Generalized Laplace-Type Transform Method for Solving Multilayer Diffusion Problems

Abstract

Multilayer diffusion problems have found significant importance that they arise in many medical, environmental, and industrial applications of heat and mass transfer. In this article, we study the solvability of a one-dimensional nonhomogeneous multilayer diffusion problem. A new generalized Laplace-type integral transform is used, namely, the ${\mathbb{M}}_{\rho ,m}$ -transform. First, we reduce the nonhomogeneous multilayer diffusion problem into a sequence of one-layer diffusion problems including time-varying given functions, followed by solving a general nonhomogeneous one-layer diffusion problem via the ${\mathbb{M}}_{\rho ,m}$ -transform. Hence, by means of general interface conditions, a renewal equations’ system is determined. Finally, the ${\mathbb{M}}_{\rho ,m}$ -transform and its analytic inverse are used to obtain an explicit solution to the renewal equations’ system. Our results are of general attractiveness and comprise a number of previous works as special cases.