2. Analytical Methods in Heat Transfer

In this chapter, classical and modern analytical methodologies for partial differential equations are reviewed, with emphasis on the linear and nonlinear heat diffusion equation. The aim is to provide both the introductory and advanced perspectives, by gradually evolving from separation of variables, to the classical integral transform method, and finally to the hybrid numerical-analytical approach known as the generalized integral transform technique (GITT), illustrating each approach through examples in the corresponding section. The last part of the chapter is dedicated to a few recent developments that have opened up new perspectives to the GITT extension and computational enhancement, including an integral balance scheme for convergence acceleration in the solution of eigenvalue problems, the proposition of convective eigenfunction basis for convection-diffusion problems, and the adoption of nonlinear eigenvalue problems in the solution of nonlinear problems.