High-accuracy difference schemes for solving non-stationary fourth-order equations and their application to non-classical partial differential equations

Abstract

In the article, high-accuracy difference schemes for the Cauchy problem for a system of fourth-order equations are obtained. Based on the method of energy inequalities, the stability of the scheme is proved, a priori estimates of the solution to difference schemes are obtained, and their convergence and accuracy are proved. The results obtained for the system are applied to solve the first initial-boundary value problem for the equation of ion-acoustic waves in a "magnetized" plasma for the generalized potential of the electric field. The schemes constructed for this problem have second-order accuracy in spatial variables and fourth-order accuracy in time variables. In energy norms, convergence and accuracy estimates are obtained in classes of smooth solutions.