NUMERICAL STUDY OF THE UNIQUE SOLVABILITY OF THE SHOWALTER -- SIDOROV PROBLEM FOR A MATHEMATICAL MODEL OF THE PROPAGATION OF NERVE IMPULSES IN THE MEMBRANE

Abstract

The article is devoted to the study of the existence of one or more solutions of a mathematical model of the propagation of a nerve impulse in a membrane based on a degenerate system of Fitz Hugh -- Nagumo equations, given on a certain domain with a smooth boundary or on a connected directed graph with the Showalter -- Sidorov initial condition. A nondegenerate mathematical model of the propagation of a nerve impulse in the membrane is widespread and is studied using the theory of singular perturbations. A feature of the process of the described investigated mathematical model is that the rate of change of one of the components of the system can significantly exceed the other, which means that the rate of the derivatives, which is much lower, can be considered equal to zero. Hence, it becomes necessary to study precisely the degenerate system of Fitz Hugh -- Nagumo equations. The degenerate system of Fitz Hugh -- Nagumo equations belongs to a wide class of semilinear Sobolev type equations. To investigate the existence of solutions of this system of equations, the phase space method will be used, which was developed by G.A.~Sviridyuk to study the solvability of semilinear Sobolev-type equations. Conditions for the existence and uniqueness or multiplicity of solutions to the Showalter -- Sidorov problem for the model under study are revealed, depending on the parameters of the system. The obtained theoretical results made it possible to develop an algorithm for the numerical solution of the problem based on the modified Galerkin method. The results of computational experiments are presented.