# Semi-analytical solution of multilayer diffusion problems with time-varying boundary conditions and general interface conditions

## TL;DR

This paper introduces a semi-analytical method for solving complex multilayer diffusion problems with time-varying boundaries and multiple layers, offering improved accuracy and efficiency over existing methods.

## Contribution

A novel semi-analytical approach that handles time-varying boundary conditions and multiple layers, surpassing previous methods in accuracy and applicability.

## Key findings

- The method converges faster with more eigenvalues.
- It outperforms the unified transform method in accuracy and efficiency.
- It can handle problems with many layers and time-varying conditions.

## Abstract

We develop a new semi-analytical method for solving multilayer diffusion problems with time-varying external boundary conditions and general internal boundary conditions at the interfaces between adjacent layers. The convergence rate of the semi-analytical method, relative to the number of eigenvalues, is investigated and the effect of varying the interface conditions on the solution behaviour is explored. Numerical experiments demonstrate that solutions can be computed using the new semi-analytical method that are more accurate and more efficient than the unified transform method of Sheils [Appl. Math. Model., 46:450-464, 2017]. Furthermore, unlike classical analytical solutions and the unified transform method, only the new semi-analytical method is able to correctly treat problems with both time-varying external boundary conditions and a large number of layers. The paper is concluded by replicating solutions to several important industrial, environmental and biological applications previously reported in the literature, demonstrating the wide applicability of the work.