Numerical study of the non-uniquity solutions phenomenon to the Showalter–Sidorov problem for the model of nerve impulse propagation in a rectangular membrane

This article presents a numerical study of a model of nerve impulse propagation in a rectangular area based on the Fitz Hugh–Nagumo system of equations. This model belongs to the class of reaction-diffusion systems describing a wide range of physico-chemical processes, including chemical reactions with diffusion and transmission of nerve impulses. Given the asymptotic stability of the model and a significant difference in the rates of change of its components, the initial problem can be reduced to a problem for a semilinear Sobolev type equation.The study touches upon the issues of non-uniqueness of the solution of the Showalter–Sidorov problem. The paper develops a computational algorithm implemented in the Maple environment based on the Galerkin method, which correctly takes into account the degeneracy of one of the equations of the system. The article provides an example of a numerical experiment for the Fitz Hugh–Nagumo system on a rectangle, illustrating the behavior of solutions depending on the parameters of the problem.