Boundary value problems for degenerate and degenerate fractional
order differential
equations with non-local linear source and difference methods for their numerical implementation
Abstract:
In the paper we study non-local boundary value problems for differential and partial differential equations of fractional order with a non-local linear source being mathematical models of the transfer of water and salts in soils with fractal organization. Apart of the Cartesian case, in the paper we consider one-dimensional cases with cylindrical and spherical symmetry. By the method of energy inequalities, we obtain apriori estimates of solutions to nonlocal boundary value problems in differential form. We construct difference schemes and for these schemes, we prove analogues of apriori estimates in the difference form and provide estimates for errors assuming a sufficient smoothness of solutions to the equations. By the obtained apriori estimates, we get the uniqueness and stability of the solution with respect to the the initial data and the right par, as well as the convergence of the solution of the difference problem to the solution of the corresponding differential problem with the rate of $O(h^2+\tau^2)$.
Keywords:boundary value problem, apriori estimate, the equation of moisture transfer, the differential equation of fractional order, Gerasimov-Caputo fractional derivative.
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