ROTHE METHOD IN COMBINATION WITH TWO-SIDED APPROXIMATIONS METHOD FOR SOLVING THE INITIAL-BOUNDARY PROBLEMS FOR THE SEMI-LINEAR HEAT EQUATION - Sidorov, M. V. (Computer Science and Applied Mathematics, 2018)
In this paper, we consider the first initial-boundary problem for the semi-linear heat equation. Problems of this type (with the search for a positive solution) often arise in the mathematical modeling of processes in chemical kinetics, combustion theory, biology, and others. Based on the modified Rothe method, the original non-stationary problem is replaced at each time layer by a nonlinear boundaryvalue problem for an equation with an elliptic operator. Next, for finding a positive solution of this nonlinear boundary value problem, a method of successive approximations with a two-sided character of convergence is constructed. To construct such a method and study the boundary value problem, methods of the theory of nonlinear operators in semi-ordered spaces are used. Using the Green’s functions method of nonlinear boundary value problems for an elliptic equation, a transition to an equivalent Hammerstein integral equation is considered, which is investigated as a nonlinear operator equation with a heterotone operator in the space of continuous functions that is semi-ordered by a cone of nonnegative functions. The properties of a nonlinear integral operator included in the equation are studied. Next, a strongly invariant cone segment and two iterative sequences are constructed which start from the corresponding ends of a strongly invariant cone segment. The first of these sequences is monotonically increasing and approximates the desired solution from below, and the second is monotonically decreasing and brings the desired solution from above. Conditions for the existence of a common limit of these sequences are given, that is, the conditions for uniqueness of the solution of nonlinear boundary value problems of the Rothe method on each time layer. A priori and a posteriori estimation of the error of the approximate solution of the problem was obtained. A computational experiment was carried out for a heterotone power nonlinearity problem. An approximate solution of a non-stationary problem at time moment 0.1 was obtained in two ways: with a step of 0.1 and a step of 0.05, which made it possible to increase the order of approximation of the method per unit and to clarify the approximate solution according to the Runge rule. The results of the computational experiment are presented in the form of figures and tables with numerical data.