Asymptotic Properties and Numerical Results of Solutions for a System of Degenerate Parabolic Equations with Nonlinear Boundary Conditions

Abstract

In this article, a system of parabolic equations with nonlinear boundary conditions
is transformed into a system of self-similar (automodel) equations by applying a shapeshifting. An
asymptotics for a compact supporting solution of the problem is proposed. The obtained asymptotic
formulas are used as an initial approximation in the numerical solution of the problem. In numerical
experiments k = 3 is taken, a finite-difference scheme is proposed, and a Python program is developed
to perform computations for various parameter values, corresponding graphs obtained.
Numerical experiments have shown that the iterative processes converge within 3 to 5 steps. The
compactly supported self-similar asymptotic solution used as the initial approximation played an
important role in ensuring the convergence of the iterative process. The graphs illustrate that, for
different numerical parameter values, the reaction–diffusion processes occur at a finite speed.