A new analytical method to solve the heat equation for a multi-dimensional composite slab

Abstract

A novel analytical approach has been developed for heat conduction in a multi-dimensional composite slab subject to time-dependent boundary changes of the first kind. Boundary temperatures are represented as Fourier series. Taking advantage of the periodic properties of boundary changes, the analytical solution is obtained and expressed explicitly. Nearly all the published works necessitate searching for associated eigenvalues in solving such a problem even for a one-dimensional composite slab. In this paper, the proposed method involves no iterative computation such as numerically searching for eigenvalues and no residue evaluation. The adopted method is simple which represents an extension of the novel analytical approach derived for the one-dimensional composite slab. Moreover, the method of 'separation of variables' employed in this paper is new. The mathematical formula for solutions is concise and straightforward. The physical parameters are clearly shown in the formula. Further comparison with numerical calculations is presented.