A METHOD FOR AN APPROXIMATE SOLUTION TO A PARABOLIC EQUATION WITH A POWER-LAW NONLINEARITY

Abstract

Aim. The purpose is to find an approximate solution to the first initial boundary value problem for a parabolic equation with a power-law nonlinearity. The problem is solved using an approximate analytical method based on the application of an a priori estimation of the solution to the problem for the linearization of the original equation. Methodology. The first step in applying the method is to reduce the nonlinear equation to the loaded equation, by replacing the nonlinear member with its integral in the spatial variable. Following this, an a priori estimate of the obtained problem is established in a suitable functional space. By integrating the loaded equation with respect to the spatial variable, a transition is made to the nonlinear ordinary differential equation associated with it. The latter is linearized using the a priori estimate of the loaded problem, in which the upper bound of inequality is chosen. Results. A formula is obtained that expresses the solution to the loaded equation in terms of its norm and the solution to the associated ordinary differential equation. Approximation of the solution to a nonlinear equation is proposed to be performed by using an iterative process for solving a sequence of linear problems. An example illustrating the application of the method to a model problem is presented. Research implications. The applied procedure makes it possible to obtain an analytical expression for an approximate solution to a nonlinear problem. The described method can be applied to partial differential equations of any type and order, containing the natural degree of the desired function or its derivative.