Mathematical Methods for Physics and Engineering : A Comprehensive Guide

INTEGRAL EQUATIONS

23.5 Solve forφ(x) the integral equation

φ(x)=f(x)+λ

∫ 1

0

[(

x

y

)n

+

(y

x

)n]

φ(y)dy,

wheref(x) is bounded for 0<x<1and−^12 <n<^12 , expressing your answer

in terms of the quantitiesFm=

∫ 1

0 f(y)y

mdy.

(a) Give the explicit solution whenλ=1.

(b) For what values ofλare there no solutions unlessF±nare in a particular

ratio? What is this ratio?

23.6 Consider the inhomogeneous integral equation

f(x)=g(x)+λ

∫b

a

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

for which the kernelK(x, y) is real, symmetric and continuous ina≤x≤b,

a≤y≤b.

(a) Ifλis one of the eigenvaluesλiof the homogeneous equation

fi(x)=λi

∫b

a

K(x, y)fi(y)dy,

prove that the inhomogeneous equation can only a have non-trivial solution

ifg(x) is orthogonal to the corresponding eigenfunctionfi(x).

(b) Show that the only values ofλfor which

f(x)=λ

∫ 1

0

xy(x+y)f(y)dy

has a non-trivial solution are the roots of the equation

λ^2 + 120λ−240 = 0.

(c) Solve

f(x)=μx^2 +

∫ 1

0

2 xy(x+y)f(y)dy.

23.7 The kernel of the integral equation

ψ(x)=λ

∫b

a

K(x, y)ψ(y)dy

has the form

K(x, y)=

∑∞

n=0

hn(x)gn(y),

where thehn(x) form a complete orthonormal set of functions over the interval

[a, b].

(a) Show that the eigenvaluesλiare given by

|M−λ−^1 I|=0,

whereMis the matrix with elements

Mkj=

∫b

a

gk(u)hj(u)du.

If the corresponding solutions areψ(i)(x)=

∑∞

n=0a

(i)

nhn(x), find an expression

fora(ni).