Abstract:Background. One of the important sections of the theory of differential equations is loaded equations. They allow us to model processes in which the influence of external factors significantly changes the behavior of the system. This is especially important in fields such as mechanics, hydrology, and materials science. The study of loaded equations contributes to the creation of more accurate models that are used to analyze the stability and reliability of structures, as well as to predict various phenomena in natural and engineering systems. New difference schemes of an increased order of accuracy are constructed for an approximate solution of the first boundary value problem for an unsteady loaded moisture transfer equation in one-dimensional and multidimensional regions. Loaded integral equations allow for a deeper understanding of the distribution of loads and the interaction of elements in complex systems. The equations studied in this paper play a significant role in solving urgent problems of ecology, agriculture, construction and climatology. Accurate modeling of moisture transfer processes makes it possible to effectively manage water resources, predict groundwater levels, optimize irrigation, ensure the stability of building structures, and predict the effects of climate change. In addition, the development of such models contributes to progress in hydrology and related sciences. Materials and methods. For an approximate solution of the tasks set, the finite difference method and the energy inequality method are used to obtain a priori estimates of the solutions of the proposed difference schemes. Results. A high-order approximation difference scheme is constructed for each problem. An a priori estimate is obtained by the method of energy inequalities for solving each difference problem. The obtained estimates imply the uniqueness and stability of the solution on the right side and the initial data, as well as the convergence of the solution of the difference problem to the solution of the corresponding initial differential problem with a speed equal to the order of approximation of the difference scheme. Conclusions. New high-order difference schemes of approximation have been developed for the approximate solution of the tasks set.
Keywords:the first boundary value problem, multidimensional moisture transfer equation, loaded equation, a priori estimate, difference scheme, high-order accuracy scheme, stability and convergence of the scheme
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