Mathematical Methods for Physics and Engineering : A Comprehensive Guide

INTEGRAL EQUATIONS

By examining the special casesx=0andx= 1, show that

f(x)=

2

(e+3)(e+1)

[(e+2)ex−ee−x].

23.13 The operatorMis defined by

Mf(x)≡

∫∞

−∞

K(x, y)f(y)dy,

whereK(x, y) = 1 inside the square|x|<a,|y|<aandK(x, y) = 0 elsewhere.

Consider the possible eigenvalues ofMand the eigenfunctions that correspond

to them; show that the only possible eigenvalues are 0 and 2aand determine the

corresponding eigenfunctions. Hence find the general solution of

f(x)=g(x)+λ

∫∞

−∞

K(x, y)f(y)dy.

23.14 For the integral equation

y(x)=x−^3 +λ

∫b

a

x^2 z^2 y(z)dz,

show that the resolvent kernel is 5x^2 z^2 /[5−λ(b^5 −a^5 )] and hence solve the
equation. For what range ofλis the solution valid?
23.15 Use Fredholm theory to show that, for the kernel

K(x, z)=(x+z)exp(x−z)

over the interval [0,1], the resolvent kernel is

R(x, z;λ)=

exp(x−z)[(x+z)−λ(^12 x+^12 z−xz−^13 )]

1 −λ− 121 λ^2

,

and hence solve

y(x)=x^2 +2

∫ 1

0

(x+z)exp(x−z)y(z)dz,

expressing your answer in terms ofIn,whereIn=

∫ 1

0 u

nexp(−u)du.

23.16 This exercise shows that following formal theory is not necessarily the best way
to get practical results!

(a) Determine the eigenvaluesλ±of the kernelK(x, z)=(xz)^1 /^2 (x^1 /^2 +z^1 /^2 )and

show that the corresponding eigenfunctions have the forms

y±(x)=A±(

√

2 x^1 /^2 ±

√

3 x),

whereA^2 ±=5/(10± 4

√

6).

(b) Use Schmidt–Hilbert theory to solve

y(x)=1+^52

∫ 1

0

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

(c) As will have been apparent, the algebra involved in the formal method used

in (b) is long and error-prone, and it is in fact much more straightforward

to use a trial function 1 +αx^1 /^2 +βx. Check your answer by doing so.