Journal Article

On the Analytical and Numerical Study of a Two-Dimensional Nonlinear Heat Equation with a Source Term

Spevak, Lev, Lempert, Anna, Kazakov, Alexander, Nefedova, Olga

2020

Overview

The paper deals with two-dimensional boundary-value problems for the degenerate nonlinear parabolic equation with a source term, which describes the process of heat conduction in the case of the power-law temperature dependence of the heat conductivity coefficient. We consider a heat wave propagation problem with a specified zero front in the case of two spatial variables. The solution existence and uniqueness theorem is proved in the class of analytic functions. The solution is constructed as a power series with coefficients to be calculated by a proposed constructive recurrent procedure. An algorithm based on the boundary element method using the dual reciprocity method is developed to solve the problem numerically. The efficiency of the application of the dual reciprocity method for various systems of radial basis functions is analyzed. An approach to constructing invariant solutions of the problem in the case of central symmetry is proposed. The constructed solutions are used to verify the developed numerical algorithm. The test calculations have shown the high efficiency of the algorithm.

• Publisher: MDPI AG
• Subject: Algorithms / Analytic functions / Boundary element method / Boundary value problems / Conduction heating / Conductive heat transfer / Construction / Existence theorems / Neighborhoods / Numerical analysis / Ordinary differential equations / Physics / Power series / Radial basis function / Reciprocity / Researchers / Temperature dependence / Thermal conductivity / Thermodynamics / Uniqueness theorems / Wave propagation