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 <∞.
