MATH 348 - Advanced Engineering Mathematics March 24, 2008
Homework 7, Spring 2008 Due: April 2, 2008
partial differential equations - the one-dimensional heat equation - generalizations
Consider the one-dimensional heat equation,
∂u
∂t =c2∂2u
∂x2,(1)
x∈(0, L), t ∈(0,∞), c2=K
σρ .(2)
Equations (1)-(2) model the time-evolution of the temperature, u=u(x, t), of a heat conducting medium in one-
dimension. The ob ject, of length L, is assumed to have a homogenous thermal conductivity K, specific heat σ, and
linear density ρ. That is, K , σ,ρ ∈R+.
1. Consider the one-dimensional heat equation, (1)-(2), with the boundary conditions1,
ux(0, t)=0, u(L, t)=0,(3)
and initial condition
u(x, 0) = f(x).(4)
(a) Assume that the solution to (1)-(2) is such that u(x, t) = F(x)G(t) and use separation of variables to find
the general solution to (1)-(2), which satisfies (3)-(4). 2
(b) What is the temperature in the medium for when tis large? That is, determine lim
t→∞
u(x, t).
2. Consider the one-dimensional heat equation, (1)-(2), with the boundary conditions3,
ux(0, t) = 0, ux(L, t)=0,(5)
u(x, 0) = f(x).(6)
(a) Assume that the solution to (1)-(2) is such that u(x, t) = F(x)G(t) and use separation of variables to find
the general solution to (1)-(2), which satisfies (5)-(6). 4
(b) Describe how the long term behavior of the general solution to (1)-(6) changes as the thermal conductivity,
K, is increased while all other parameters are held constant. Also, describe how the solution changes when
the linear density, ρ, is increased while all other parameters are held constant.
Define,
f(x) = x, 0< x ≤L
−x+ 2L, L < x < 2L(7)
3. For the following questions we consider the solution, u, to the heat equation given by, (1)-(2), which satisfies the
initial condition given by (7). 5
(a) Suppose this solution also satisfies the boundary conditions are given by u(0,t) = u(2L, t) = 0. Determine
the unique solution the PDE satisfying all of these conditions. What is lim
t→∞
u(x, t)?
(b) For L= 2L, find the particular solution to (1)-(2) with boundary conditions (5)-(6) for when the initial
temperature profile of the medium is given by (7). Show that lim
t→∞
u(x, t) = favg =.25.6
1Here the boundary conditions correspond to making the left endpoint perfectly insulated while the right endpoint is fixed at zero degrees
2This is done for different boundary conditions in Kreyszig pp.552-556.
3Here the boundary conditions correspond to perfect insulation of both endpoints
4An insulated bar is discussed in examples 4 and 5 on page 557.
5When solving for the following problems it would be a good idea to go back through your notes and the homework looking for similar
calculations.
6It is interesting here to note that though the initial condition fdoesn’t appear to satisfy the boundary conditions its periodic Fourier
extension does. That is, if you draw the even periodic extension of the initial condition then you would see that the slope is not well defined
at the end points. Remembering that the Fourier series averages the right and left hand limits of the periodic extension of the function f
at the endpoints shows that the boundary conditions are, in fact, satisfied, since the derivative of an average is the average of derivatives.
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