General solution to dispersive wave equation and its application to propagation

The treatment of propagation in a linear dispersive medium is a problem that is outlined in many electromagnetic texts which consider the continuous time case. These texts don't deal with initial value problems, but instead refer to Stratton for a complete treatment of the problem. Stratton uses both the Fourier transform and the Laplace transform to solve the initial value problems. The usage of both methods can cause both mathematical and conceptual problems.