Analytic Approximation of Solutions of Parabolic Partial Differential Equations with Variable Coefficients

Vladislav V Kravchenko,Sergii M Torba,Josafath A Otero

doi:10.1155/2017/2947275

Abstract

A complete family of solutions for the one-dimensional reaction-diffusion equation, uxx(x,t)-q(x)u(x,t)=ut(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 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.

Highlights

In the present work a complete system of solutions of a one-dimensional reaction-diffusion equation with a variable coefficient uxx (x, t) − q (x) u (x, t) = ut (x, t) (1)considered on Ω fl (−b, b) × (0, τ) is obtained
In the present work using a mapping property of the transmutation operators discovered in [7] we show that the construction of the complete systems of solutions for equations of form (1), representing transmuted heat polynomials, can be realized with no previous construction of the transmutation operator
The use of the mapping property leads to an explicit solution of the noncharacteristic Cauchy problem for (1) with Cauchy data belonging to a Holmgren class [1]

Summary

Introduction

In the present work using a mapping property of the transmutation operators discovered in [7] we show that the construction of the complete systems of solutions for equations of form (1), representing transmuted heat polynomials, can be realized with no previous construction of the transmutation operator. We illustrate the implementation of the complete system of the transmuted heat polynomials by a numerical solution of an initial boundary value problem for (1). The approximate solution is sought in the form of a linear combination of the transmuted heat polynomials and the initial and boundary conditions are satisfied by a collocation method. Advances in Mathematical Physics initial boundary value problems for (1) implementing the transmuted heat polynomials is discussed.

Transmutation Operators and Formal Powers

Transmuted Heat Polynomials

Numerical Illustration

Conclusions