Asymptotics of solutions of a reaction-diffusion system with nonlinear boundary conditions

This paper investigates the asymptotic properties of a system of parabolic equations with nonlinear boundary conditions. Systems of parabolic equations describe the behavior of various processes over time, such as heat propagation and material diffusion. Nonlinear boundary conditions involve complex properties that differ from the usual local conditions at the boundaries of the system, which influence the solutions of the system. In this article, a system of parabolic differential equations associated with nonlinear boundary conditions is reduced to a system of self-similar equations by means of certain transformations using the self-similar approach, and the asymptotic behavior of solutions with compact support is found. In addition, the reaction-diffusion system is approximated using implicit difference schemes and an iterative process for the numerical solution is constructed. It is known that the main problem for the iterative process is the choice of the best initial approximation. The article proposes to use the constructed asymptotic formula to solve this problem. Computational experiments were carried out with different values of numerical parameters and the results are presented graphically. The numerical results obtained in the case of slow diffusion p>2 of a nonlinear reaction-diffusion system show that the reaction-diffusion process proceeds at a finite rate.

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