MECHANICS RESEARCH COMMUNICATIONS Vol. 7(3),159-164, 1980. Printed in the USA. 0093-6413/80/030159-06502.00/0 Copyright (c) Pergamon Press Ltd.

LAPLACE TRANSFORM METHOD IN BOUNDARY VALUE PROBLEMS FOR THE HEL~OLTZ EQUATION

R. Souchet Laboratoire de M~canique Th~orique Universit~ de Poitiers, France

(Received and accepted for print 15 February 1980)

Abstract

The purpose of this present note is to obtain boundary integral equation formulations of boundary value problems for the two-dimensional Helmholtz equation. The integral equations are derived using the Laplace transform method.

Introduction

Many texts present the derivation of boundary integral equations on the ba-

sis of the elementary solution for the partial differential equation con-

nected with the physical problem II, 2, 3 I . However this method provides

two real disadvantages :

| - The required elementary solutions depend on the partial differen-

tial operators.

2 - The integral equations involve singular kernels, thereby hampering

numerical analysis.

In the present paper, we propose an idea, originally posed by Picone 141

and Amerio 151, to eliminate the above difficulties. A formulation of boun-

dary value problems using the Laplace transform is shown to lead to bounda-

ry integral equations. No attempt has been made to include general partial

differential equations. Our object is to present the method on a particular

case, namely the two-dimensional Helmholtz equation. Guided by this example,

we expect that other boundary value problems, e.g. in elasticity, will be

solved in such a way.

Moreover, it is of interest to test the method, using a computational pro-

cedure. This was made on a Neumann problem for the Laplace operator 161.

159

160 R. SOUCHET

The numerical results are simplified for two reasons :

I - The kernels of integral equations do not depend in essential way

on the differential operator.

2 - The integral equations involve regular kernels. So the actual me-

thod does not possess the above disadvantages of the singular integral

equations.

~erio bound.ayy formula

Let 12 denote a bounded domain in ~2 with a piece-wise smooth boundary ~.

Let (xl, x 2) be cartesian coordinates and let (xl, x2) be the outward unit

normal on ~. We consider the Helmholtz equation

- ~u + k2u = f , k ~ 0 (I)

where f represents a given, continuous function. In what follows, we shall

made use of the second Green formula

(u .&v - v.Au) dXldX 2 = (u "~-~n - v . ds . (2) ~ ~ dn

We should therefore require that any solution u of equation (I) is availa-

ble for formula (2). Let E denote this class of solutions.

Let u be a solution of class E. It is well known 141 that u can be charac-

terized by its Laplace transform

I xl + 2) u(x ,x2) dxldx 2 u ( p l , P 2 ) = e x p (Pl P2 x " 1 (3) ~

I t s h o u l d b e n o t e d t h a t p l , P 2 a r e two i n d e p e n d a n t c o m p l e x v a r i a b l e s . I n o r -

d e r t o d e t e r m i n e u , we f o l l o w a p r o c e d u r e due t o P i c o n e 14[ . The w e i g h t i n g

f u n c t i o n v i n f o r m u l a (2 ) i s a s s u m e d t o b e t h e L a p l a c e k e r n e l

L ( P l , p 2) ( X l , X 2) = e x p ( P l X l + P2X2 ) (4 )

H e n c e we f i n d

(p l 2 2 u • I - + P2 - k2) (PI'P2) = j f" L(PI'P2) dXldX2

I d L ( P l ' P 2 ) I an - u . dn ds + - - . L ( P l , p 2) ds (5 ) ~ j ~ dn

k 2 ~ . 2 2 _ # O , i t i s c l e a r t h a t we c a n f i n d u zn t e r m s o f t h e b o u n - When Pl + P2 du

d a r y v a l u e s u and ~-~. N e x t , when we h a v e

2 _ k 2 Pl 2 + P2 = O (6)

i t f o l l o w s t h a t a s o l u t i o n u o f c l a s s E s a t i s f i e s t h e i d e n t i t y

LAPLACE TRANSFORMS FOR HELMHOLTZ EQUATION 161

J d L ( P I ' P 2 ) f du I u . dn ds - d--~ " L ( P l ' P 2 ) d s - f . L ( P l , p 2) dXldX 2 = 0 (7) ~ ~ ~

This i m p o r t a n t i d e n t i t y was d e r i v e d w i t h o u t undue e f f o r t . However, Amerio

151 p o i n t s ou t the need f o r a supp l emen ta ry c o n d i t i o n in o r d e r to g u a ran -

t e e un iquenes s : the domain ~ must be s imply c o n n e c t e d . Thus we have the

f o l l o w i n g theorem

Theorem 1 - A f u n c t i o n u o f c l a s s E i s s o l u t i o n of e q u a t i o n (1) in a s im-

p ly c o n n e c t e d bounded domain ~, i f and on ly i f the boundary v a l u e s u and

du 2 P22 k 2" d-~ s a t i s f y the Amerio boundary fo rmula (7) when Pl + =

We c onc lude t h a t e q u a t i o n (7) p r o v i d e s i n t e g r a l e q u a t i o n s f o r boundary

v a l u e p rob lems . Thus, g i v e n u on 2£ ( D i r i c h l e t p r o b l e m ) , the fo rm u la becomes du

an i n t e g r a l e q u a t i o n f o r ~nn " N e v e r t h e l e s s , e q u a t i o n (7) i s no t a boundary

i n t e g r a l e q u a t i o n i n the usua l s e n s e .

Boundary integral equations

2 P2 2 k 2 For simplicity, we shall assume that k # O. It follows that Pl + =

if and only if

Pl = ~I k( ~p + P) ' P2 = ~ k( - p) , p # 0 (8)

where p is a complex variable• Hence the Amerio boundary formula reduces

to

k [ (! -- k (i z + p ~)} ds ~ J u. n + p n). exp {7 p

- ~ P

_ [ du k 1 ~ ~-~n" exp {7 (~ z + p ~)} as

f k (~ + ~)} = 0 - f . exp {7 p Z P dXldX 2 (9)

(z = x! + i x2, ~ = x I - i x 2 ; n = n I + i n2, ~ = n! - i n2).

Next we wish to introduce the following function of the complex variable p

(p # O)

f k 1 -- _ du} k (.~ + ~)} as F(p) = {~ u(~ n + p n) dn " exp {~ P Z P ~

f k (! -- • z + p z)} dXldX 2 (IO) - f exp {7 p

162 R. SOUCHET

Now we consider the boundary ~ as a finite number of arcs C, so that the

integrand in (;0) possesses the required properties on each C in order to

have an analytic function. Hence the function F is analytic throughout

the complex plane except at the isolated singularity p = 0. Thus the

Amerio boundary formula takes the reduced form

F ( p ) = 0 , p ~ ~ 2 \ { 0 } . ( 1 1 )

Now we attempt to find a boundary integral equation. We derive from (11)

Re F(p) = O , p ~ ~, (O ¢ ~)" (12)

So, the truth of (12) is a necessary condition for the truth of (ll). But

it is also a sufficient condition. Indeed, if we take O # ~ u ~, ReF is

a harmonic function in ~, such that (12).

Hence, we have a Dirichlet problem ; therefore

Re F(p) = O , p ~ ~. (13)

~:~

ff

-~ ~:~

a ) F i r s t c a s e : 0 ~ S~ ~ SS2 ; ~ i s a s i m p l y - c o n n e c t e d r e g i o n .

It follows that

F(p) = i K , p ~ ~ (14)

where K is a real constant. Then

F(p) = iK , p ~ C\{O} (15)

using the principle of analytic continuation. But we observe that F(p) is a

real number provided that p = I (in fact IPl = I) ; thus K = O.

In the case 0 E ~, the proof must be modified. We emphasize that

Im F(p) = O , IPl = I , (16)

LAPLACE TRANSFORMS FOR HELMHOLTZ EQUATION 163

so that

d__ Re F(p) = 0 Ipl = i. (17) dn '

If ~ coincides with the unit circle, from the two conditions (12), (16),

(i.e. F(p) = 0 , IPl = I), all the coefficients of the Laurent series of

F are zero. Thus F = O.

If ~ is an arbitrary contour, with the two conditions (12), (17), i.e.

d Re F(p) = 0 , p ~ ~ and ~n Re F(p) = 0 , IPl = I ,

we have a mixed problem for Re F in each region ~' limited by ~ and the

circle IPl = ]" Hence

Re F(p) = 0 , P 6 ~' , (18)

and clearly

Re F(p) = 0 , IPl = I . (19) ~,

~

b) An example of the second case : O ~ ~ ; ~ is a simply connected region.

Hence all the coefficients of the Laurent series of F are~ zero. Thus again

F = O.

Let us now introduce the following kernels

k (! k (_; z + p 7)}} (20) = n + p ~) exp {-~ P R(p,x) -~ Re { P

k (± z + p 7)}} (21) S(p,x) = Re {exp {~ P

where x = (x 1,x 2). Then we find the following basic result.

Theorem 2 - A function u of class E is solution of equation (I) in a simply du

connected domain ~ if an only if the boundary values u and ~n satisfy the

boundary integral formula

R(p,x) . u(x) ds- S(p,x).~-~n (x) ds - S(p,x).f(x) dx= O (22)

x ~ ~ x ~ ~ x ~!~

164 R. SOUCHET

where p is a (complex) variable specifying points on the boundary $~.

We observe two important facts : the kernels (20), (21) are regular when

p and x describe ~ and were trivially derived from the Laplace kernel

exp (plx! + P2X2 ). Since the formula (22) involves the boundary values u du

and ~ , we conclude that formula (22) provides boundary integral equations

for boundary value problems. Thus, given u on ~ , the formula (22) becomes du

a boundary integral equation for the basic unknown ~ . We would like to

repeat that (22) is sufficient for a simply connected domain.

Finally, one should note the following remarks :

; - In theorem 2, we can replace the condition p d ~ by p 6 ~, where

~ is a domain having the same properties that ~ 161.

2 - For the Laplace equation - &u = f , a similar boundary integral

equation 161 can be derived from the Amerio boundary formula (7).

3 - If the domain ~ is multiply connected, we must consider simply con-

nected sub-regions : the boundary integral equation (22) is used for each

sub-domain and the continuity of the functions and their normal derivatives

is assigned along common boundaries.

Conclusion

The formulation discussed in the present paper has proved successful with a

torsion problem of Saint-Venant involving computational procedure 161. The

absence of any singularities in the integrand simplifies the numerical treat-

ment, since no analytical approximation need be made to the integral over

some neighborhood of a singularity. It should also be pointed out that the

actual method can be extended to others boundary value problems, e.g. in

classical elasticity.

References

I! M.A. Jaswon, Proc. Roy. Soc. A, 275, 23 (!963)

12 F.T. Rizzo, Quart. Appl. Math., 25, 83 (|967)

13 J.Co Lachat and J.O. Watson, Intern. J. for Num. Meth. in Engin., IO, 991, (1976)

14 M. Picone, Rend. Acc. Lincei, 2, 365, (1947)

15 L. Amerio , Rend. Ist. Lombardo di Sci. e Lettere, 78, 79, (!944)

16 R. Souchet and D. Robinaud, to be communicated at the Second Int. Semo on Recent Advances in Boundary Element Methods, Southampton, England, (1980).