1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

210 Chapter 2 The Heat Equation

24.Solve the eigenvalue problem by settingφ(ρ)=ψ(ρ)/ρ:

1

ρ^2

(

ρ^2 φ′

)′

+λ^2 φ= 0 , 0 <ρ<a,

φ( 0 ) bounded,φ(a)= 0.

Is this a regular Sturm–Liouville problem? Are the eigenfunctions or-

thogonal?

25.Solve this problem for heat conduction in a sphere. (Hint: Letu(ρ,t)=

v(ρ,t)/ρ.)

1

ρ^2

∂

∂ρ

(

ρ^2 ∂∂ρu

)

=^1 k∂∂ut, 0 <ρ<a, 0 <t,

u( 0 ,t) bounded, u(a,t)= 0 , 0 <t,

u(ρ, 0 )=T 0 , 0 <ρ<a.

26.State and solve the eigenvalue problem associated with

e−x∂

∂x

(

ex∂u

∂x

)

=^1

k

∂u

∂t

, 0 <x<a, 0 <t,

u( 0 ,t)= 0 , ∂u

∂x

(a,t)= 0.

27.Find the steady-state solution of the problem

∂^2 u

∂x^2 +γ

2 (T(x)−u)=^1

k

∂u

∂t,^0 <x<a,^0 <t,

u( 0 ,t)=T 0 ,

∂u

∂x(a,t)=^0 ,^0 <t,

whereT(x)=T 0 +Sx.

28.Determine whether or notλ=0isaneigenvalueoftheproblem

φ′′+λ^2 xφ= 0 , 0 <x<a,

φ′( 0 )= 0 ,φ(a)= 0.

29.Same question as Exercise 28, but with boundary conditions

φ′( 0 )= 0 ,φ′(a)= 0.