# Analytic approximation of solutions of parabolic partial differential equations with variable coefficients

## TL;DR

This paper develops an explicit solution method for one-dimensional reaction-diffusion equations with variable coefficients, enabling accurate numerical solutions and extending classical heat polynomial techniques through transmutation operators.

## Contribution

It introduces a new family of solutions derived via transmutation operators, facilitating explicit solutions and numerical approximations for variable-coefficient parabolic PDEs.

## Key findings

- Solutions are explicitly constructed using transmutation operators.
- The method achieves high accuracy in numerical solutions.
- Applicable to various boundary conditions beyond Dirichlet.

## Abstract

A complete family of solutions for the one-dimensional reaction-diffusion equation \[ u_{xx}(x,t)-q(x)u(x,t) = u_t(x,t) \] with a coefficient $q$ depending on $x$ is constructed. The solutions represent the images of the heat polynomials under the action of a transmutation operator. Their use allows one to obtain an explicit solution of the noncharacteristic Cauchy problem for the considered equation with sufficiently regular Cauchy data as well as to solve numerically initial boundary value problems. In the paper the Dirichlet boundary conditions are considered however the proposed method can be easily extended onto other standard boundary conditions. The proposed numerical method is shown to reveal good accuracy.