1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

168 Chapter 2 The Heat Equation

u( 0 ,t)=T 0 , ∂∂ux(a,t)= 0 , 0 <t,

u(x, 0 )=T 0 , 0 <x<a.

6.Solve this problem for the temperature in a rod in contact along the lateral

surface with a medium at temperature 0.

∂^2 u

∂x^2 =

1

k

∂u

∂t+γ

(^2) u, 0 <x<a, 0 <t,
u( 0 ,t)= 0 ,
∂u
∂x(a,t)=^0 ,^0 <t,
u(x, 0 )=T 0 , 0 <x<a.
7.Solve the problem
∂^2 u
∂x^2 =

1

k

∂u

∂t,^0 <x<a,^0 <t,

∂u

∂x(^0 ,t)=^0 , u(a,t)=T^0 ,^0 <t,

u(x, 0 )=T 1 , 0 <x<a.

8.Compare the solution of Exercise 7 with Eq. (20). Can one be turned into

the other?

9.Solve the problem in Exercise 7 takingT 0 =0and

u(x, 0 )=T 1 cos

(πx

2 a

)

, 0 <x<a.

10. a.Show that the eigenfunctions found in this section are orthogonal. That
    is, prove that
    ∫a

0

sin(λnx)sin(λmx)dx=

{ 0 (m=n),

a

2 (m=n)

whenλn=(^2 n− 2 a^1 )π.

b. Use the orthogonality relation in partato justify the formula in

Eq. (18).

11.To justify the expansion of Eq. (17), for an arbitrary sectionally smooth

g(x),

∑∞

n= 1

bnsin

(

( 2 n− 1 )πx

2 a

)

=g(x), 0 <x<a,