Setting up finite transforms using Sturm--Liouville Theory

Abstract Initial and boundary value problems are very common in physics, mathematics, and engineering. They can model many types of problems related to heat diffusion and membrane vibration, for example. When the analytical solution of these problems is needed one may find extra difficulties when the boundary conditions that describe the phenomena are nonhomogeneous. In this way, in this work is presented a technique for solving initial and boundary problems by means of integral transformations. The differential of the presentation is in the construction of the integral transform appropriate to the solution of the problem. These transformations are known as finite transforms and in this case they are related to a Sturm--Liouville problem associated with the differential operator connected to the differential equation. As an example, we solve two problems of heat diffusion in different spatial coordinates systems. The presentation of the work follows in pedagogically and self-contained fashion. Therefore, we expect the reader to understand the technique and can use it in solving other problems involving partial differential equations.