1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

184 Chapter 2 The Heat Equation

6.Show that, for the function in Exercise 5,

∫r

l

f^2 (x)p(x)dx=

∑∞

n= 1

b^2 n.

7.What are the normalized eigenfunctions of the following problem?

φ′′+λ^2 φ= 0 , 0 <x< 1 ,

φ′( 0 )= 0 ,φ′( 1 )= 0.

2.9 Generalities on the Heat Conduction Problem

On the basis of the information we have about the Sturm–Liouville problem,
we can make some observations on a fairly general heat conduction problem.
We take as a physical model a rod whose lateral surface is insulated. In order
to simplify slightly, we will assume that no heat is generated inside the rod.
Since material properties may vary with position, the partial differential
equation that governs the temperatureu(x,t)in the rod will be

∂

∂x

(

κ(x)∂u

∂x

)

=ρ(x)c(x)∂u

∂t

, l<x<r, 0 <t. (1)

Any of the three types of boundary conditions may be imposed at either
boundary, so we use as boundary conditions

α 1 u(l,t)−α 2 ∂∂xu(l,t)=c 1 , t> 0 , (2)

β 1 u(r,t)+β 2

∂u

∂x(r,t)=c^2 , t>^0. (3)

If the temperature is fixed, the coefficient of∂u/∂xis zero. If the boundary is
insulated, the coefficient ofuis zero, and the right-hand side is also zero. If
there is convection at a boundary, both coefficients will be positive, and the
signs will be as shown.
We already know that in the case of two insulated boundaries, the steady-
state solution has some peculiarities, so we set this aside as a special case. As-
sume, then, that eitherα 1 orβ 1 or both are positive. Finally we need an initial
condition in the form

u(x, 0 )=f(x), l<x<r. (4)

Equations (1)–(4) make up an initial value–boundary value problem.