The Rothe method and the method of two-sided approximations in the numerical analysis of problems for one-dimensional quasilinear parabolic equations
Abstract
In this paper, we consider the first and the second initial-boundary problem for the one-dimensional 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 boundary-value problem for an ordinary differential equation. 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 two-sided approximations to the positive solution of the problem, methods of the theory of semi-ordered spaces are used on each time layer, in particular, the results of V.I. Opoĭcev on the solvability of operator equations with a heterotone operator are used. Using the Green’s functions method of nonlinear boundary value problems for an ordinary differential 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 non-negative functions. 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 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.
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References
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