Generalized integral transform method for solving multilayer diffusion problems

TL;DR

This paper introduces a generalized integral transform method to solve complex multilayer diffusion problems, providing explicit solutions and extending previous approaches in heat and mass transfer applications.

Contribution

The paper develops a new generalized integral transform approach for solving nonhomogeneous multilayer diffusion problems, extending existing methods and enabling explicit solutions.

Findings

Explicit solutions for multilayer diffusion problems are derived.

The method generalizes previous approaches and handles nonhomogeneous conditions.

Renewal equations are formulated and solved using the new transform.

Abstract

Multilayer diffusion problems have found significant important that they arise in many medical, environmental and industrial applications of heat and mass transfer. In this article, we study the solvability of one-dimensional nonhomogeneous multilayer diffusion problem. We use a new generalized integral transform, namely, $ \bb M_{\rho,m} -$transform [Srivastava et al., https://doi.org/10.1016/S0252-9602(15)30061-8]. 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 $ \bb M_{\rho,m} -$transform. Hence, by means of general interface conditions, a renewal equations' system is determined. Finally, the $ \bb M_{\rho,m} -$transform and its analytic inverse are used to obtain an explicit solution to the renewal equations' system. Our results generalize those ones in [Rodrigo and Worthy, http://dx.doi.org/10.1016/ j.jmaa.2016.06.042].