Numerical solution of Showalter–Sidorov and Cauchy problems of ion–acoustic waves propagation mathematical model

Abstract

The paper deals with the problem of numerical investigation of semilinear mathematical model of ion-acoustic waves propagation in plasma. The research is based on the previous study of solvability of the Cauchy problem for an abstract semilinear Sobolev type equation of higher order. The theory of relatively polynomially bounded operator pencils, the theory of differentiable Banach manifolds, and the phase space method are used for analytical study of the model. Projectors splitting spaces into direct sums of subspaces are constructed. Given equation is reduced to a system of two equations. One of them determines the phase space of the initial equation, and its solution is a function with values from the eigenspace of the operator at the highest time derivative. The solution of the second equation is the function with values from the image of the projector. Moreover, in the second section, the sufficient conditions for the solvability of the abstract problem under study are presented. These results are applied to the mathematical model of ion-acoustic waves in plasma which is based on the fourth-order equation with a singular operator at the highest time derivative. Reducing the matematical model to an abstract problem, we obtain sufficient conditions for the existence of unique solution. The results of analytical investigation of the Showalter – Sidorov problem which is more natural for Sobolev type equations are also presented in this section. The Galerkin method is used for numerical study of the model. An algorithm for the numerical solution of the Showalter – Sidorov problem for the model of ion-acoustic waves in plasma is described in the last section.