1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

182 Chapter 2 The Heat Equation

mis a fixed integer) and integrating fromltoryields

∫r

l

f(x)φm(x)p(x)dx=

∑∞

n= 1

cn

∫r

l

φn(x)φm(x)p(x)dx.

The orthogonality relation says that all the terms in the series, except that one
in whichn=m, must disappear. Thus
∫r

l

f(x)φm(x)p(x)dx=cm

∫r

l

φm^2 (x)p(x)dx

gives a formula for choosingcm.
We can now cite a convergence theorem for expansion in terms of eigen-
functions. Notice the similarity to the Fourier series convergence theorem. Of
course, the Fourier sine or cosine series are series of eigenfunctions on a regu-
lar Sturm–Liouville problem in which the weight functionp(x)is 1.

Theorem. Letφ 1 ,φ 2 ,...be eigenfunctions of a regular Sturm–Liouville problem
Eqs.(1)–(3),inwhichtheα’s andβ’s are not negative.
If f(x)is sectionally smooth on the interval l<x<r, then

∑∞

n= 1

cnφn(x)=

f(x+)+f(x−)

2 , l<x<r, (6)

where

cn=

∫r

l∫fr(x)φn(x)p(x)dx

lφ^2 n(x)p(x)dx

.

Furthermore, if the series

∑∞

n= 1

|cn|

[∫r

l

φn^2 (x)p(x)dx

] 1 / 2

converges, then the series Eq.(6)converges uniformly, l≤x≤r.

EXERCISES

1.Ve r i f y t h a t

λ^2 n=

( nπ

ln(b)

) 2

,φn=sin

(

λnln(x)

)

are the eigenvalues and eigenfunctions of