The solution of a singular integral equation with some applications in potential theory

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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 equationwhichgovernssome two-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 equationwithkernel q +log

sin(O-)

Shail

[3]

transfoedthisequationinto a well known Cademan type

Adifferenttechniquehas beenused by Sampathand Jain

[4]

basedondecouplingtheequationinto

twosingular equationswhich can be solvedusing eigennctions expansion metod The sametechnique

(2)

In

section 2 we use theapproach

of[4]

tosolve equation(1,

l),and

insection3 weapplytheresultsto solvelaplaceequation associated with Dirichlit andNeumann boundaryconditions Asanillustrative

exampleweconsider 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)

(3)

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).

(4)

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)

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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 the n-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

./(O)

-a<

O

<at (36)

11 ./

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

(6)

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-#

4N0)

(3 20)

/1.

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)

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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

-K

#(x)

I,

(0)=

,

g(x)

.

(o)

o

Substitutingthesevalus in(2 32)wefind that

K,

cos(n

2

Tofindthe valueof

K

weuse the condition that thetotal charge perunitheighton 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(

na 2)

2a

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)

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ACKNOWLEDGMENT.

The present work issupported bytheuniversityof Jordan underproject

#356

REFERENCES

[1]

GAUTESEN,A

K

&

OLMESTEAD,W

E., On

the solution ofthe integral equationforpotential 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

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&

Math.Set. 11(1988)751-762

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&

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

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22

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JAIN,S

& JAIN,D

L,Diffractionofan

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&

JAIN,DL,

Some

electrostaticproblems oftwoequalco-axial circularstrips,.1.

Math.Ph)’.

Sc’t25 1991),217-230

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S

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&

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

[9]

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&

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