Page 135 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

SOLUTIONS WITH LAPLACE TRANSFORMS 125
Problems
1. Solve the above vibration problem when

y(0,τ) = 0
y(1,τ) = g(τ)

Hint: To make use of convolution see Example 8.3.
2. Solve the problem

2
2
∂ y ∂ y
=
∂t 2 ∂x 2
y x (0, t) = y(x, 0) = y t (x, 0) = 0
y(1, t) = h

using the Laplace transform method.

8.2 DIFFUSION OR CONDUCTION PROBLEMS
We now consider the conduction problem

Example 8.3.

u τ = u ςς
u(1,τ) = f (τ)
u(0,τ) = 0
u(ς, 0) = 0

Taking the Laplace transform of the equation and boundary conditions and noting that
u(ς, 0) = 0,

sU(s ) = U ςς
solution yields
√ √
U = A sinh s ς + B cosh s ς
U(0, s ) = 0

U(1, s ) = F(s )

The first condition implies that B = 0 and the second gives
√
F(s ) = A sinh s
√
s ς
and so U = F(s ) sinh √ .
sinh s