The Application of the Green-Rvachev Quasifunction Method for Constructing Two-Sided Approximations to the Solution of the Dirichlet Problem for a Nonlinear Heat Equation

• Дата: 2018
• Автор: Сидоров, Максим Вікторович
• Формат: Стаття
• Мова: Ukrainian
• Опубліковано: Кам'янець-Подільський національний університет імені Івана Огієнка 2018
• Онлайн доступ: http://mcm-math.kpnu.edu.ua/article/view/159393
• Назва журналу: Mathematical and computer modelling. Series: Physical and mathematical sciences

The problem of mathematical modeling of nonlinear stationary heat conduction processes leads to the necessity for an effective solution of boundary value problems for an elliptic equation with a coefficient that is nonlinearly dependent on temperature. In this paper, the Dirichlet problem for the heat equation with a nonlinear function of power of heat sources and a heat conductivity coefficient with power law dependence on temperature, is considered. To find a positive solution of the problem under consideration it is proposed the using of the two-sided approximations method, constructed on the basis of the application of the Green-Rvachev’s quasi-function method. For this, the unknown function was replaced in order to obtain a nonlinear problem for the equation with the Laplace operator. This problem was replaced by the equivalent Uryson integral equation using the Green-Rvachev’s quasi-function. The investigation of this equation was carried out by methods of nonlinear analysis in semi-ordered spaces, in particular, using the theory of heterotone operators by V.I. Opoǐcev. An integral operator entering the Uryson equation is considered as a heterotone operator acting in the space of continuous functions, which is semi-ordered by a cone of non-negative functions. This made it possible to find out the conditions for the existence of a unique positive solution of the problem under consideration and to construct a two-sided iterative process to search out it. This process begins at the ends of a strongly invariant for a heterotone operator cone segment and allows one to build two sequences of functions that approximate the desired solution from below and above. The advantage of the constructed two-sided iterative process is the availability of a convenient a posteriori error estimate for the approximate solution at each iteration. The efficiency of the developed method was illustrated by a computational experiment in a unit square for the case of the exponential dependence of the power of thermal sources on temperature. The results of the experiment are presented in the form of graphical (contour lines and the surface of an approximate solution) and numerical (values of an approximate solution at some points in the area) information.