Grid method for approximate solution of initial-boundary value problems for generalized convection-diffusion equations

M. Kh. Beshtokov   ; Z. V. Beshtokova

Vladikavkazskij matematičeskij žurnal, Tome 23 (2021) no. 3, pp. 28-44

Résumé

In a rectangular domain, we study an initial-boundary value problems for one-dimensional generalized convection-diffusion equations with the Bessel operator and fractional derivatives in the sense of Riemann–Liouville and Caputo of order $\alpha$ ($0<\alpha<1$) with boundary conditions of the first and third kind. The fractional-order convection-diffusion equation with the Bessel operator arises when passing from the three-dimensional fractional-order convection-diffusion equation to cylindrical (spherical) coordinates, in the case when the solution $u=u(r)$ does not depend on either $z$ or $\varphi$. For the numerical solution of the problems under consideration, monotone difference schemes of the second order of accuracy with respect to the grid parameters are constructed, which approximate these problems on uniform grids. Using the method of energy inequalities for solving initial-boundary value problems, a priori estimates are obtained in differential and difference interpretations under the assumption of the existence of a regular solution to the original differential problem. The obtained a priori estimates imply the uniqueness and stability of the solution with respect to the right-hand side and the initial data, as well as, due to the linearity of the difference problems, the convergence of the solution of the corresponding difference problem to the solution of the original differential problem with the rate $O(h^2 + \tau^2)$.

Bibliographie

[1] Mandelbrojt S., “Sulla generalizzazione del calcolo clelle variazione”, Atti Reale Accad. Naz. Lincei. Rend Cl. Sei., Fis. Mat. e Natur., 6:1 (1925), 151–156 | Zbl

[2] O'Shaughnessy L., Post E. L., “Solutions of problems: Calculus: 433”, Amer. Math. Month., 25:4 (1918), 172–173 | DOI | MR

[3] Al-Bassam M. A., “On fractional calculus and its applications to the theory of ordinary differential equations of generalized order”, Nonlinear Analysis and Applications, Lect. Notes Pure Appl. Math., 80, Dekker, New York, 1982, 305–331 | MR

[4] Al-Abedeen A. Z., Arora H. L., “A global existence and uniqueness theorem for ordinary differential equations of generalized order”, Canad. Math. Bull., 21:3 (1978), 267–271 | DOI | MR | Zbl

[5] Samko S. G., Kilbas A. A., Marichev O. I., Integraly i proizvodnye drobnogo poryadka i nekotorye ikh prilozheniya, Nauka i tekhnika, Minsk, 1987, 688 pp.

[6] Nakhushev A. M., Drobnoe ischislenie i ego primenenie, Fizmatlit, M., 2003, 272 pp.

[7] Goloviznin V. M., Kiselev V. P., Korotkin I. A., Yurkov Yu. I., Nekotorye osobennosti vychislitelnykh algoritmov dlya uravnenii drobnoi diffuzii, Preprint No IBRAE-2002-01, IBRAE RAN, M., 2002, 57 pp.

[8] Goloviznin V. M., Kiselev V. P., Korotkin I. A., Chislennye metody resheniya uravneniya drobnoi diffuzii s drobnoi proizvodnoi po vremeni v odnomernom sluchae, Preprint No IBRAE-2002-10, IBRAE RAN, M., 2002, 35 pp.

[9] Taukenova F. I., Shkhanukov-Lafishev M. Kh., “Raznostnye metody resheniya kraevykh zadach dlya differentsialnykh uravnenii drobnogo poryadka”, Zhurn. vychisl. matem. i mat. fiz., 4:10 (2006), 1871–1881 | MR

[10] Diethelm K., Walz G., “Numerical solution of fractional order differential equations by extrapolation”, Numer. Algorithms, 16:3–4 (1997), 231–253 | DOI | MR | Zbl

[11] Alikhanov A. A., “Apriornye otsenki reshenii kraevykh zadach dlya uravnenii drobnogo poryadka”, Dif. uravneniya, 46:5 (2010), 658–664 | MR | Zbl

[12] Alikhanov A. A., “A new difference scheme for the time fractional diffusion equation”, J. Comput. Phys., 280 (2015), 424–438 | DOI | MR | Zbl

[13] Beshtokov M. Kh., “Kraevye zadachi dlya nagruzhennykh psevdoparabolicheskikh uravnenii drobnogo poryadka i raznostnye metody ikh resheniya”, Izv. vuzov. Matematika, 2019, no. 2, 3–12 | DOI | MR | Zbl

[14] Beshtokov M. Kh., Vodakhova V. A., “Nelokalnye kraevye zadachi dlya uravneniya konvektsii-diffuzii drobnogo poryadka”, Vestn. Udmurt. un-ta. Matem. Mekh. Kompyut. nauki, 29:4 (2019), 459–482 | DOI | MR | Zbl

[15] Beshtokov M. Kh., “Raznostnyi metod resheniya nelokalnoi kraevoi zadachi dlya vyrozhdayuschegosya psevdoparabolicheskogo uravneniya tretego poryadka s peremennymi koeffitsientami”, Zhurn. vychisl. matem. i mat. fiz., 56:10 (2016), 1780–1794 | DOI | MR | Zbl

[16] Beshtokov M. Kh., “Kraevye zadachi dlya vyrozhdayuschikhsya i nevyrozhdayuschikhsya uravnenii Sobolevskogo tipa s nelokalnym istochnikom v differentsialnoi i raznostnoi traktovkakh”, Dif. uravneniya, 54:2 (2018), 249–266 | DOI | MR | Zbl

[17] Beshtokov M. Kh., “Kraevye zadachi dlya uravneniya vlagoperenosa s drobnoi proizvodnoi Kaputo i operatorom Besselya”, Dif. uravneniya, 56:3 (2020), 353–365 | DOI | MR | Zbl

[18] Kumykova S. K., “Ob odnoi kraevoi zadache dlya uravneniya ${\rm sign}~y^m u_{xx} + u_{yy} = 0$”, Dif. uravneniya, 12:1 (1976), 79–88 | MR | Zbl

[19] Ladyzhenskaya O. A., Kraevye zadachi matematicheskoi fiziki, Nauka, M., 1973, 407 pp. | MR

[20] Malshakov A. V., “Uravneniya gidrodinamiki dlya poristykh sred so strukturoi porovogo prostranstva, obladayuschei fraktalnoi geometriei”, Inzh.-fiz. zhurn., 62:3 (1992), 405–410 | MR

[21] Samarskii A. A., Teoriya raznostnykh skhem, Nauka, M., 1983, 617 pp.

[22] Beshtokov M. Kh., “K kraevym zadacham dlya vyrozhdayuschikhsya psevdoparabolicheskikh uravnenii s drobnoi proizvodnoi Gerasimova — Kaputo”, Izv. vuzov. Matematika, 2018, no. 10, 3–16 | MR | Zbl