Page 103 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

SOLUTIONS USING FOURIER SERIES AND INTEGRALS 93
we find
∞
y 1 1
u(x, y) = f (ς) − d ς (5.115)
2
2
π (ς − x) + y 2 (ς + x) + y 2
0
Problem
Consider the transient heat conduction problem

u t = u xx + u yy x ≥ 0, 0 ≤ y ≤ 1, t ≥ 0
with boundary and initial conditions

u(t, 0, y) = 0
u(t, x, 0) = 0

u(t, x, 1) = 0

u(0, x, y) = u 0
and u(t, x, y) is bounded.
Separate the problem into two problems u(t, x, y) = v(t, x)w(t, y) and give appropriate
boundary conditions. Show that the solution is given by

4 x ∞ sin(2n − 1)πy 2 2

u(t, x, y) = erf √ exp[−(2n − 1) π t]
π 2 t 2n − 1
n=1
FURTHER READING
V. Arpaci, Conduction Heat Transfer. Reading, MA: Addison-Wesley, 1966.
J. W. Brown and R. V. Churchill, Fourier Series and Boundary Value Problems. 6th edition. New
York: McGraw-Hill, 2001.