Kuwait University Kuwait FoundationDepartment of Mathematics & for the Advancement

Computer Science of Sciences (KFAS)

PROCEEDINGS OF THE

INTERNATIONAL CONFERENCE

ON MATHEMATICS AND ITS

APPLICATIONS

(ICMA 2004)

April 5-7, 2004

State of Kuwait

ORGANIZING COMMITTEE

Dr. Mansour A. Al-Zanaidi, Chairman

Prof. Bader N. Al-Saqabi, Treasurer Prof. Ali Al-Zamel

Prof. Adnan H. Al-Aqeel Prof. Shyam L. Kalla

Prof. Reyadh R. Khazal Prof. Man M. Chawla, Secretary

ii

FOREWORD

This volume contains papers and invited talks presented at the “International

Conference on Mathematics and Its Applications” (ICMA04-Kuwait), which was

held at Kuwait University from April 5 to 7, 2004. All papers contained in this

volume are refereed/peer reviewed.

The Conference covered active research fields of the Department of Mathematics

and Computer Science:

(A) Computational Differential Equations and Linear Algebra.

(B) Integral Transforms and Special Functions with Fractional Calculus.

(C) Groups, Rings & Categories, and Differential Geometry.

The ICMA04-Kuwait was jointly sponsored and organized by the Department of

Mathematics and Computer Science, Kuwait University, and the Kuwait Foundation

for the Advancement of Sciences (KFAS). The Conference was attended by sixty

three participants from twenty two countries.

Keynote speakers at the conference were: G. V. Berghe (Belgium), S. Caenepeel

(Belgium), L. Debnath (USA), D. J. Evans (UK), V. D. Mazurov (Russia), A. J.

Scholl (UK) and B. Wegner (Germany).

The organizers would like to thank all the participants for their cooperation in

preparing their contributions for this Proceedings and their active interest in the

peer reviewing process for the volume. Thanks are also due to all colleagues who

helped in the reviewing process.

All papers in this volume are arranged alphabetically according to the name of

the first author.

We would like to take this opportunity to express our gratitude to the administra-

tion at Kuwait University and Kuwait Foundation for the Advancement of Sciences,

for sponsoring this Conference and publication of the Proceedings.

Finally we offer our sincere thanks to the faculty and staff of the Department of

Mathematics and Computer Science at Kuwait University for taking active interest,

shouldering several responsibilities during the conference and publication of these

proceedings.

Mansour A. Al-Zanaidi, Shyam L. Kalla and Man M. Chawla

Chairman Editors

Kuwait, December 2004.

iii

CONTENTS

Organizing Committee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

M. Al-Hajri and S. L. Kalla On An Integral Transform Involving Bessel Func-

tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

N. Alias, M. S. S. Mohamed, A. R. Abdullah PVM-Based Implementation

Of A New ADI Technique (IADEI) In Solving One Dimensional Parabolic Equation

Using PC Cluster System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

H. Alinejad, J. Mahmoodi, S. Sobhanian, M. Momeni A New Nonlinear

Differential Equation For Describing Ion-Acoustic Waves In Plasma . . . . . . . . . . . . . . 27

G. Alobaidi and R. Mallier Using Monte Carlo Methods To Evaluate Sub-

Optimal Exercise Policies For American Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

A. Azizi Group Theory And The Capitulation Problem For Certain Biquadratic

Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

P. Balasubramaniam and A. V. A. Kumar Runge-Kutta Neural Networks For

Delay Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

V. Beniash-Kryvets On Rosenberger’s Conjecture For Generalized Triangle Groups

Of Types (2, 10, 2) and (2, 20, 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

G. V. Berghe Exponential Fitting: General Approach And Applications For ODE-

Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

A. Bounaım, S. Holm, W. Chen, A. Ødegard Mathematics In Medicine: Bioa-

coustic Modeling And Computations For An Ultrasonic Imaging Technique . . . . . . 92

L. Boyadjiev and R. Scherer On The Fractional Heat Equation . . . . . . . . . . . . . . 105

S. Caenepeel and E. De Groot Galois Corings Applied To Partial Galois The-

ory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

iv

S. Caenepeel and M. Iovanov Comodules Over Semiperfect Corings . . . . . . . . . . 135

D. K. Callebaut, G. K. Karugila and A. H. Khater Nonlinear Fourier Analysis

Of Systems Of Partial Differential Equations With Computer Algebra: Survey And

New Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

U. C. De and G. C. Ghosh On Quasi Einstein And Special Quasi Einstein

Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

L. Debnath Four Major Discoveries In Applied Mathematics During The Second

Half Of The Twentieth Century . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

D. J. Evans The Generalised Coupled Alternating Group Explicit (CAGE)

Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

B. I. Golubov Dyadic Fractional Differentiation And Integration Of Walsh Trans-

form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

G. Heinig and K. Rost Fast “Split” Algorithms For Toeplitz And Toeplitz-Plus-

Hankel Matrices With Arbitrary Rank Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

M. N. M. Ibrahim Productivity Of Oil Wells In Arbitrarily Shaped Reservoirs 313

M. Lehn and R. Scherer Mathematical Modeling And Scientific Computing Ex-

emplary For Chemical Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318

M. Lenard Weighted (0,2)-Interpolation With Additional Interpolatory Condi-

tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

V. D. Mazurov Groups With Prescribed Orders Of Elements . . . . . . . . . . . . . . . . . . . 343

P. Miskinis Integrable Nonlocal Burgers Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

S. Naik and S. Ponnusamy Univalency Of A Convolution Operator Concerning

Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370

M. A. Pathan On Unified Elliptic-Type Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375

M. Savsar Modeling Of An Unreliable Production Merging Process With Interme-

diate Storage Tank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388

v

N. El Saadi, M. Adioui and O. Arino A Mathematical Analysis For An Aggre-

gation Model Of Phytoplankton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397

S. M. El-Sayed Iterative Methods For Some Nonlinear Matrix Equations . . . . . . . 406

A. M. A. El-Sayed Fractional Calculus And Intermediate Physical Processes . . . 417

M. Sayed Coset Enumeration Algorithm For Symmetrically Generated Groups . . 424

S. V. Tikhonov and V. I. Yanchevskiı Pythagoras Numbers Of Function Fields

Of Genus Zero Curves Defined Over Hereditarily Pythagorean Fields . . . . . . . . . . . . . 438

B. Wegner Mathematics On Its Way Into Information Society . . . . . . . . . . . . . . . . . . 444

M. Wojtowicz Algebraic Structures Arising From Euler’s And Pythagorean Equa-

tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452

M. A. Al-Zanaidi, C. Grossmann and A. Noack Implicit Taylor Methods For

Parabolic Equations With Jumps In Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458

vi

ON AN INTEGRAL TRANSFORM INVOLVING BESSELFUNCTIONS

M. Al-Hajri and S. L. Kalla

Department of Mathematics & Computer Science

Kuwait university, P. O. Box. 5969, Safat 13060, KUWAIT

email: kalla@mcs.sci.kuniv.edu.kw

Abstract

This paper deals with a new integral transform, involving a combination of Bessel

functions as a kernel. The inversion formula is established and some properties

are given. This transform can be used to solve some mixed boundary value prob-

lems. We consider here a problem of heat conduction in an infinite and semi-infinite

cylinder a ≤ r ≤ b, with radiation-type boundary conditions.

AMS Subj. classification: 44A20, 35K20, 80A20.

Keywords: Integral transform, Bessel Functions, Differential equations, Boundary

conditions.

1. INTRODUCTION

Let f(t) be a given function defined on an interval [a, b], that belongs to a certain

class of functions. An integral transform of f(t) is a mapping of the form,

T [f(t); s] = f(s) =

∫ b

a

K(s, t)f(t)dt,

provided that the integral exists. K(s, t) is a prescribed function, called the ker-

nel of the transform [2,5,6,11]. Among the well known transforms are the Laplace,

Fourier, Hankel, Stieltjes and Mellin transforms. The most versatile of these, the

Laplace transform has been widely used to solve differential equations, and par-

ticularly problems related to heat transfer and electrical circuits. On the other

hand for problems in which there is an axial symmetry, the Hankel transforms are

found to be most appropriate. The Mellin transform being closely related to the

Fourier transform, has its own peculiar uses, as for deriving expansion and solving

problems with wedge shape boundaries. In general, the use of an integral transform

often reduces a partial differential equation in n independent variables to (n − 1)

variables, that provides a simplification of the problem.

1

The success of the use of integral transforms to solve boundary value problems

and to exclude a variable with range (0,∞) or (−∞,∞) led investigators to con-

sider finite integral transforms. Doetsch considered finite Fourier transforms, and

Sneddon [11] extended the idea of Bessel function kernel, called ’Finite Hankel

Transforms’. Using the Sturm-Liouville theory [3,], a number of integral transforms

can be implemented, according to the prescribed boundary conditions.

Recently Khajah [9] has considered a modified Hankel transform in the form,

Jµ[f(z); s, λ] =

∫ b

0

zλf(z)Jµ(zs)dz

where f(z) satisfies Dirichlet’s conditions on the interval [a, b]. He has derived

the inversion formula, Parseval-type identities, transform of derivatives, as well as

transforms of products of the form zλf(z).

Using the Sturm-Liouville theory Kalla and Villalobos [7,8] have defined and

studied an integral transform defined as,

T [f(x), a, b, ν; λi] = fν(λi) =

∫ b

a

xf(x)Cν(λix)dx,

where

Cν(λix) = {Yν(λia) + Bν(λib)}Jν(λix)

−{Jν(λia) + Aν(λib)}Yν(λix)

and

Aν(λx) = Jν(λx) + hλJ′ν(λx)

Bν(λx) = Yν(λx) + hλY′ν (λx)

and λi are the positive roots of equation,

Jν(λa)Bν(λb) − Yν(λa)Aν(λb) = 0

This transform has been used to solve a heat conduction problem in an infinite

cylinder bounded by the surface r = a, r = b (b > a).

In this paper we define and study a new integral transform involving Bessel

functions of first and second kind, by invoking the Sturm-Liouville theory. Inversion

formula is established and some properties are mentioned. The transform has been

2

used to solve a heat conduction problem in an infinite and a semi- infinite circular

cylinder, bounded by surfaces r = a and r = b (b > a), with radiation-type boundary

conditions on both surfaces.

2. DEFINITION AND INVERSION FORMULA

Consider Bessel’s differential equation

x2y′′

+ xy′+ (λ2x2 − ν2)y = 0 , x ∈ [a, b] (1)

with homogeneous boundary conditions:

y(a) + h1y′(a) = y(b) + h2y

′(b) = 0 (2)

The general solution of (1) is given by:

y(x) = c1Jν(λx) + c2Yν(λx) (3)

where c1 , c2 are arbitrary constants, and Jν(x), Yν(x) are the Bessel functions of first

and second kind respectively. To obtain a solution of (1) that satisfies conditions

(2), we have

c1

[Jν(λa) + h1λJ

′ν(λa)

]+ c2

[Yν(λa) + h1λY

′ν (λa)

]= 0 (4)

c1

[Jν(λb) + h2λJ

′ν(λb)

]+ c2

[Yν(λb) + h2λY

′ν (λb)

]= 0 (5)

from which we deduce

c1

c2

= −Yν(λa) + h1λY′ν (λa)

Jν(λa) + h1λJ ′ν(λa)

= −Yν(λb) + h2λY′ν (λb)

Jν(λb) + h2λJ ′ν(λb)

(6)

Let

Aν(λx, hk) = Jν(λx) + hkλJ′ν(λx) , k = 1, 2

Bν(λx, hk) = Yν(λx) + hkλY′ν (λx) , k = 1, 2

Then, the function given by (3) is a solution of equation (1), subject to the conditions

(2), if λ is a root of the transcendental equation,

Bν(λa, h1)Aν(λb, h2) − Aν(λa, h1)Bν(λb, h2) = 0 (7)

Henceforth, we take λi (i = 1, 2, . . .) to be the positive roots of equation (7). Then,

from (4–5), we have

yi(x) =c1

Bν(λia, h1)[Jν(λix)Bν(λia, h1) − Aν(λia, h1)Yν(λix)] (8)

=c1

Bν(λib, h2)[Jν(λix)Bν(λib, h2) − Aν(λib, h2)Yν(λix)] (9)

3

If we define

Zi = Bν(λia, h1) + Bν(λib, h2) , Wi = Aν(λia, h1) + Aν(λib, h2)

then the following functions are taken to be solutions of (1–2):

Mν(λix) = Zi Jν(λix) − Wi Yν(λix) (10)

By Sturm-Liouville theory [3], the functions of the system (10) are orthogonal on

the interval [a, b] with weight function x, that is

∫ b

a

xMν(λix) Mν(λjx)dx =

0 , i �= j

Mν(λi) , i = j

(11)

where Mν(λi) = ‖√xMν(λix)‖22 — the weighted L2 norm. If a function f(x) and

its first derivative are piecewise continuous on the interval [a, b], then the relation

T [f(x), a, b, ν; λi] = fν(λi) =

∫ b

a

xf(x)Mν(λix)dx (12)

defines a linear integral transform. To derive the inversion formula for this transform,

given the series expansion,

f(x) =∞∑i=1

aiMν(λix) (13)

we multiply (13) by xMν(λjx) and integrate both sides with respect to x to get the

coefficients:

ai =1

Mν(λi)

∫ b

a

x f(x)Mν(λix) dx =fν(λi)

Mν(λi), i = 1, 2, . . . (14)

and the inversion formula becomes

f(x) =∞∑i=1

fν(λi)

Mν(λi)Mν(λix) (15)

Using some well known properties of Bessel functions [12] we can easily derive the

following relation:

2Mν(λi) = Z2i

[b2P (λi, b, ν) − a2P (λi, a, ν)

] − 2ZiWi

[b2Q(λi, b, ν) − a2Q(λi, a, ν)

]

+W2i

[b2R(λi, b, ν) − a2R(λi, a, ν)

](16)

4

in which

P (λi, µ, ν) = J2ν (λiµ) − Jν−1(λiµ)Jν+1(λiµ)

R(λi, µ, ν) = Y 2ν (λiµ) − Yν−1(λiµ)Yν+1(λiµ)

and

Q(λi, µ, ν) = J′ν(λiµ)Yν−1(λiµ) − 1

λiµJν−1(λiµ)Yν(λiµ) − J

′ν−1(λiµ)Yν(λiµ)

and µ stands for a or b . It is not difficult to verify some properties of the transform

from definition (12). For example,

T [αf(x) + βg(x), a, b, ν; λi] = αf(λi) + βg(λi) (17)

T [f(px), a, b, ν; λi] =

∫ b

a

xf(px)Mν(λix)dx =1

p2T [f(x), pa, pb, ν; λi/p ] (18)

3. TRANSFORM OF A DIFFERENTIAL OPERATOR

We derive the transform of the following operator

Df(x) =d2

dx2f(x) +

1

x

d

dxf(x) − ν2

x2f(x) , a ≤ x ≤ b (19)

Let I be the transform of the first two terms of D , that is

I =

∫ b

a

x[f ′′(x)+

1

xf ′(x)

]Mν(λix) dx =

∫ b

a

xf ′′(x)Mν(λix)dx+

∫ b

a

f ′(x)Mν(λix)dx

Integration by parts of the first integral leads to,

∫ b

a

xf ′′(x)Mν(λix)dx = xMν(λix)f ′(x)∣∣∣ba−

∫ b

a

f ′(x)[xλiM

′ν(λix) + Mν(λix)

]dx,

and hence,

I = xMν(λix)f ′(x)∣∣∣ba− λi

∫ b

a

xf ′(x) M ′ν(λix) dx

Integrating by parts once again leads to,

I = x [f ′(x)Mν(λix)−λif(x)M ′ν(λix)]

∣∣∣ba+

∫ b

a

x−1[λ2

i x2M ′′

ν (λix)+λixM ′ν(λix)

]f(x) dx

Since Mν satisfies (1)we have

λ2i x

2M ′′ν (λix) + λixM ′

ν(λix) = (ν2 − λ2i x

2) Mν(λix)

5

and∫ b

a

x−1[λ2

i x2M ′′

ν (λix) + λixM ′ν(λix)

]f(x) dx =

∫ b

a

x[ν2

x2− λ2

]f(x) Mν(λix) dx

Furthermore, it follows from the boundary conditions (2) that

λiM′ν(λia) =

1

h1

Mν(λia) , λiM′ν(λib) =

1

h2

Mν(λib)

Hence

I =b

h2

Mν(λib)[f(b)+h2f

′(b)]− a

h1

Mν(λia)[f(a)+h1f

′(a)]−λ2

i f(λi)+T

[ν2

x2f(x)

]

and the transform of the operator D in (19) becomes

T [Df(x)] =b

h2

Mν(λib)[f(b) + h2f

′(b)] − a

h1

Mν(λia)[f(a) + h1f

′(a)] − λ2

i f(λi)

(20)

Transform of xν

From definition (12) we have

T [xν , a, b, ν; λi] =

∫ b

a

xν+1Mν(λix) dx

Using a result of [12], namely∫xρ+1Zρ(x)dx = xρ+1Zρ+1(x)

where Zρ(x) is any of the Bessel functions, we obtain

T [xν , a, b, ν; λi] =1

λi

[bν+1 Mν+1(λib) − aν+1 Mν+1(λia)

]

Since

Mν+1(cz) =ν

czMν(cz) − M ′

ν(cz)

the transform becomes,

T [xν , a, b, ν; λi] =bν+1

λi

[ν

λibMν(λib) − M ′

ν(λib)

]− aν+1

λi

[ν

λiaMν(λia) − M ′

ν(λia)

]

Then, from the boundary conditions (2) this reduces to,

T [xν , a, b, ν; λi] =bν+1

λ2i

[ν

b+

1

h2

]Mν(λib) − aν+1

λ2i

[ν

a+

1

h1

]Mν(λia) (21)

6

In particular, the transform of a constant (where ν = 0) is found to be

T [c, a, b, 0; λi] =c

λ2i

[b

h2

M0(λib) − a

h1

M0(λia)

]= c T [1, a, b, 0; λi] (22)

4. HEAT CONDUCTION IN AN INFINITE CYLINDER

Consider a long hollow cylinder of inner radius a and outer radius b , with radiation

type boundary conditions in both outer and inner surface, and a prescribed initial

temperature. The differential equation of the phenomena is:

1

K

∂U

∂t=

∂2U

∂r2+

1

r

∂U

∂r, (23)

where a < r < b , t > 0 and U(r, t) denotes the temperature at any radial position r

at time t; K is a constant that depends on the material of the cylinder. The initial

and boundary conditions are as follows:

U(r, 0) = I(r) , a < r < b (24)

U + h1∂U

∂r

∣∣∣∣r=a

= f(t) , U + h2∂U

∂r

∣∣∣∣r=b

= g(t) , t > 0 (25)

Taking ν = 0 , we consider the transform of U with respect to the radial variable,

that is

U(λi, t) =

∫ b

a

r U(r, t) M0(λir) dr (26)

Referring to (20) and (23), we obtain

1

K

∂U

∂t=

b

h2

M0(λib)

[U + h2

∂U

∂r

]r=b

− a

h1

M0(λia)

[U + h1

∂U

∂r

]r=a

− λ2i U

From the boundary conditions (25) we have this reduced to

1

K

∂U

∂t=

b

h2

M0(λib) g(t) − a

h1

M0(λia) f(t) − λ2i U

and the following ODE is obtained

∂U

∂t+ Kλ2

i U = K

[b

h2

M0(λib) g(t) − a

h1

M0(λia) f(t)

](27)

whose solution is given by

U(λi, t) = exp(−Kλ2i t)

[K

∫ t

0

exp(Kλ2i s)

[ b

h2

M0(λib) g(s) − a

h1

M0(λia) f(s)]ds + C

]

7

Taking the transform of the initial condition (24), namely

U(λi, 0) = I(λi)

leads to C = I(λi) , hence

U s(λi; t) = exp(−Kλ2i t)[

K

∫ t

0

exp(−Kλs[ b

h2

M0(λib) g(s) − a

h1

M0(λia) f(s)]ds + I(λi)

](28)

The solution of (23) follows after applying the inversion formula to the above,

thus

U(r, t) =∞∑i=1

U(λi, t)

Mν(λi)M0(λir) (29)

with the understanding that the summation is taken over all the positive roots of

the equation:

B0(λa, h1)A0(λb, h2) − A0(λa, h1)B0(λb, h2) = 0 (30)

Special cases:

(i) Let us consider the previous problem with the following initial and boundary

conditions;

U(r, 0) = 0 , a < r < b (31)

U + h1∂U

∂r

∣∣∣∣r=a

= U0 (constant), U + h2∂U

∂r

∣∣∣∣r=b

= U1(constant), t > 0

(32)

Then equation (28) will become

U(λi; t) = exp(−Kλ2i t)[

K

∫ t

0

exp(Kλ2i x)

{b

h2

M0(λib)U1 − a

h1

M0(λia)U0

}dx

](33)

=1 − exp(−Kλ2

i t)

λ2i

[bU1

h2

M0(λib) − aU0

h1

M0(λia)

]

And according to (29) the solution will become

U(r, t) =∞∑i=1

1 − exp(−Kλ2i t)

λ2iMν(λi)

[bU1

h2

M0(λib) − aU0

h1

M0(λia)

]M0(λir), i = 1, 2, 3, ...

(34)

where the summation is taken over all positive roots of (30).

8

(ii) If h1 −→ 0 in (25), that is U |r=a = f(t) , our result (29) reduces to a result of

Kalla and Villalobos [8, p.41, eq.(20].

5. HEAT CONDUCTION IN A SEMI-INFINITE CYLINDER

Let us consider the problem of finding the temperature distribution U(r, z, t) in a

hollow semi-infinite cylinder with outer radius b and inner radius a. This problem

is expressed by the differential equation,

1

K

∂U

∂t=

∂2U

∂r2+

1

r

∂U

∂r+

∂2U

∂z2(35)

where a < r < b ; z , t > 0 , and the initial/boundary conditions are taken to be:

U(r, z, 0) = I(r, z) (36)

U(r, 0, t) = V (r, t) , limz→∞

U(r, z, t) = 0 (37)

U + h1∂U

∂r

∣∣∣∣r=a

= f(z, t) , U + h2∂U

∂r

∣∣∣∣r=b

= g(z, t) (38)

Consider the integral transform,

U(λi, z, t) =

∫ b

a

r U(r, z, t)M0(λir) dr (39)

and the Fourier sine transform

U s(λi, p, t) =

∫ ∞

0

U(λi, z, t) sin(p z) dz (40)

Following a similar argument as in the previous section, the transformed equation

of (35) becomes

1

K

∂U

∂t=

b

h2

M0(λib)g(z, t) − a

h1

M0(λia)f(z, t) − λ2i U +

∂2U

∂z2

whose Fourier sine transform is found to be,

1

K

dU s

dt=

b

h2

M0(λib)gs(p, t) − a

h1

M0(λia)fs(p, t) − λ2i U s − p2U s + pV (λi, t)

and which is expressed as,dU s

dt+ K(λ2

i + p2)U s

= K

[b

h2

M0(λib)gs(z, t) − a

h1

M0(λia)fs(z, t) + pV (λi, t)

](41)

9

Now we have the solution of the above ODE given by

U s(λi, p, t) = e−K(λ2i +p2)t

[K

∫ t

0

eK(λ2i +p2) τ

[ b

h2

M0(λib)gs(z, τ)

− a

h1

M0(λia)fs(z, τ) + pV (λi, τ)]dτ + C

]

and from the initial condition, we have C = Is(λi, p) , so that

U s(λi, p, t) = e−K(λ2i +p2)t

[K

∫ t

0

eK(λ2i +p2) τ

[b

h2

M0(λib)gs(z, τ)

− a

h1

M0(λia)fs(z, τ) + pV (λi, τ)]dτ + Is(λi, p)

](42)

Finally, taking respective inverse transforms will lead to the solution:

U(r, z, t) =2

π

∞∑i=1

1

Mν(λi)

[∫ ∞

0

U s(λi, p, t) sin(pt) dp

]M0(λir). (43)

Special cases:

(i) We consider the previous partial differential equation with the following con-

ditions

(U + h1

∂U

∂r

)r=a

= 0, z > 0, t > 0

(U + h2

∂U

∂r

)r=b

=1

z, z > 0, t > 0

U(r, 0, t) = U0, (constant) a < r < b, t > 0

U(r, z, 0) = 0 a < r < b, z > 0

U(r, z, t) → 0 as z → ∞

then (42) will become

Us(λi, p, t) = e−K(λ2i +p2)t

K

t∫0

e−K(λ2i +p2)x

{b

h2

M0(λib)π

2

+pU0

λ2i

[b

h2

M0(λib) − a

h1

M0(λia)

]}dx

]

=1 − e−K(λ2

i +p2)t

λ2i + p2

{p

h2

[π

2+

PU0

λ2i

]M0(λib) − aPU0

λ2i h1

M0(λia)

}

10

and according to (43)

U(r, z, t) =∞∑i=1

2

π

∞∫0

1 − e−K(λ2i +p2)t

λ2i + p2

{p

h2

[π

2+

PU0

λ2i

]M0(λib) − aPU0

λ2i h1

M0(λia)

}

sin(pt)dz} × M0(λir)

Mν(λi).

(ii) For h1 −→ 0 in (38), that U |r=a = f(z, t) , our result (43) reduces to one

considered in [8, p.43; eq. (36)].

6. CONCLUSIONS

Here we have introduced a new finite integral transform (Hankel-type) involving

product of Bessel functions as the Kernel. This transform can be used to solve

certain class of mixed boundary value problems. As indicated in previous sections

4 and 5, this transform is suitable to solve heat conduction problems in hollow

cylinders with radiation (mixed) conditions on both surfaces (r = a and r = b).

The Hankel-type finite transforms considered earlier in [5, 7, 8, 11] were able to

solve problems with both surfaces of the cylinder kept at prescribed temperature or

radiation condition on only one surface, r = b [8].

Numerical treatment of the results obtained here can be done by using “Matem-

atica” and [S. L. Kalla and S. Conde: Tables of Bessel Functions and Roots of

Transcendental Equations, Univ. Zulia, Venezuela, 1978].

ACKNOWLEDGEMENT

The authors are thankful to the referee for some useful comments and suggestions

for a better presentation of this paper.

REFERENCES

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certain partial differential equations. Jour. Australian Math. Soc. 41 B

(1999), 105-117.

[2] Andrews, L. C. and Shivamoggi, B. K., Integral Transforms for Engineers,

SPIE, Bellingham, Washington, (1999).

11

[3] Birkhoff, G. and Rota, G. C., Ordinary Differential Equations, Wiley, New

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[4] Churchill, R. V., Operational Mathematics, McGraw-Hill, New York, (1972).

[5] Debnath, L. Integral Transforms and their Applications. CRC Press, Boca

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