A direct integral equation method for the solution of dual-or triple-series equations with applications to heat conduction and diffusion–reaction systems

Abstract

A direct integral equation method is presented for the solution of dual- or triple-series equations obtained from separation-of-variables solutions to mixed boundary- value problems. The approach is based upon transformation of the dual- or triple- series to a single or set of Fredholm integral equations of the first kind whose kernel and forcing function aren an infinite series that can be systematically obtained from generalized formulas. Solution values for the integral equation are ontained by application of an appropriate quadrature method that accounts for the presense of logarithmic singularities in the kernel. The integral equation method is applied to several application-type problems such as heat conduction and simultaneous diffusion with chemical reaction. Comparisons are made to exact where available and also to other approximate solutions based upon the method of wieghted residuals. The results of various numerical experiments suggest that the integral equation method can yield results of the same or superior accuracy with less computational effort than those based upon MWR.