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

Vladislav V. Kravchenko, Josafath A. Otero, Sergii M. Torba

Published 2017-06-19Version 1

A complete family of solutions for the one-dimensional reaction-diffusion equation \[ \frac{\partial^2 u}{\partial x^2}-q(x)u(x,t)=\frac{\partial u}{\partial t}(x,t) \] with a coefficient $q$ depending on $x$ is constructed. The solutions represent the images of the heat polynomials under the action of a transmutation (transformation) operator. Their use allows one to obtain an explicit solution of the so-called noncharacteristic Cauchy problem for the considered equation with sufficiently regular Cauchy data as well as to solve numerically initial boundary value problems. The proposed numerical method is shown to reveal good accuracy.