Integral Solution of Diffusion Equation: Part 2—Boundary Conditions of Second and Third Kinds

Abstract

An analytical solution of the diffusion equation using the Galerkin method to calculate the eigenvalues is currently available for boundary conditions of the first kind. This paper includes algebraic techniques to accommodate boundary conditions of the second and third kinds. Several case studies are presented to illustrate the utility and accuracy of the procedure. Selected examples either have no exact solutions or their exact solutions have not been cited because of the mathematical or numerical complexity. The illustrations include transient conduction in hemielliptical solids with either external convective surfaces or convective bases, and buried pipe in a square enclosure. Whenever possible, symbolic programming is used to carry out the differentiations and integrations. In some cases, however, the integrations must be strictly numerical. It is also demonstrated that a Green’s function can be defined to accommodate many geometries with nonorthogonal boundaries subject to more complex boundary conditions for which an exact Green’s function does not exist.