TABLE OF CONTENTS The Laplace transform provides a useful technique for solving linear differential equations. The basic idea is to first rewrite the equation in the transformed space, where the original unknown is replaced by its Laplace transform. Next, this equation is solved leading to the Laplace transform, and finally the solution is obtained through the inverse transform operation. This latter step is usually the most difficult and sometimes requires numerical techniques. The main advantage of the method of Laplace transform, as well as of any other integral transform such as the finite Fourier transform (see Section 3.19 and Chapter 7), is that the dimension of the differential equation decreases by one in the transformed space. Thus, ordinary differential equations (ODEs) reduce to algebraic equations, two-dimensional partial differential equations (PDEs) reduce to ODEs, and, in general, m-dimensional PDEs reduce to ( m ? 1)-dimensional PDEs.
In this chapter we first define the Laplace transform and introduce some of its most useful properties. We then proceed to discuss the application of these transforms to solve ODEs and PDEs.
In general, the integral transform 
of a function f( t) is defined by
where K( s, t) is called the kernel of the transformation and we have assumed that the integral exists. The relation (8.1.1) establishes a correspondence between the function f( t) and its transform F( s); that is, it transforms a function of the independent...
