BIT 20 (1980)+ 251-253

AN A P P L I C A T I O N O F

E X T R A P O L A T I O N TO T H E L I M I T

JAMES CALDWELL

The analysis described below can be used to construct solutions to potential problems of the Dirichtet type, that is the problem of solving Laplace's equation V2<P = 0 inside a region where 4> is specified at each point of the boundary (for example, the boundary curve of a plane region, or the surface of a three dimensional region). It can also be used for problems of the Neumann type with values of the normal derivative O~/dn prescribed on the surface and can be further extended to solve problems of the mixed boundary value type.

We shall now show how the Dirichlet problem can be reduced to an integral equation. The problem is to solve Laplace's equation

(1) V2~ = 0

in some closed volume V with the condition

(2) <I, = q~p(rs)

on the boundary surface S, ~p being prescribed (see Figure 1).

V 0 r'~ r~ ~ en n

Figure 1. Vo lume V bounded by a surface S.

A Green's function type of solution of (1) is given by

~(r) = [" o'(r~)

(3) dS' 3

Received February 4, 1980. Revised March 13, 1980.

252 JAMES CALDWELL

where the surface charge density is

(4) a(rs) = -4--n ~ n s = 4~ \ ~ n On "

Replacing the derivatives in (4) by simple differences gives

(5) a(rs) = - (4n~)- 1 { q~ (r s + en) + cb (r s - en) - 2q~p(rs) } + O (e) .

Using (3) this then reduces to

(6) 4rcsa(rs)+ s a(r's) ]r,s_rs_~n] + [r,s_rs+~nl dS' = 2cbp(rs)+O(e ) .

This is a Fredholm integral equation of the second kind and we are interested in solutions in the limit e ---, 0. Proceeding to this limit we obtain

f s a(r~) dS' = (7) tr's-rst ~p(rs)

which is a Fredholm integral equation of the first kind. Having solved (7) for a we may then compute ~ using quadrature from (3).

The integral equation (7) and its relation to solutions of potential problems is well known (see Garabedian [1]) and is the basis for much analysis. We now show how it can be used to obtain numerical solutions of three dimensional potential problems.

The difficulties associated with the numerical solution of (7) are illustrated by considering the one dimensional equation

(8) f 'oK(X,Y)f(y)dy=g(x)

and the corresponding eigenvalue problem

(9) f loK(X,y) f (y)dy=2f(x) .

Equation (9) possesses a set of eigenfunctions f~ and corresponding eigenvalues 2 i such that 2 i --, 0 as i --* ~ . This means that if we replace (8), in an obvious notation, by

(10) ~, wjKijf~ = gi J

where wj are weights corresponding to a numerical quadrature, then K~ will be singular in the limit. In other words, our problem is ill-conditioned.

These difficulties do not arise in the solution of (6) since it is an equation of the second kind, and this suggests a possible line of approach, namely, that we solve (6) for different values of e and extrapolate to the limit. We must expect, however, that to obtain accurate results it is necessary to solve (6) for small values of e.

AN APPLICATION OF EXTRAPOLATION TO THE LIMIT 253

Furthermore, as e decreases we will find that more and more points must be included in the quadrature scheme. Note also that as e ~ 0 the kernel associated with (6) has a sharp peak near r~ = r s which will further complicate our numerical quadrature.

It should be pointed out that Bakushinskii [2] has considered the numerical solution of Fredholm integral equations of the first kind, namely

l K(x,y)f(y)dy = g(x) by examining the limit case of the equation

~f(x)+ K(x,y)f(y)dy = g ( x ) a s ~ - , O . a

This technique of extrapolation to the limit already described has been applied successfully to the Dirichlet problem in the special case when the volume V is a sphere of magnetic material of unit radius.

In the first instance no allowance was made for axial symmetry and so the following full three dimensional problem was considered:

V2q~ = 0 inside the unit sphere r = 1 (11) and cb = q~p(O, q~) on the surface.

All the difficulties outlined above are well illustrated by taking the symmetric case as a test example. This involves the use of Gaussian quadrature to solve the integral equation (6) for a(O) and then obtaining ~(r) from equation (7). Results have been obtained for the cases e=0.1, 0.2, 0.3 and 0.4 and extrapolated to the limit. In this way high accuracy has been obtained using the simple finite- difference approximation in equation (5) without resorting to very high order

Gaussian quadrature.

Acknowledgement. The author would like to thank Professor R. D. Gibson (Newcastle-upon-Tyne

Polytechnic) for both his help and interest in this work.

R E F E R E N C E S

t. P. R. Gara.bedian, Partial Differential Equations, (Wiley) 1964.

2. A. B. Bakushinski i , A numerical method for solving Fredhofm integral equations of the first kind, Zh.

vychisl. Mat . mat. Fiz. 5, 4 (1965), 744-749.

DEPARTMENT OF MATHEMATICS AND COMPUTER STUDIES

SUNDERLAND POLYTECHNIC SUNDERLAND

ENGLAND