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DOI: 10.46698/p2286-5792-9411-x

# Finite-Difference Method for Solving of a Nonlocal Boundary Value Problem for a Loaded Thermal Conductivity Equation of the Fractional Order

Beshtokov, M. Kh. , Beshtokova, Z. V.  , Khudalov, M. Z.
Vladikavkaz Mathematical Journal 2020. Vol. 22. Issue 4.
Abstract:
We study a nonlocal boundary value problem in a rectangular area for a one-dimensional in a spatial variable of the loaded heat fractional conductivity equation with a heat capacity concentrated at the boundary. The problem is considered as a mathematical model, arising, in particular, in the practice of regulating the salt regime of soils with a fractal organization, when the lamination of the upper layer is achieved by drain layer of the water from the surface of an area flooded for some time. The main research method is the method of energy inequalities. An a priori estimate is obtained by the assumption of the existence of a regular solution to the differential problem, which implies the uniqueness and continuous dependence of the solution from the input data of the problem. A difference scheme of the second order of approximation by the grid parameters is put on a uniform grid by correspondence with the differential problem. Under the assumptions of the existence of a regular solution to the differential problem, an a priori estimate is obtained, which implies the uniqueness and continuous dependence of the solution on the right side and the initial data. By virtue of the linearity of the problem under consideration, the received inequality allows us to assert the convergence of the approximate solution to the exact one (assuming that the latter exists in the class of sufficiently smooth functions) with a rate equal to the order of the approximation error. The numerical experiments are carried out to illustrate the recieved theoretical results.
Keywords: heat equation, fractional Caputo derivative, lumped heat capacity, difference schemes, stability, convergence