Mathematical Methods for Physics and Engineering : A Comprehensive Guide

23.4 CLOSED-FORM SOLUTIONS

23.4.1 Separable kernels

The most straightforward integral equations to solve are Fredholm equations

withseparable(ordegenerate) kernels. A kernel is separable if it has the form

K(x, z)=

∑n

i=1

φi(x)ψi(z), (23.8)

whereφi(x)areψi(z) are respectively functions ofxonly and ofzonly and the

number of terms in the sum,n, is finite.

Let us consider the solution of the (inhomogeneous) Fredholm equation of the

second kind,

y(x)=f(x)+λ

∫b

a

K(x, z)y(z)dz, (23.9)

which has a separable kernel of the form (23.8). Writing the kernel in its separated

form, the functionsφi(x) may be taken outside the integral overzto obtain

y(x)=f(x)+λ

∑n

i=1

φi(x)

∫b

a

ψi(z)y(z)dz.

Since the integration limitsaandbare constant for a Fredholm equation, the

integral overzin each term of the sum is just a constant. Denoting these constants

by

ci=

∫b

a

ψi(z)y(z)dz, (23.10)

the solution to (23.9) is found to be

y(x)=f(x)+λ

∑n

i=1

ciφi(x), (23.11)

where the constantscican be evalutated by substituting (23.11) into (23.10).

Solve the integral equation

y(x)=x+λ

∫ 1

0

(xz+z^2 )y(z)dz. (23.12)

The kernel for this equation isK(x, z)=xz+z^2 , which is clearly separable, and using the
notation in (23.8) we haveφ 1 (x)=x,φ 2 (x)=1,ψ 1 (z)=zandψ 2 (z)=z^2. From (23.11)
the solution to (23.12) has the form

y(x)=x+λ(c 1 x+c 2 ),

where the constantsc 1 andc 2 are given by (23.10) as

c 1 =

∫ 1

0

z[z+λ(c 1 z+c 2 )]dz=^13 +^13 λc 1 +^12 λc 2 ,

c 2 =

∫ 1

0

z^2 [z+λ(c 1 z+c 2 )]dz=^14 +^14 λc 1 +^13 λc 2.