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Русская версия

**Zaryana Vladimirovna Beshtokova**

Junior Researcher, Department of Computational Methods,

Institute of Applied Mathematics and Automation, KBSC RAS

**Murat Khamidbievich Beshtokov**

Leading Researcher, Department of Computational Methods,

Institute of Applied Mathematics and Automation, KBSC RAS

Boundary value problems are studied for an integro-differential equation with a time fractional derivative and a Bessel operator. To solve the problems under consideration, a priori estimates are obtained in a differential interpretation, which implies the uniqueness and stability of the solution with respect to the initial data and the right-hand side. For the numerical solution of boundary value problems, monotone difference schemes with directed differences are constructed, analogs of a priori estimates are proved for them, and error estimates are given under the assumption that the solutions of the equations are sufficiently smooth. The obtained a priori estimates in the difference form imply the uniqueness and stability of the solution with respect to the initial data and the right-hand side, and also, by virtue of the linearity of the difference problems, the convergence with the second order in the parameters of the grid. An algorithm for the approximate solution of a boundary value problem with a condition of the third kind is proposed, and numerical calculations are carried out for a test example illustrating the theoretical results obtained in this work concerning the convergence and the order of approximation of the difference scheme.

- a priori estimate
- boundary value problems
- fractional-order differential equation
- Gerasimov-Caputo fractional derivative
- integro-differential equation
- monotone schemes

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**.**Local and nonlocal boundary value problems for degenerating and nondegenerating pseudoparabolic equations with a Riemann-Liouville fractional derivative // Differential Equations, 54:6 (2018), 758-774 - Beshtokov, M. KH. To boundary-value problems for degenerating pseudoparabolic equations with Gerasimov-Caputo fractional derivative, Russian Mathematics, 62:10 (2018), 1-14
- Beshtokov, M. KH. Boundary-value problems for loaded pseudoparabolic equations of fractional order and difference methods of their solving, Russian Mathematics, 63:2 (2019), 1-10
- Beshtokov, M. KH. Difference method for solving a nonlocal boundary value problem for a degenerating third-order pseudo-parabolic equation with variable coefficients // Comput. Math. Math. Phys., 56:10 (2016), 1763-1777
- Beshtokov, M. KH. Boundary value problems for degenerating and nondegenerating Sobolev-type equations with a nonlocal source in differential and difference forms // Differential Equations, 54:2 (2018), 250-267
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