On an explicit difference method for solving one nonlinear parabolic equation with double degeneration and nonlocal spatial operator

Abstract

We consider the initial-boundary value problem for nonlinear parabolic equation. This type of equation can be classified as a parabolic equation with double degeneration: degeneration can be present in space operator, and a nonlinear function which is under the derivative sign with respect to the variable t, may not be separated from zero. The space operator of the considered equation nonlinearly depends on the sought function, its gradient and the non-local (integral) solution characteristic. This problem has an applied nature. Such equations appear, for example, in modeling the process of bacteria population spreading. In the present paper we propose and investigate the explicit differential scheme. A priori estimates are obtained, and the convergence of constructed algorithm is proved. The current work is a continuation of the research begun in the works [1], [2], [3], where the existence and uniqueness theorems for the generalized solution have been proved, the convergence of the finite-element method scheme and the explicit difference scheme in the case when nonlinearity is present only in the spatial operator have been investigated. In paper [4] for a problem with double degeneration, an approximate method has been studied. That method was constructed with the use of semidiscretization with respect to a variable t and the finite element method in the space variable with lowering nonlocality to the lower layer, the existence of an approximate solution and the convergence of the constructed algorithms were proved.