants the solution is

T(r, t>=nC [k;SiJ1(Sib)-k2J,(~,b)]{k,[k;CJl(rib)-k;J,(rib)] ti

-n~exp(-~rft)[k;TiJ1(Tib)-k;Jo(tib)]{ksCk;riJl(i;ib)-k;Jo(eib)]

-k;[k,riJ,(iiU)+k,J,(SiU)]}‘~ (6% 0 I

Fo(5i)=[k~2+5~][k~~iJ~(5iU)+k~Jo(C~)]2-[k~+~?][k;~iJ~(~ib)-k;Jo(5ib)]2 (70)

COO-, Tr) = Jo(Sir)[klti Yi(<ia> + k2 G(tia)] - Yo(Sir)[klSiJ1(Tia) + 4 Jo(Sia)] (71)


[k,5i~(5iu)+k2Yo(tiu)][k;SiJ1(Tib)-k;Jo(rib)]

=[k;giY,(Cb>-k;Y,(6,b)][k,giJl<eiu)+k2Jo(giu)]

The solution given in [3] is

(72)

T(r, t)= - uk,[k; - bk,ln(r/b)] + bk$[k, +uk,ln(r/u)]

uk,k; + bk,k; +abk,k;ln(b/u)

-n~exp(-~TZt)[k;TiJ,(g,b)-k;J,(5ib)]

{k,[k~~d~(Cb)-k;Jo(iibl]-k;[k,~iJ~(Ei~)+k,Jo(~i~,Bc~ (73) 0 L

The solutions in equations (69) and (73) are the same except for the steady-state expressions. It is shown* that the steady-state term in equation (69) is actually the Fourier-Bessel series expansion of the steady-state term in equation (73). Hence, the two solutions are identical thereby proving that the general solution in equation (54) contains equation (73) as a special case.

* Appendix 2.

550 G. CINELLI

REFERENCES

[l] H. S. CARSLAW and J. C. JAEGER, Conduction of Heat in Solids (second edition). Oxford University Press (1959).

[2] I. N. SNEDDON, Fourier Transforms. McGraw Hill (1951). [3] S. KAPLAN and G. SONNEMANN, Fourth Midwest Conference on Solid Mechanics, pp. 497-513,

University of Texas (1959). [4] R. V. CHURCHILL, Operational Muthemntics (second edition). McGraw Hill (1958). [5] A. C. ERINGEN, Quart. J. Math. 2, (5) 120-129 (1954). [6] W. P. REID, SIAM Rev. 1, (1) 44-46 (1959). [7] H. KRAUS and G. SONNEMANN, J. Engng. Pwr, Trans. ASME, Series A, 81,449-454 (1959). [8] J. E. SCHMIDT and G. SONNEMANN, J. Engng. Pwr, Trans. ASME, Series A, 82,273-278 (1960). [9] E. JAHNKE and F. EMDE, Tables of Functions (second edition). Dover Publications (1945).

APPENDIX 1

The integral in equation (9) is derived from a formula given in [9] which is

s Z2 zw,(~z)WV(~z)dz=4[2~y(~~)Wy(az)-w,+,(Ctz)W,_1(~~)- W~-I(~z)~,+l(ci~)] (74)

where W,(CIZ), W,(ctz) are any Bessel functions of the first, second, or third kind and of order v.

Using the recurrence relationships

it follows that

w”*l@z)w”~l(~z)= ; 2 0 w,(az)W,(az) - w;(uz)W;(az)

Therefore,

w,+l(az)W,_&z)+w,_~(uz)W”+&z)=2 v 2 K > E w,(a2)W,(uz)-w:(uz)W;(uz)

1

Placing this expression into the integral in equation (74) and collecting terms gives

s zw,(~~z)W,(az)dz = w:(~z)w;(az)+ l- [ (-$Jw.(az) w&7”,)

(75)

(76)

(77)

(78)

(79)

which is equation (9).


APPENDIX 2

In order to show that the steady-state term in equation (69) is the Fourier-Bessel series expansion of the steady-state term in (73) it is necessary to express the infinite series in equation (18) in terms of the quantities in equation (73). Using equation (65), equation (18) becomes

s b

T(ti) = rW)Co(r, 53dr (81) (I

The steady-state term in equation (69) is rewritten as

Comparing equations (82) and (73) the basic question which must be answered is what is the finite Hankel transform of

- ak; + bk;ln(r/b)

akik,+bk,k;+abk,k;ln(b/a)

and

bkI + akJn(r/a)

ak~k,+bk,k~+abk2k~ln(b/a)

(83)

(84)

To obtain the transforms of equations (83) and (84) the following integrals must be evaluated.

11~ s

‘rC,(r, 5i)dl a (85)

I, = s

b rln(r/a)Co(r, &)dr 0

(86)

I, = s

b rln(r/b)C,(r, 5,)dr a (87)

Using the indefinite integrals

s 1 rJo(&r)dr = &rJl(<ir) (88)

552 G. CINELLI

the integrals in equations (85), (86) and (87) are

The finite Hankel transform of equation (83) is

- uk; + bkiln(r/b)

uk,k; + bk,k$ +abk,k;ln(b/u) Cdr, 5Jdr

Expanding equation (93) gives

I,= -uk;I, +ubk;l,

uk,k~+bk,k~+abk,k~ln(b/u)’

Using equations (90) and (92) and collecting terms gives

1

*~~[uk,k~+bk,k;+ubk,k~ln(b/u)]

(90)

(91)

(92)

(93)

(94)

(9%

The first bracket is zero by virtue of equation (72) thereby reducing equation (95) to

14~~[~~~Siu~y~~Sia~~Jo~Siu~y~ll(Siu~]

L

Using the relationships

Jb(ti”)= -Jl(ti”> Y6(ti”)= - yl(ti”)

(96)

(97)


and equation (1 l), equation (96) becomes

Putting equation (98) into equation (80) gives as a final result

- ak; + bk;ln(r/b)

ak,k; + bk,k;+abk,k;ln(b/a) =.Z[k,rir,(rib)-k,J,(5ib,l’Y;;:i’

ei 0 i

(98)

(99)

In a similar manner it can be shown that

bkI + ak,ln(r/a)

ak,k; + bk,k;+abk,k!Jn(b/a) = -XC [k;5iJl(eib)-k;Jo(Sib)]

ri

Comparing equations (73), (82), (99) and (100) it follows that the steady-state in equation (69) is indeed the Fourier-Bessel series expansion of the steady-state term in equation (73).

APPENDIX 3

The new finite Hankel transforms, inverse infinite series, and operational properties on the derivatives for all eight cases are given in this appendix. Since each case has two series expansions, results are given for both. In addition, the boundary conditions that hold are given in order to facilitate the choosing of the appropriate transform to solve a given problem.

As stated previously the constants (h, k) can be positive, negative, or zero. The bar over the lower case letter indicates a finite Hankel transform whereas a prime on a letter indicates differentiation. The linear integral operator is again indicated by the letter H.

Case 1

x = a, Dirichlet Boundary Conditions

x = b, Dirichlet IA

(101)

(102)

where ri is a positive root of

(103)

554 G. CINELLI

IB

f((rJ=HlIf (X)1 = s

‘xf(x)[J,(tix)Y,(5,b,-~~(5ib)Y,(r,x)]dx a

” tfJ2(5ia13(ti) f(x) = T; J;(&7) - J,2(&b) CJJ6x) Yp(tib) - J,(tib>yfl(5ix)]

where ti is a positive root of equation (103).

(104)

(105)

(106)

(107)

Case 2

x = a, Neumann Boundary Conditions

x= b, Dirichlet IIA

f’(tJ = H[f (X)] = s

:Xf (x)[Jp(CX)Y,‘(tiu) -Jb(tia) Y,(t,x)]dx (1’33)

7L2 S?JE(S’ib)f(tJ f(x)= yg {[J~(SiU)]2-[1-(~/~iU>2]J~(~ib)} * CJ&Six) y,1(5i”> - Jk<ti"> yp(5ix)] (109)

where 5i is a positive root of

J,(5ib)Y,‘(51u)-J1(Siu)Y,(5ib)=O W)

H ;;;+;g - $f 1 = --$f’(u)+;$$$f(b)-t;f(g,) (111) I P 1 ITB

f(x)= H[f (X)] = s

:~f(x)[J,(Six)Y~(e,b)-J,(5ib)Y,(Six)ldx (112)

.[J,(Six)Y,(5,b)--J,(5,b)Y,(Six)] (113)

where ti is a positive root of equation (110).

H dtf

C

ldf P’ dj+;& - x>f 1 =;f(b)- (114)


Case 3

x = a, Dirichlet Boundary Conditions

x = b, Neumann IIIA

T~(5i)=HlY(x)l= * s

xf(x)[J,(rix)y,(gia)-J,(Sia)Y,(5ix!ldx a

where t1 is a positive root of

IIIB

where 5i is a positive root of equation (117).

Case 4

(121)

x = a, Neumann Boundary Conditions

x= b, Neumann IVA

f(rJ = HCfCX>I = s

bXf(X)CJp(riX)Y;(riu)-J;(Sia)Y~(eix)]dx 0 uw

f(x& C;,‘CJ~(5ib)]2fCtJ 2 it (CJl(eP>I’[l -(d5ib)21 - [Jh(4’ib)]‘[ I- (p/Cia)‘]}

556 G.CINELLI

where Ti is a positive root of

1 =

IVB

- [~~(~ix)Y~(~Ib) - ~~(~~b) Y&W] where Ti is a positive root of equation (124).

x = a, Cauchy Boundary Conditions

x= b, Dirichlet VA

1’(~J=Htf(x)]= f

1 xf(x)t~,(Six)Y,(eib)-J,(Cb)Y,(gix)ldx

where li is a positive root of

(124)

U 25)

1127)

(128)

(129)

(130)

(1311

(132)


where ti is a positive root of equation (13 1).

Case 6

r = a, Cauchy Boundary Conditions

r = b, Neumann VIA


Jh({ib)[<iY,‘(<,a) + h Y,(tia>] - YL(<ib)[tiJl(i2ia) + IzJp(t,a>I =O (138)

.,f(<J=H[f(X>]= :xf(x)iJr(SiX)CSiY,:(Sin)+hY~(clP)I s

where ci is a positive root of equation (138).

f’(b) - $fW f W’(a)1 - <?3&> (142)

558 G. CINELLI

Case 7 f-r = a, Dirichlet

Boundary Conditions

I r = b, Cauchy VIIA

.Rf(5i)=H[f(x>I= :wf(x)V,~(i’ix)Y,(eiY)-J,(Tia)Y,(eixlldx s

7c2 5f[5iJjX5ib> + kJp(tib)]2_f(f(ti)

fcx’=TF {J~(t*a>(k2 +5iL[1-(~/5*b)2])-[~iJ:(Sib)+kJ~(5jb)]2}

‘~J~(5ix)Y~(~ia)-J~(~ia>y~(5ix)l


Yp(S’ia)[SJh(cib) + kJ,(tib)] -JJcia)[SiYL(<ib) + kYp(rib)l =O

1 =:[~iJ~~~~~~J~(r,blI[ff(b)+kf(b)I-~f(a)-5tl(ri)

VIIB

J;(SrJ=HCf(x)] = s

1 xf (X){J,(5iX)[SiY~(gib) + kY,t(tib)I

- Yp(Six)[SiJ~(Tib)+ kJ&Sib)l)dx

t12J~(5ia).fC(5*)

f (x)zG z (Ji(tia)(k’ + <:[I- (~/5ib)‘]>- [tiJh(tib) + kJp(tib)12)

* (JJ5ix)[tiY~(tib) + kYJtib)] - Yp(Six)[tiJ~(<ib> + kJp(Cib>]>

where li is a positive root of equation (145).

1 =- 2 [5dh(5ib)+ kJJ‘ccib)lf @)+$f’(b)+ kf(b)] -t?f((5.)

i7 J,(&a) I ,

7-t

Case 8 r = a, Neumann

Boundary Conditions r = b, Cauchy

VIIIA

_T(&) = wY(~)l= s

b xf (x)[Jp(Six)Y~(tia)- J~(Sia)Yp(<G>Id~ ll

(143)

(144)

(145)

(146)

(147)

(148)

(149)

(150)

. [J:J,(tP) Yi(<P) - J;(Sia)Yp(tF)I (151)



YL(tia)[<iJh(tib) + kJN(tib>] -Ji(tib)[ti Y,‘(tib) + kYs(t&)I =O (152)

Jb(SiU)

x [tiJL(Sib) + kJp(tib>I [f’(b) + kf (b)] - $f ‘(a)- t;i’f(<‘(w (153) I

VIIIB

f?(tJ=H[f(x)]= ~x~(x)o,(SiX)[~iY;(si~)+kU,(~,b)I s

- Yp(5ix)[5iJL(5ib>+ kY&tib)I}dx (154)

* {J,(CG)[St Y;(Tib)+ kYJ<ib)] - Yj(SiX)[TiJ~(Sib) + kJJ(ib)]} (155)

where ti is a positive root of equation (152),

(Received 3 1 December 1964)

R&utre--L’auteur utilise la nouvelle transform% finie de Hankel avec noyaux de la forme [J,(&x) Y&u) - J&U) Y,(&x)] et les series infmies correspondantes, pour ramener la solution de l’equation de Bessel, avec conditions terminales asymetriques, dans le cadre de la theorie des transform&s integrales et il donne la solution &n&ale de la temperature de transition provoquee dam un cylindre creux, thri, par une source calorifique asymetrique, interieure lorsqu’il se produit du rayonnement sur les quatre surfaces du cylindre.

L’auteur montre que cette solution englobe, dam un cas particulier, les resultats trouves par Carslaw et Jaeger [l] sur des cyhndres creux.

Zusammenfassuag-Es werden neue endliche Hankeltransformationen und die entsprechenden unendlichen Reihen eingeftirt. Die ersteren be&en die Kernform [J,(&tx)Y,(eca)--J,(&ra)Y,(~{x)] und bringen di Lijsung der Resselgleichung mit asymmetrischen Endpunktbedingungen in den Rereich der integralen Transformationstheorie. Fi.ir ubergangstemperaturen, welche von einer asymmetrischen intemen W&ma- quelle erzeugt we&n, wenn in einem begrenzten Hohlzylinder auf allen vier Oberflachen Abstrahhmg stattfindet wird eine allgemeine Usung gefunden und gezeigt, dass diese L&sung, die von Carslaw und Jaeger fur Hohlzylinder erzielten Ergebnisse [l] als Speziallfalle enthalt.

Soramarfo-S’introducono nuove trasformazioni fmite di Hankel di forma [J,(&) YP(&u) -J,(&z) Y,(ix)] e la serie infinita corrispondente, the fanno rientrare la soluzione dell’equazione di Bessel con condizioni asimmetriche d’estremita nel camp0 della teoria delle trasformazioni integrali. Si giunge a una soluzione generale per le temperature transitorie prodotte in un cilindro finito e cave da una sorgente sorgente inteme asimmetrica quando la radiazione awiene su tutte e quattro le super6ci. Si dimostra come questa soluzione contenga i risultati de1 Carslow e de1 Jaeger [I] sui cilindri cavi come case speciale.

A6c~pe~r--Bsomrrc~ HOBOS, Konemioe npeo6pa3oBamre EarrKena c aKpo~ B Brine [J,(&x) Y,,(&u)- 44&u) Y&G41 H CxxJ TBCTCTB~~ 6ecKOHe¶Bb& pffA, C nOMOIIzbH) KOTOpbD( MOXClia penlTiTb YpaBHerXBe Eeccero~ c riecm+rerprfs~bt~B KpaeBbrbm yc~~o~rrahoi B o6BacrB reopmi mrrerpam,nbrx npeo6pa3o~am#& Aaercrr o6ruee pememre 3ana=, neycrartoarmmettca rehmeparypbt B IIOIIOM mr,~rm~pe nOA BJIBaEtBeM BAyTpeHHHx HecRMMeTpawbM BIXOBBBKOB remta, KorKa BsnyserrBe rr~eer MecTO Ba ~cex rerrrpex no~epx~ocrax. ~orasbmaerca, ¶rO pe3ynbrar Kapcnoya Erepa [I] ruurserca ‘lacrrrbrbf cnysaeM nprmo- -or0 pemetisia .

an extension of the finite hankel transforms*

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