Open AccessOn an exact solution to the nonlinear heat equation with a source
Author(s) -
А. Л. Казаков,
A. A. Lempert,
Л. Ф. Спевак
Publication year - 2021
Publication title -
journal of physics. conference series
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.21
H-Index - 85
eISSN - 1742-6596
pISSN - 1742-6588
DOI - 10.1088/1742-6596/1847/1/012006
Subject(s) - mathematics , mathematical analysis , cauchy problem , singularity , heat equation , initial value problem , boundary value problem , nonlinear system , partial differential equation , power series , cauchy distribution , cauchy boundary condition , series (stratigraphy) , free boundary problem , physics , quantum mechanics , paleontology , biology
The paper is devoted to the study of a singular nonlinear second-order parabolic equation, which is called the porous medium equation or the nonlinear heat equation. One of the important classes of its solutions is heat waves propagating over a zero background with a finite velocity. This property is not typical for parabolic equations and is a consequence of singularity. The main object of study is exact solutions of mentioned type. A new way of separating variables is used to represent them. We obtain conditions when it is possible to make a reduction to the Cauchy problem for an ordinary second-order differential equation with a singularity. It is shown that the Cauchy problem describes a heat wave whose front moves exponentially. We construct a solution to the Cauchy problem as a power series and determine the cases when the series breaks off, i.e. the solution has the form of a polynomial, and the corresponding heat wave can be written explicitly. If the Cauchy problem cannot be explicitly integrated, the solution is constructed numerically. An algorithm based on the boundary element method is proposed. We perform a computational experiment and conclude the properties of the found solutions. Besides, the accuracy of the calculation results is analyzed.