Mathematical Methods for Physics and Engineering : A Comprehensive Guide

EIGENFUNCTION METHODS FOR DIFFERENTIAL EQUATIONS

and we assume that we may interchange the order of summation and integration,

then (17.49) can be written as

y(x)=

∫b

a

{∞

∑

n=0

[

1

λn

yˆn(x)yˆ∗n(z)

]}

f(z)dz.

The quantity in braces, which is a function ofxandzonly, is usually written

G(x, z), and is theGreen’s functionfor the problem. With this notation,

y(x)=

∫b

a

G(x, z)f(z)dz, (17.50)

where

G(x, z)=

∑∞

n=0

1

λn

yˆn(x)yˆ∗n(z). (17.51)

We note thatG(x, z) is determined entirely by the boundary conditions and the

eigenfunctionsyˆn, and hence byLitself, and thatf(z) depends purely on the

RHS of the inhomogeneous equation (17.44). Thus, for a givenLand boundary

conditions we can establish, once and for all, a functionG(x, z) that will enable

us to solve the inhomogeneous equation foranyRHS. From (17.51) we also note

that

G(x, z)=G∗(z, x). (17.52)

We have already met the Green’s function in the solution of second-order dif-

ferential equations in chapter 15, as the function that satisfies the equation

L[G(x, z)] =δ(x−z) (and the boundary conditions). The formulation given

above is an alternative, though equivalent, one.

Find an appropriate Green’s function for the equation

y′′+^14 y=f(x),

with boundary conditionsy(0) =y(π)=0.Hence,solvefor(i)f(x)=sin2xand(ii)

f(x)=x/ 2.

One approach to solving this problem is to use the methods of chapter 15 and find
a complementary function and particular integral. However, in order to illustrate the
techniques developed in the present chapter we will use the superposition of eigenfunctions,
which, as may easily be checked, produces the same solution.
The operator on the LHS of this equation is already Hermitian under the given boundary
conditions, and so we seek its eigenfunctions. These satisfy the equation

y′′+^14 y=λy.

This equation has the familiar solution

y(x)=Asin

(√

1

4 −λ

)

x+Bcos

(√

1

4 −λ

)

x.