ABSTRACT
The Laplace transform method is normally used to find the response of a linear system at any time t to the initial data at t=0 and the disturbance f(t) acting for t≥ 0. If the disturbance or input function is f(t) = exp(at2), a > 0, the usual Laplace transform cannot be used to find the solution of an initial value problem because the Laplace transform of f(t) does not exist. From a physical point of view, there seems to be no reason at all why the function f(t) cannot be used as an acceptable disturbance for a system. It is often true that the solution at times later than t would not affect the state at time t. This leads to the idea of introducing the finite Laplace transform in 0≤ t≤ T in order to extend the power and usefulness of the usual Laplace transform in 0≤ t <∞. This chapter deals with the definition and basic operational properties of
the finite Laplace transform. In Section 11.4, the method of the finite Laplace
and the boundary problems. This chapter is essentially based on papers by Debnath and Thomas (1976) and Dunn (1967).
