Finite-difference method for solving of a nonlocal boundary value problem for a loaded thermal conductivity equation of the fractional order. (Russian. English summary) Zbl 1474.65268

Beshtokov, Murat Khamidbievich; Beshtokova, Zar’yana Vladimirovna; Khudalov, Marat Zakharovich

Vladikavkaz. Mat. Zh. 22, No. 4, 45-57 (2020).

Summary: 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.

MSC:

65M06	Finite difference methods for initial value and initial-boundary value problems involving PDEs
65N06	Finite difference methods for boundary value problems involving PDEs
65N12	Stability and convergence of numerical methods for boundary value problems involving PDEs
35K05	Heat equation
26A33	Fractional derivatives and integrals
35R11	Fractional partial differential equations
35B45	A priori estimates in context of PDEs
35B65	Smoothness and regularity of solutions to PDEs
35B30	Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs

Keywords:

heat equation; fractional Caputo derivative; lumped heat capacity; difference schemes; stability; convergence