2k WORDS

VOL. 21 NO. (1998) 189-196

RESEARCH NOTES

THE SOLUTION OF A SINGULAR INTEGRAL

^EQUATION

^WITH SOME APPLICATIONS IN POTENTIAL THEORY

N.T. SHAWAGFEH

Department

of mathematics

Universityof Jordan Amman,

JORDAN

(Received January 6, 1995 and in revised form December 9, 1996)

ABSTRACT.

An analyticalsolution is derivedforasingular integral equationwhichgovernssometwo- dimensionalpotential boundaryvalueproblemsinaregionexteriorton-infinite co-axial circular strips

An

applicationin electrostatics isdiscussed

KEY

WORDS

AND

PHRASES:Singular integral equation,electrostatics, Laplace’s equation 1991

AMS SUBJECT CLASSIFICATION CODES:

31Bl0, 45B05, 78A30

INTRODUCTION

Inthispaperwederive a solutiontothe Fredhoimsingular integral equation

where q and a are constantsand I"(0)isadifferentiable nction for

]0-

(2k#

n)l

^{< a,k 0 n-I This}

integral equationgovernsthe solutionof various two-dimensional Difichlit andNeumann potential

bounda

^valueproblems fortheregion consistingof the whole

(r, O)

planeoutside the circularstrips 2k

Theprevious investigationsinpotential problems ofcircularstrips

[1-8]

^wereconcerned mainlywith the caseoftwostrips,whereGreensnction approach leadstoasingular integral equation^withkernel

q +log

sin(O-)

^Shail

^[3]

^{transfoedthisequation}into a well known Cademan type

Adifferenttechniquehas beenused by Sampathand Jain

[4]

basedondecouplingtheequationinto twosingular equationswhich can be solvedusing eigennctions expansion metod The sametechnique hasbeen usedtosolve various

bounda

^valueproblems involving^twocircularstrips[5-8

In

section 2 we use theapproach

of[4]

tosolve equation(1,

l),and

ⁱⁿsection3 weapplytheresultsto solvelaplaceequation associated with Dirichlit andNeumann boundaryconditions Asanillustrative example^weconsider insection 4 the electrostaticsproblemin aregion externalto

perfectly

conducting

n-circular strips Formulae for thesurface charge density andconcentration factor are derived,and these arebelievedtobenew

2.SOLUTIONOF

THE

INTEGRAL EQUATION

To

obtain the solutionoftheintegral equation(1 l)we firstreduceittosimpler singular integral equations,sointerchangingthe sum andintegral signsandusingsomeproperties ofthe function in the kernel we obtain

1() qn--iog2

⁺₂

g’j(l- ---t_0 ^cs(O- _J

Usingtheidentity

[9],

2r))

(]

cos(O- -

^--;-i-(l-

cos(n(O- ))),

(2.2)

intheequation (2 we get

()[Q

^+og(1

^cos,,(8- )]d ^2F(a), I<a

^{(2 3)}

where

(.)

2qn (2n l)log2

Nowwewriteeach of the functions

l(0)and F(0)

as

l(O) ie(8)+ lo(e

(2 4a)

l,’(O)

I.,

(/9)+

I, ^(O)

^(24b)

where thesubscripts eand o standforthe even andoddpartsrespectively. Substituting

(2

4)into(23) weobtainthefollowingdecoupled singular integral equations

/,,(#)log n 2

inordertosolve eq(25)weintroducethe transformation

cos(,,) sin:()cosx

⁺

cos:(),

cos(n#) ^sin

()cosy

^{+ cos}

()

whichtransforms the equationintotheform

j H(y)[ ^R

⁺

^log(21cosx cosyiy ^h(x),

where

0<8<a (25)

O<O

<or (26)

O< x<n" (27a) O< y<n" (27b)

O<x<r (2 8)

R=Q+log sin

z(,,a/2)

sin" (-2-)

^siny

and H(y)=

I())

nsin(n#)

Expanding

h(x)

as Fourier cosineseries

where

.

^cos(rex)

h(x) -a

⁺

a. [ h(x) cos(mx)dx.

andmakinguseofthe bilinearform

10]

log(

21cos

^{x cos}

^Yl)

^-2 cos(mx) cos(my) m wefindthat the solution of(28)^canbe written inthe form

H(y) 4,, +

,4.

cos(my), where

a,,

A.. ^---a,.

^m>

A,,

2M? r

thususing (2 11 wefindthatthe solution of(2 5)isgiven by

.-ncos(n

2)

A

+

A.,

cos(my)

!,,()

/cos(,,)-

^cos(,,a)

where we have used the relation

sioo. Csc:

Nowtosolveeq (2 6)^wefirst differentiatebothsides with respectto8

nlo()

sin(nO

0

cos(,,)-cos(-O) d ^b"(e)

Applyingthe transformation(2 7)into(2 19)reduces itto G(y) dy g(x).

cosy cos x

O<x,y

where

g(x)

1."(o)

G(y)

1o(#)sin

^y Expanding (;(y)inFourier cosine series

(;(y)

B,,

+

B..

^cos(my).

O<X<K

m>_O

O<y< zr O<

O<x<

(29) (2 10)

(2 I1)

(2 12)

(2 13)

(2 14)

(2

(2 16)

(2 17)

(2 18)

(2 19)

(2 20)

(221) (2 22)

(223)

andnotingtheintegral

cos(my)

dr’=

zcsin(mx)csc(x)

cos)’ cosx

whichfollows by putting w=

e"

andintegrating aroundthe circle

]w[

^Then

Thus if weexpand

itfollowsthat

and hence

G(y) dy

B,

sin(mx)csc(x),

cosy COS X

g(x)sin x

(’,

^sin(mx)

g(x)sinxsin(mx)dx

m_>l

O<X<⁷

I,,() _ml _B,,

_{+ (" cos(my)}

slrly zr

Toevaluate theconstant

B,,

^weusethefinitenesscondition that G(y) --,0 as y

-

^{0 thisgives}

Substituting (2 30)into(229)wefindfrom (2 22)and the relation(2 18)that

/(0)

siny.,

(’(1-cosmy)

^-a

<

^<a

Finally summingup the results we write the solution ofeq (1 l)as

/() cosO,)-

cos(,,a)

(’= (1-

cosmy) -a< <a nsiny

4n’[qn

^+log(2

m>l

a., h(

x

cos(mx)dx,

m>O

("’ =--re2 !

^{g(x)sin x}

^sin(mx)dx

where

(224)

(225)

(226)

(2.27)

(228)

(2 29)

(230)

(231)

(232)

(233)

(2 34) (235)

(236)

h(x)

!.i,(0

g(x)

!,i;(0

^{(2 37)}

3.APPLICATION IN POTENTIAL

THEORY

The singularintegraleq(11)governsthe solutions of various Dirichlit andNeumann problemsfor two-dimensionalLaplace equationinthe domain

D

consisting ofallthepoints (r,0)lyingooutside then- circularinfinitestrips

(

^r=a,

[0-zk

^{<a,k 0 n-l, (3 I)}

These types ofproblems appearindifferentphysicalfieldssuchas electrostatics,magnetostatics, steady stateheatflow,and others

A

Dirichlitproblem Weseek a function

(1)(r, 0)that

satisfies thefoilwing boundary value

problem

V:(1) 0 (32)

(1)(a,O): f(O) lo_2krl<a

^{k 0 n_1 (33)}

!-"

(l)(r,

0)1---

^{logr r---> (3 4)}

and are continuous across the arcs

2k;r r

(’;

^{r=a +a} <0< 2(k+l)---at (35)

ll

The function

./’(0)

satisfies the symmetry condition

.[(o

⁺

^2kn’l

11 ^./

^./(O)

^-a<

^O

^{<at (36)}

c,

--

^{ds (37)}

Using

Green’s

functionapproachand the symmetry in theproblem thesolution can be

represented

as

*(r. O)

-- ^{a ,, i /r..,}

^i(,)log

+a"-2racos(O-b-2kZC)dqb,,

^{(3 8)}

wherethe density funcUon

l(b)

is defined as

Theboundarycondition(33)leadstothefollowing integral equation

(3 which iseq (1 1)with q log(2a)

F(O) -2" _f(O),

andhence it has the solution(232)^withthe

a substitutionofthese values

B.

Neumann

problem

We

seek a function

(r, O)

satisfyingtheboundaryvalueproblem

V2T

=0

T(a,O)= p(O) ^---<at ,k 0 n-I (312)

and

q(r,O)-O ,as r

-

arecontinuousacross the arcs The function

I’(0)

satisfies the symmetry condition

andthe consistency condition

(3 13)

i ^p(O)dO

^{0 (3 15)}

Green’s

functionapproach leadstothefollwing

representation

W(r,O)

--N _,.](#)

^log

r: +/9: 2r,ocos(0- #- 2kzt) d# (3

16)

II

where./()=[V(r,]: -a.

^{(3 17)}

Fuhermore

.1()

satisfiestheedgecondition

./(a)

⁰

(3

8)

Usingthe

bounda

^{condition(3 12)weobtain}

’/(’)csc:

₂

^7(0- - ^{2k)d’ 80)}

^{(3 19)}

Integrating by

pans

usingtheedgecondition(3 18)^{we obtain}

J

(#)cot(0-#

^/1.

^4N0)

^{(3 20)}

Upon

integrationwith respectto we get

,/’()oglsin(O- ^{#- ,, ) d# -2’(0)}

^{+ (321)}

where 1’() p(O)dOand (" is an

arbitra

^constant

^Eq

⁽³21)is speciseofeq (1 1)withq 0

F(O)

^(’-2p(O),and

I(#) .1’(#) To

deteine theconstant (’weuse theedgecondition(3 18) Finally thedensity

.1()

isgivenby

.1(0) .I’()d#

^a

^O

^{a (3 22)}

4.1LLUSTTIVE

EXAMPLE

As

anexampletoillustrate our results we consider the electostaticspotenti problem ofthen-equal infinite co-axial

perfectly

conducting stripsinflee space chargedso thatthetotal charge per heighton eachstripisunity.ln cylindricalcoordinates

(r, 0,z)It

^{the stipsbe definedby}

r=a,

O-

<a

,

⁰ n-l,,

-

⁽⁴¹⁾

p(o

⁺

2kzCl ^p(O)

^-ct<

^O

^{< (3 14)}

then the electrostatics potential satisfies the Dirichlit

problem

A, where on theboundaryweassume that has theconstantpotentialK Thus

"’i [r ^2krC)dq

(1)(r.0)

-a

^{(q) log +}

^a’-

^2ra

^{cos(0 q}

and l(q)

satist

^theintegral equation(3 10)with

f(0)=K

,and hence

#(x)

I, (0)= , -K

g(x)

. ^{(o) o}

Substitutingthesevalus in(2 32)wefind that K,

cos(n

²

Tofindthe valueof

K

weuse the condition that thetotal charge per^unitheighton eachstripisunity.that

is.

Inseing(4 4)into(4

5)

and evaluatingtheresulting integralwefindthat

K -tog"}sin

^{,,a 2}

I)

and thecharge densityin this case is

, _cos(n

2)

/(0) m$co,0)-

cos(-a) ^’-a

<0

^{< a}

whilethechargeconcentrationfactor isgiven by

N

limCa ^-0) ’ ^{<0) 4ncot(}

^{2a na 2)}

Allthe above formulae are believedtobe new Forn=4theformulae(4 6).(4

7).and

(4 8) agreewith those in

[5].

andwhen a werecovertheknown results forachargedinfinite cylinder when the totalcharge perunitheightisn

(4 2)

(4

3)

(4 4)

(4 5)

(47)

(48)

(49)

ACKNOWLEDGMENT.

The present work issupported by^theuniversityof Jordan underproject

#356

REFERENCES

[1]

GAUTESEN,A

K &

OLMESTEAD,W

E., On

the solution ofthe integral equation^forpotential of

twostrip SlAMJ.Math.Anal 2 1971),293-306

[2]

GOEL,G C

& JAIN,D

L,Anoteonelectostaticproblem involvingtwostrips,.l.Pure

Appl.Math

7 (1976),751-756

[3]

SHAlL,R,

A

class of singular integral equationwith someapplication, Int..I.Math. Edu. l’ech. 15 (! 984),359-374

[4]

^SAMPATH,C

&

JAIN,D L., Onsolutionof theintegral equationsfor thepotential problems oftwo

circular strips, lnternat .1. Math.

&

Math.Set. 11(1988)751-762

[5]

^SAMPATH,C

&

JAIN,D L, Some boundaryvalueproblemsin electrostatics,.I.Math.

l’hA’.Sct

²²

(1988)

[6]

^JAIN,S

& JAIN,D

L,Diffractionofan

H-polarized

electromagneticwavebytwoequalinfinite circular strips, RadtoSet 24(1989),443-454

[7]

^SAMPATH,C

&

JAIN,DL,

Some

electrostaticproblems oftwoequalco-axial circularstrips,.1.

Math.Ph)’.

Sc’t25 1991),217-230

[8] VARMA,

S

K &

JAIN,D L,DiffractionofelasicPwavesbytwoequalco-axial circularstrips Eur MechA 11(1992),157-168

[9]

GRADSHTEYN,I S

&

RYZHIK, M, Tables

?fhltegrals Sertes

andProducts,Academic Press NewYork, 1965

[10]

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R

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Mathematical Problems in Engineering

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