Mathematical Methods for Physics and Engineering : A Comprehensive Guide

23.5 NEUMANN SERIES

23.5 Neumann series

As mentioned above, most integral equations met in practice will not be of the

simple forms discussed in the last section and so, in general, it is not possible to

find closed-form solutions. In such cases, we might try to obtain a solution in the

form of an infinite series, as we did for differential equations (see chapter 16).

Let us consider the equation

y(x)=f(x)+λ

∫b

a

K(x, z)y(z)dz, (23.34)

where either both integration limits are constants (for a Fredholm equation) or

the upper limit is variable (for a Volterra equation). Clearly, ifλwere small then

a crude (but reasonable) approximation to the solution would be

y(x)≈y 0 (x)=f(x),

wherey 0 (x) stands for our ‘zeroth-order’ approximation to the solution (and is

not to be confused with an eigenfunction).

Substituting this crude guess under the integral sign in the original equation,

we obtain what should be a better approximation:

y 1 (x)=f(x)+λ

∫b

a

K(x, z)y 0 (z)dz=f(x)+λ

∫b

a

K(x, z)f(z)dz,

whichisfirstorderinλ. Repeating the procedure once more results in the

second-order approximation

y 2 (x)=f(x)+λ

∫b

a

K(x, z)y 1 (z)dz

=f(x)+λ

∫b

a

K(x, z 1 )f(z 1 )dz 1 +λ^2

∫b

a

dz 1

∫b

a

K(x, z 1 )K(z 1 ,z 2 )f(z 2 )dz 2.

It is clear that we may continue this process to obtain progressively higher-order

approximations to the solution. Introducing the functions

K 1 (x, z)=K(x, z),

K 2 (x, z)=

∫b

a

K(x, z 1 )K(z 1 ,z)dz 1 ,

K 3 (x, z)=

∫b

a

dz 1

∫b

a

K(x, z 1 )K(z 1 ,z 2 )K(z 2 ,z)dz 2 ,

and so on, which obey the recurrence relation

Kn(x, z)=

∫b

a

K(x, z 1 )Kn− 1 (z 1 ,z)dz 1 ,