1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

326 Chapter 5 Higher Dimensions and Other Coordinates

several series to form the combination:

u(r,θ,t)=

∑

n

a 0 nJ 0 (λ 0 nr)cos(λ 0 nct)

+

∑

m,n

amnJm(λmnr)cos(mθ)cos(λmnct)

+

∑

m,n

bmnJm(λmnr)sin(mθ)cos(λmnct)

+

∑

n

A 0 nJ 0 (λ 0 nr)sin(λ 0 nct)

+

∑

m,n

AmnJm(λmnr)cos(mθ)sin(λmnct)

+

∑

m,n

BmnJm(λmnr)sin(mθ)sin(λmnct). (25)

Whent=0, the last three sums disappear, and the cosines oftin the first three
sumsareallequalto1.Thus

u(r,θ, 0 )=

∑

n

a 0 nJ 0 (λ 0 nr)+

∑

m,n

amnJm(λmnr)cos(mθ)

+

∑

m,n

bmnJm(λmnr)sin(mθ)

=f(r,θ), 0 <r<a, −π<θ≤π. (26)

We expect to fulfill this equality by choosing thea’s andb’s according to
some orthogonality principle. Since each function present in the series is an
eigenfunction of the problem

∇^2 φ=−λ^2 φ, 0 <r<a,

φ(a,θ)= 0 ,

φ(r,θ+ 2 π)=φ(r,θ), 0 <r<a,

we expect it to be orthogonal to each of the others (see Section 5.4, Exercise 7).
This is indeed true: Any function from one series is orthogonal to all of the
functions in the other series and also to the rest of the functions in its own
series. To illustrate this orthogonality, we have
∫∫

R

J 0 (λ 0 nr)Jm(λmnr)cos(mθ)dA

=

∫a

0

J 0 (λ 0 nr)Jm(λmnr)

∫π

−π

cos(mθ)dθrdr= 0 , m= 0. (27)