### NUMERICAL STUDY OF A MATHEMATICAL MODEL OF AN AUTOCATALYTIC REACTION WITH DIFFUSION IN A TUBULAR REACTOR

#### Abstract

The article analyzes analytically and numerically the model of the autocatalytic reaction with diffusion in the degenerate case on a finite connected directed graph G with the Showalter -- Sidorov condition. The mathematical model of the autocatalytic reaction with diffusion is based on the system of distributed Brusselator equations. The system of degenerate equations of a distributed Brusselator whose functions satisfy the conditions of continuity and flow balance belongs to a wide class of semilinear Sobolev type equations. To investigate the existence of a solution of this system of equations, the phase space method which was developed by G.A.Sviridyuk and his students to study the solvability of Sobolev type equations will be used. We will show the simplicity of the phase space and the existence of a unique local solution of the given Showalter -- Sidorov problem. The theoretical results of this article served as the basis for developing an algorithm for numerical study of the model in a Maple environment. The algorithm of numerical investigation is based on the Galerkin method, which allows us to take into account the phenomenon of degeneracy of the equation. The article gives several examples illustrating the results of the computational experiment obtained on the three-ribbed and five-ribbed graphs.

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