1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

356 Chapter 5 Higher Dimensions and Other Coordinates

hasasitssolution

φ(x)=

∫x

0

1

1 −y^2

∫ 1

y

f(z)dz dy,

provided that the functionfsatisfies

∫ 1

− 1

f(z)dz= 0.

11.Suppose that the functionsf(x)andφ(x)in the preceding exercise have

expansions in terms of Legendre polynomials

f(x)=

∑∞

k= 0

bkPk(x), − 1 <x< 1 ,

φ(x)=

∑∞

k= 0

BkPk(x), − 1 <x< 1.

What is the relation betweenBkandbk?

12.By applying separation of variables to the problem

∇^2 u= 0 , 0 <ρ<a, 0 ≤φ<π,

withubounded atφ=0,πanduperiodic( 2 π)inθ,derivethefollow-

ing equation for the factor function(φ):

sin(φ)

(

sin(φ)′

)′

−m^2 +μ^2 sin^2 (φ)= 0 ,

wherem= 0 , 1 , 2 ,...comes from the factor(θ).

13.Using the change of variablesx=cos(φ),(φ)=y(x)on the equation

of Exercise 12, derive a differential equation fory(x).

14.Solve the heat conduction problem

1

r

∂

∂r

(

r∂∂ur

)

=^1 k∂∂ut, 0 <r<a, 0 <t,

∂u

∂r(a,t)=^0 ,^0 <t,

u(r, 0 )=T 0 −

(r

a

) 2

, 0 <r<a.