Page 131 - Essentials of applied mathematics for scientists and engineers

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CHAPTER 8

Solutions with Laplace Transforms

In this chapter, we present detailed solutions of some boundary value problems using the Laplace
transform method. Problems in both mechanical vibrations and diffusion are presented along
with the details of the inversion method.

8.1 MECHANICAL VIBRATIONS
Example 8.1. Consider an elastic bar with one end of the bar fixed and a constant force F per
unit area at the other end acting parallel to the bar. The appropriate partial differential equation
and boundary and initial conditions for the displacement y(x, t) are as follows:

y ττ = y ζζ , 0 <ζ < 1, t > 0
y(ζ, 0) = y t (ζ, 0) = 0
y(0,τ) = 0
y ς (1,τ) = F/E = g

We obtain the Laplace transform of the equation and boundary conditions as

2
s Y = Y ςς
Y (s, 0) = 0
Y ς (s, 1) = g/s

Solving the differential equation for Y (s , ζ),

Y (s ) = (A sinh ςs + B cosh ς s )

Applying the boundary conditions we find that B = 0and
g
= As cosh s
s
g
A =
2
s cosh s
g sinh ς s
Y (s ) =
2
s cosh s