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DOI: 10.46698/v2914-8977-8335-s
On a Difference Scheme for Solution of the Dirichlet Problem for Diffusion Equation with a Fractional Caputo Derivative in the Multidimensional Case in a Domain with an Arbitrary Boundary
Abstract: In this paper, we study the Dirichlet problem for the diffusion equation with a fractional Caputo derivative in the multidimensional case in a domain with an arbitrary boundary. Instead of the original equation, we consider the diffusion equation with a fractional Caputo derivative with a small parameter. A~locally one-dimensional difference scheme of A. A. Samarsky, the main essence of which is to reduce the transition from layer to layer to the sequential solution of a number of one-dimensional problems in each of the coordinate directions. Moreover, each of the auxiliary problems may not approximate the original problem, but in the aggregate and in special norms such an approximation takes place. These methods have been called splitting methods. Using the maximum principle, we obtain an a priori estimate in the uniform metric norm. The stability of the locally one-dimensional difference scheme and the uniform convergence of the approximate solution of the proposed difference scheme to the solution of the original differential problem for any \(0 < \alpha < 1\) are proved. An analysis is made of the choice of optimal values of \(\varepsilon\), at which the rate of uniform convergence of the approximate solution of the considered difference scheme to the solution of the original differential problem will be determined in the best way.
Keywords: generalized equation, convection-diffusion equation, fractional order equation, fractional derivative in the sense of Caputo, maximum principle, locally one-dimensional scheme, stability and convergence, boundary value problems, a priori estimate
For citation: Beshtokova, Z. V., Beshtokov, M. Kh. and Shkhanukov-Lafishev, M. Kh. On a Difference Scheme for Solution of the Dirichlet Problem for Diffusion Equation with a Fractional Caputo Derivative in the Multidimensional Case
in a Domain with an Arbitrary Boundary, Vladikavkaz Math. J., 2022, vol. 24, no. 3, pp. 37-54. DOI 10.46698/v2914-8977-8335-s
1. Oldham, K. B., Spanier, J. The Fractional Calculus, New York-London, Academic Press, 1974, 234 p.
2. Miller, K. S. and Ross, B. An Introduction to the Fractional Calculus and Fractional Differential Equations, New York, John Wiley and Sons. Inc., 1993, 376 p.
3. Nakhushev, A. M. Drobnoe ischislenie i ego primenenie [Fractional Calculus and its Application], Moscow, Fizmatlit, 2003, 272 p. (in Russian).
4. Shogenov, V. Kh., Kumykova, S. K. and Shkhanukov-Lafishev, M. Kh.
Generalized Transport Equation and Fractional Derivatives, Adyghe International
Scientific Journal, 1996, vol. 2, no. 1, pp. 43-45. (in Russian).
5. Podlubny, I. Fractional Differential Equations, San-Diego, Academic Press, 1999, 368 p.
6. Samko, S. G., Kilbas, A. A. and Marichev, O. I. Integraly i proizvodnye drobnogo poryadka i nekotorye ikh prilozheniya [Fractional Integrals and Derivatives and Some of Their Applications], Minsk, Nauka i Tekhnika, 1987, 688 p. (in Russian)
7. Kochubey, A. Yu. Fractional-Order Diffusion, Differential Equations, 1990, vol. 26, no. 4, pp. 485-492.
8. Mal'shakov, A. V. Hydrodynamic Equations for Porous Media with a Pore Space Structure with Fractal Geometry, Inzhenerno-Fizicheskii Zhurnal, 1992, vol. 62, no. 3, pp. 405-410 (in Russian).
9. Uchaykin, V. V. Metod drobnykh proizvodnykh [Fractional Derivatives Method], Ul'yanovsk, Publishing House "Artishok'', 2008, 512 p. (in Russian).
10. Тarasov, V. E. Modeli teoreticheskoy fiziki s integro-differentsirovaniyem drobnogo poryadka [Theoretical Physics Models with Integro-Differentiation of Fractional Order], Izhevsk, Izhevsk Institute of Computer Science, 2011, 568 p. (in Russian).
11. Douglas, J. and Rachford, H. H. On the Numerical Solution of Heat Conduction Problems in Two and Three Space Variables, Transactions of the American Mathematical Society, 1956, vol. 82, no. 2, pp. 421-43.
DOI: 10.1090/s0002-9947-1956-0084194-4.
12. Peaceman, D. W. and Rасhfоrd, H. H. The Numerical Solution of Parabolic and Elliptic Differential Equations, Journal of the Society for Industrial and Applied Mathematics, 1955, vol. 3, no 1, pp. 28-41. DOI: 10.1137/0103003.
13. Yanenko, N. N. Metod drobnykh shagov resheniya mnogomernykh zadach matematicheskoy fiziki [The Method of Fractional Steps for Solving Multidimensional Problems of Mathematical Physics], Novosibirsk, Nauka, Siberian Branch, 1967, 196 p. (in Russian).
14. Samarskii, A. A. On an Economical Difference Method for the Solution of a Multidimensional Parabolic Equation in an Arbitrary Region, USSR Computational Mathematics and Mathematical Physics, 1963, vol. 2, no 5, pp. 894-926. DOI: 10.1016/0041-5553(63)90504-4.
15. Samarskii, A. A. Teoriya raznostnykh skhem [The Theory of Difference Schemes], Moscow, Nauka, 1983, 616 p. (in Russian).
16. Samarskii, A. A. and Gulin, A. B. Ustoychivost' raznostnykh skhem [The Stability of Difference Schemes], Moscow, Nauka, 1973, 415 p. (in Russian).
17. Marchuk, G. I. Metody rasshchepleniya [Splitting Methods], Moscow, Nauka, 1988, 264 p. (in Russian).
18. Dyakonov, Ye. G. Difference Schemes with a "Disintegrating'' Operator for Multidimensional Problems, USSR Computational Mathematics and Mathematical Physics, 1963, vol. 2, no 4, pp. 581-607. DOI: 10.1016/0041-5553(63)90531-7.
19. Lafisheva, M. M. and Shkhanukov-Lafishev, M. Kh. Locally One-Dimensional Difference Schemes for the Fractional Order Diffusion Equation, Computational Mathematics and Mathematical Physics, 2008, vol. 48, no. 10, pp. 1875-1884. DOI: 10.1134/S0965542508100102.
20. Ashabokov, B. A., Beshtokova, Z. V. and Shkhanukov-Lafyshev, M. Kh. Locally One-Dimensional Difference Scheme for a Fractional Tracer Transport Equation, Computational Mathematics and Mathematical Physics, 2017, vol. 57, no. 9, pp. 1498-1510. DOI: 10.1134/S0965542517090044.
21. Bazzaev, A. K. and Shkhanukov-Lafishev, M. Kh. Locally One-Dimensional Schemes for the Diffusion Equation with a Fractional Time Derivative in an Arbitrary Domain, Computational Mathematics and Mathematical Physics, 2016, vol. 56, no. 1, pp. 106-115. DOI: 10.1134/S0965542516010061.
22. Beshtokova, Z. V. and Shkhanukov-Lafyshev, M. Kh. Locally One-Dimensional Difference Scheme for the Third Boundary Value Problem for a Parabolic Equation of the General Form with a Nonlocal Source, Differential Equations, 2018, vol. 54, no. 7, pp. 870-880.
DOI: 10.1134/S0012266118070042.
23. Beshtokova, Z. V., Lafyshev, M. M. and Shkhanukov-Lafyshev, M. Kh.
Locally One-Dimensional Difference Schemes for Parabolic Equations in Media Possessing Memory, Computational Mathematics and Mathematical Physics, 2018, vol. 58, no. 9, pp. 1477-1488. DOI: 10.1134/S096554251809004X.
24. Beshtokov, M. Kh. and Vodakhova, V. A. Nonlocal Boundary Value Problems for a Fractional-Order Convection-Diffusion Equation, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2019, vol. 29, no. 4, pp. 459-482 (in Russian). DOI: 10.20537/vm190401.
25. Nakhusheva, F. M., Vodakhova, V. A., Kudaeva, F. Kh. and Abaeva, Z. V. Locally One-Dimensional Difference Scheme for the Fractional-Order Diffusion Equation with a Concentrated Heat Capacity, Modern Problems of Science and Education, 2015, no. 2-1, p. 763 (in Russian).
26. Vishik, M. I. and Lyusternik, L. A. Regular Degeneration and Boundary Layer
for Linear Differential Equations with a Small Parameter, Uspekhi Matematicheskikh Nauk,
1957, vol. 12, no. 5(77), pp. 3-122 (in Russian).
27. Godunov, S. K. and Ryaben'kiy, V. S. Raznostnyye skhemy [Difference Schemes], Moscow, Nauka, 1977, 439 p. (in Russian).
28. Alikhanov, A. A. A Priori Estimates for Solutions of Boundary Value Problems for Fractional-Order Equations, Differential Equations, 2010, vol. 46, no 5, pp. 660-666.
DOI: 10.1134/S0012266110050058.
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