ABSTRACT
The integral transform method is an effective tool of reducing the complexity of PDEs to a clear tractable form that allows many problems of mathematical physics to be tackled straightforwardly. In this chapter we will be primarily interested in solving problems of PDEs that requires an employment of either the Fourier transform or the Laplace transform technique. The essence of the method of either transform is to reduce a given PDE into an ODE in a somewhat similar spirit as that of the separation of variables. However, here the main difference lies in the fact that while in the case of the Fourier transform, the domain of the differential operator is defined over the full-line (− ∞, + ∞) and hence the method is applicable when the given function is defined over an entire real line equipped with appropriate boundary conditions, for the Laplace transform, since the domain of the differential operator is only the half-line (0, ∞), the method becomes relevant when we are dealing with an initial value problem. Such a half-line-full-line contrast of the two transforms is due to the fact that it often happens that Fourier transforms of certain classes of functions may not exist, such as for example x 2, and a control factor is necessary to be appended to the function to ensure better convergence behaviour on at least on one side of the interval. We have discussed and collected some useful analytical results of the Fourier and Laplace transforms in Appendix B and Appendix C respectively. In the following we begin first with the Fourier transform method.
