Integral Equations

9

Integral Equations As is well-known in classical electrodynamics, it is possible to reduce the problem of solving a boundary-value problem to that of solving an integral equation. This is very advantageous since methods of field calculation based upon integral equations have gained great importance. In this chapter we shall present the general theory; details of numerical procedures are given in Chapter 10.

9.1 Integral equations for scalar potentials In the following account, we consider a domain F in three-dimensional space and its boundary OF. Inside the domain F we attempt to solve a uniquely specified boundary-value problem for Poisson's equation (7.1)

(9.1)

V2V(r) = - S ( r )

9.1.1 General theory In order to obtain an integral equation, we start from Green's theorem for a modified domain F' with boundary OF' and for a variable of integration 1, t

- v-gy F'

,/da'

(9.2)

OF'

valid for any differentiable functions a ( r ' ) and V(r'), regardless of their special meanings. The operator 0 / 0 n ' = n ' . V' is the so-called normal derivative, the derivative in the direction of the outward oriented surface normal n' on 0F'. The boundary Or' itself may consist of several distinct dosed surfaces and the integral on the right-hand side of (0.2) is then the sum of the contributions arising from the different surfaces; this summation is implicit in the notation. The function 6: in (0.2) can be chosen arbitrarily; the most suitable choice is the free-space Green's function, defined by G(r, r') - (4~'lr - r'l) -~

(9.3)