Solution of Mixed Boundary Value Problems by Integral Equations and Methods of Weighted Residuals with Application to Heat Conduction and Diffusion-Reaction Systems

Abstract

Integral equation methods are considered as an alternate approach for the solution of mixed boundary value problems in heat conduction or diffusion with chemical reaction that can be described by dual or triple series equations. The approach is based upon reduction of the dual or triple series to a single or a set of Fredholm integral equations of the first kind whose kernel $K(x,y)$ and forcing function $f(x)$ are presented in terms of an appropriate infinite series which can be derived from the separation-of-variables solution. The solution of the integral equation for the unknown function $g(y)$ is obtained by a modified quadrature method which accounts for the presence of a logarithmic singularity in the kernel for $x = y$ which is a common occurrence in problems of this type. The approach is illustrated by solving selected example problems involving either diffusion and reaction or heat conduction. Comparisons are made to exact solutions where available and also to other approximate solutions based u...