Analytical approximations in short times of exact operational solutions to reaction diffusion problems on bounded intervals

TL;DR

This paper develops analytical short-time approximations for solutions to reaction-diffusion boundary value problems on bounded intervals using integral transforms, enhancing computational efficiency for related simulations.

Contribution

It introduces a unified operational approach using Fourier and Laplace transforms to derive short-time analytical approximations for reaction-diffusion problems with various boundary conditions.

Findings

Derived new formulas for short-time solution approximations.

Validated the approximations against infinite series solutions.

Potentially improves numerical simulations of moving boundary problems.

Abstract

This paper aims at obtaining, by means of integral transforms, analytical approximations in short times of solutions to boundary value problems for the one-dimensional reaction-diffusion equation with constant coefficients. The general form of the equation is considered on a bounded generic interval and the three classical types of boundary conditions, i.e., Dirichlet as well as Neumann and mixed boundary conditions are considered in a unified way. The Fourier and Laplace integral transforms are successively applied and an exact solution is obtained in the Laplace domain. This operational solution is proven to be the accurate Laplace transform of the infinite series obtained by the Fourier decomposition method and presented in the literature as solutions to this type of problem. On the basis of this unified operational solution, four cases are distinguished where innovative formulas expressing consistent analytical approximations in short time limits are derived with respect to the behavior of the solution at the boundaries. Compared to the infinite series solutions, the analytical approximations may open new perspectives and applications, among which can be noted the improvement of numerical efficiency in simulations of one-dimensional moving boundary problems, such as in Stefan models.