Experiments with the Laplace Transform

Part 1. Introduction

Let f be a piecewise smooth function defined for t between 0 and infinity and let s be positive. Then the Laplace transform F of f is defined by

The Laplace transform is a close relative of the Fourier transform. However, the fact that the Laplace transform is defined on the semi-infinite interval from 0 to infinity rather than on the whole real line, makes it somewhat more useful for dealing with initial value problems for ordinary differential equations. In addition, the Laplace transform is useful in determining solutions of partial differential equations -- particularly where the time or spatial domains are semi-infinite, e.g., heat flow in an infinite rod where the temperature at one end is known.

1. Your computer algebra system knows how to calculate Laplace transforms and inverse Laplace transform of most common functions. Use your system to calculate the Laplace transform of each of the following functions. Then calculate the inverse Laplace transform of each result. Note that each of these Laplace transforms is a rational function in the transformed variable s, i.e., each transform is a quotient of polynomials. Assume that a is positive throughout.
   • 1

   • t

   • exp(-at)

   • cos(at)

   • sin(at)

   • H(t-a), where H is the Heaviside function.

2. Verify the differentiation rule stated above for the functions below. If you use the results of Step 1, you should not need additional computer algebra calculations for this step.
   • exp(-at)

   • cos(at)

Just as with the Fourier transform, the convolution of functions plays an important role in calculating Laplace transforms and inverse Laplace transforms. For the Laplace transform, the appropriate convolution f * g is given by

With this definition of convolution, if F is the Laplace transform of f and G is the Laplace transform of g, the Laplace transform of f * g is just the product of F and G.

3. Verify the rule for the Laplace transform of the convolution for the following pairs of functions:
   • f(t) = t and g(t) = exp(-2t)

   • f(t) = t^2 and g(t) = cos(3t)