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638 CHAPTER 17 The Heat Equation

SECTION 17.5 PROBLEMS

1. Suppose R = 1, k = 1and f (r) = r. Assume that 2. Repeat the calculations of Problem 1 with k =16, R =3
r
U(1,t) = 0for t > 0. Use a numerical integration to and f (r) = e .
approximate the coefficients a 1 ,··· ,a 5 and use these
3. Repeat the calculations of Problem 1 with k = 1/2,
numbers in the fifth partial sum of the series solu- 2
R = 3and f (r) = 9 −r .
tion to approximate U(r,t). Graph this partial sum for
different values of t.

17.6 Heat Conduction in a Rectangular Plate

We will solve for the temperature distribution u(x, y,t) in a flat, square homogeneous plate
covering the region 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 in the plane. The sides are kept at temperature zero, and
the interior temperature at (x, y) at time 0 is f (x, y).
2
To be specific, we will let f (x, y) = x(1 − x )y(1 − y) and solve the initial-boundary value
problem
2
2
∂u ∂ u ∂ u
= k + for 0 < x < 1,0 < y < 1,t > 0,
∂t ∂x 2 ∂y 2
u(x,0,t) = u(x,1,t) = 0for 0 < x < 1,t > 0,
u(0, y,t) = u(1, y,t) = 0for 0 < y < 1,t > 0,

and
2
u(x, y,0) = f (x, y) = x(1 − x )y(1 − y).
Let u(x, y,t) = X(x)Y(y)T (t), and separate variables to obtain

X + λX = 0,Y + μY = 0, T + (λ + μ)T = 0,
as in the analysis of wave motion of a membrane in Chapter 16. The boundary conditions imply
that
X(0) = X(1) = 0,Y(0) = Y(1) = 0,
so the eigenvalues and eigenfunctions are
2
2
λ n = n π , X n (x) = sin(nπx)
and

2
2
μ m = m π ,Y m (y) = sin(mπy)
for n = 1,2,··· and m = 1,2,···. The equation for T becomes
2
2
2

T + (n + m )π T = 0
2
2
2
with solutions that are constant multiples of e −(n +m )π kt . For each positive integer m and n,we
now have functions
∞ ∞
2
2
2
−(n +m )π kt
u nm (x, y,t) = c nm sin(nπx)sin(mπy)e
n=1 m=1
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October 14, 2010 15:25 THM/NEIL Page-638 27410_17_ch17_p611-640