1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

148 Chapter 2 The Heat Equation

Can you think of a physical interpretation of this problem? Note the differ-

ence between the partial differential equation in this exercise and in Exer-

cise 1. What happens ifγ=π/a?

4.State the initial value–boundary value problem satisfied by the transient

temperature distribution corresponding to Eqs. (8)–(11).

5.Find the steady-state solution of the problem

∂

∂x

(

κ∂u

∂x

)

=cρ∂u

∂t

, 0 <x<a, 0 <t,

u( 0 ,t)=T 0 , u(a,t)=T 1 , 0 <t

if the conductivity varies in a linear fashion withx:κ(x)=κ 0 +βx,where

κ 0 andβare constants.

6.Find and sketch the steady-state solution of

∂^2 u

∂x^2 =

1

k

∂u

∂t,^0 <x<a,^0 <t

together with boundary conditions

a. ∂∂ux( 0 ,t)=0, u(a,t)=T 0 ;

b. u( 0 ,t)−∂∂ux( 0 ,t)=T 0 , ∂∂xu(a,t)=0;

c. u( 0 ,t)−

∂u

∂x(^0 ,t)=T^0 , u(a,t)+

∂u

∂x(a,t)=T^1.

7.Find the steady-state solution of this problem, whereris a constant that

represents heat generation.

∂^2 u

∂x^2 +r=

1

k

∂u

∂t,^0 <x<a,^0 <t,

u( 0 ,t)=T 0 , ∂u

∂x

(a,t)= 0 , 0 <t.

8.Find the steady-state solution of

∂^2 u

∂x^2 +γ

2 (U(x)−u)=^1

k

∂u

∂t,^0 <x<a,^0 <t,

u( 0 ,t)=U 0 , ∂u

∂x

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

whereU(x)=U 0 +Sx(U 0 ,Sare constants).