On the convergence of locally one-dimensional schemes for the differential equation in partial derivatives of fractional orders in a multidimensional domain

A. K. Bazzaev

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 20 (2023) no. 2, pp. 1064-1078.

Bibliographie

[1] K.V. Chukbar, “Stochastic transport and fractional derivatives”, Zh. Éksp. Teor. Fiz., 108:11 (1995), 1875–1884

[2] V.L. Kobelev, Ya.L. Kobelev, E.P. Romanov, “Self-maintained processes in the case of nonlinear fractal diffusion”, Dokl. Phys., 44 (1999), 752–753 | Zbl

[3] V.L. Kobelev, Ya.L. Kobelev, E.P. Romanov, “Non-Debye relaxation and diffusion in fractal space”, Dokl. Phys., 43 (1998), 752–753 | MR

[4] V.M. Goloviznin, V.P. Kiselev, I.A. Korotkin, Yu.P. Yurkov, “Pryamye zadachi klassicheskogo perenosa radionuklidov v geologicheskikh formatsiyakh”, Izv. Ross. Akad. Nauk, Energ., 2004:4 (2004), 121–130

[5] V.M. Goloviznin, V.P. Kiselev, I.A. Korotkin, Yu.P. Yurkov, Chislennyye metody resheniya uravneniya diffuzii s drobnoy proizvodnoy v odnomernom sluchaye, Preprint IBRAE, No 2003-12, IBRAE RAS, M., 2003

[6] R.R. Nigmatulin, “Relaxation features in a system with remnant memory”, Fiz. Tverd. Tela, 27:5 (1985), 1583–1585

[7] V.Kh. Shogenov, A.A. Akhkubekov, R.A. Akhkubekov, “Fractional differentiation method in the theory of Brownian motion”, Izv. Vyssh. Uch. Zaved. Sev.-Kav. Reg., 2004:1 (2004), 46–49

[8] V.V. Uchaikin, “Anomalous diffusion of particles with a finite free-motion velocity”, Theor. Math. Phys., 115:1 (1998), 496–501 | DOI | MR | Zbl

[9] V.Yu. Zaburdaev, K.V. Chukbar, “Enhanced superdiffusion and finite velocity of Levy flights”, J. Exp. Theor. Phys., 94:2 (2002), 252–259 | DOI

[10] J. Klafter, M.F. Shlesinger, G. Zumofen, “Beyond brownian motion”, Phys. Today, 49:2 (1996), 33–39 | DOI

[11] F. Mainardi, Y. Luchko, G. Pagnini, “The fundamental solution of the space-time fractional diffusion equation”, Fract. Calc. Appl. Anal., 4:2 (2001), 153–192 | MR | Zbl

[12] E. Scalas, R, Gorenflo, F. Mainardi, “Uncoupled continuous-time random walks: solution and limiting behaviour of the master equation”, Phys. Rev. E, 69:1 (2004), 011107–011114 | DOI | MR

[13] Y. Zhang, D.A. Benson, M.M. Meerschaert, H.-P. Scheffler, “On using random walks to solve the space-fractional advectiona-dispersion equations”, J. Stat. Phys., 123:1 (2006), 89–110 | DOI | MR | Zbl

[14] N.G. Abrashina-Zhadaeva, I.A. Timoshchenko, “Finite-difference schemes for a diffusion equation with fractional derivatives in a multidimensional domain”, Differ. Equ., 49:7 (2013), 789–795 | DOI | MR | Zbl

[15] Bangti Jin, Raytcho Lazarov, Zhi Zhou, “A Petrov-Galerkin finite element method for fractional convection-diffusion equations”, SIAM J. Numer. Anal., 54:1 (2014), 481–503 | MR | Zbl

[16] Bangti Jin, Raytcho Lazarov, Josef Pasciak, Zhi Zhou, “Error analysis of a finite element method for the space-fractional parabolic equation”, SIAM J. Numer. Anal., 52:5 (2014), 2272–2294 | DOI | MR | Zbl

[17] Bangti Jin, Raytcho Lazarov, “An analysis of the $L_1$ scheme for the subdiffusion equation with nonsmooth data”, IMA J. Numer. Anal., 36:1 (2016), 197–221 | MR | Zbl

[18] F.I. Taukenova, M.Kh. Shkhanukov-Lafishev, “Difference methods for solving boundary value problems for fractional differential equations”, Comput. Math. Math. Phys., 46:10 (2006), 1785–1795 | DOI | MR

[19] M.M. Lafisheva, M.Kh. Shkhanukov-Lafishev, “Locally one-dimensional difference schemes for the fractional order diffusion equation”, Comput. Math. Math. Phys., 48:10 (2008), 1875–1884 | DOI | MR | Zbl

[20] A.K. Bazzaev, M.Kh. Shkhanukov-Lafishev, “Locally one-dimensional scheme for fractional diffusion equations with Robin boundary conditions”, Comput. Math. Math. Phys., 50:7 (2010), 1141–1149 | DOI | MR | Zbl

[21] A.K. Bazzaev, A.V. Mambetova, M.Kh. Shkhanukov-Lafishev, “Locally one-dimensional scheme for the heat equation of fractional order with concentrated heat capacity”, Zh. Vychisl. Mat. Mat. Fiz., 52:9 (2012), 1656–1665 | MR | Zbl

[22] A.K. Bazzaev, “Difference schemes for fractional-order diffusion equation with Robin boundary conditions in a multidimensional domain”, Ufim. Mat. Zh., 5:1 (2013), 11–16 | DOI | MR

[23] A.K. Bazzaev, M.Kh. Shkhanukov-Lafishev, “Locally one-dimensional schemes for the diffusion equation with a fractional time derivative in an arbitrary domain”, Comput. Math. Math. Phys., 56:1 (2016), 106–115 | DOI | MR | Zbl

[24] Y. Povstenko, “Axisymmetric solutions to fractional diffusion-wave equation in a cylinder under Robin boundary condition”, Eur. Phys. J. Spec. Top., 222:8 (2013), 1767–1777 | DOI | MR

[25] Y. Povstenko, “Time-fractional heat conduction in an infinite medium with a spherical hole under Robin boundary condition”, Fract. Calc. Appl. Anal., 16:2 (2013), 354–369 | DOI | MR | Zbl

[26] Ch. Tadjeran, M. Meerschaert, H. Scheffler, “A second-order accurate numerical approximation for the fractional diffusion equation”, J. Comput. Phys., 213:1 (2006), 205–213 | DOI | MR | Zbl

[27] A.V. Pskhu, Partial differential equations of fractional order, Nauka, M., 2005 | MR | Zbl

[28] Yu. Luchko, “Maximum principle for the generalized time-fractional diffusion equation”, J. Math. Anal. Appl., 351:1 (2009), 218–223 | DOI | MR | Zbl

[29] Yu. Luchko, “Boundary value problems for the generalized time-fractional diffusion equation of distributed order”, Fract. Calc. Appl. Anal., 12:4 (2009), 409–422 | MR | Zbl

[30] Yu. Luchko, “Maximum principle and its application for the time-fractional diffusion equations”, Fract. Calc. Appl. Anal., 14:1 (2011), 110–124 | DOI | MR | Zbl

[31] Yu. Luchko, “Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation”, Comput. Math. Appl., 59:5 (2010), 1766–1772 | DOI | MR | Zbl

[32] Juan J. Nieto, “Maximum principles for fractional differential equations derived from Mittag-Leffler functions”, Appl. Math. Lett., 23:10 (2010), 1248–1251 | DOI | MR | Zbl

[33] H. Ye, F. Liu, V. Anh, I. Turner, “Maximum principle and numerical method for the multi-term time-space Riesz-Caputo fractional differential equations”, Appl. Math. Comput., 227 (2014), 531–540 | MR | Zbl

[34] M.M. Meerschaert, H.-P. Scheffler, C. Tadjeran, “Finite difference methods for two-dimensional fractional dispersion equation”, J. Comput. Phys., 211 (2006), 249–261 | DOI | MR | Zbl

[35] W. Tian, H. Zhou, W. Deng, “A class of second order difference approximation for solving space fractional diffusion equations”, Math. Comput., 84:294 (2015), 1703–1727 | DOI | MR | Zbl

[36] X.-Q. Jin, F.-R. Lin, Z. Zhao, “Preconditioned iterative methods for two-dimensional space-fractional diffusion equations”, Commun. Comput. Phys., 18:2 (2015), 469–488 | DOI | MR

[37] X.-L. Lin, M.K. Ng, “A fast solver multidimensional time-space fractional diffusion equation with variable coefficients”, Comput. Math. Appl., 78:5 (2029), 1477–1489 | DOI | MR | Zbl

[38] X.-L. Lin, M.K. Ng., H.-W. Sun, “Stability and convergence analysis of finite difference schemes for time-dependent space-fractional diffusion equation with variable diffusion coefficients”, J. Sci. Comput., 75:2 (2018), 1102–1127 | DOI | MR | Zbl

[39] A.K. Bazzaev, M.Kh. Shhanukov-Lafishev, “On the convergence of difference schemes for fractional differential equations with Robin boundary conditions”, Comput. Math. Math. Phys., 57:1 (2017), 133–144 | DOI | MR | Zbl

[40] A.A. Alikhanov, “Stability and convergence of difference schemes for boundary value problems for the fractional-order diffusion equation”, Comput. Math. Math. Phys., 56:4 (2016), 561–575 | DOI | MR | Zbl

[41] A.A. Alikhanov, A.M. Apekov, A.Kh. Khibiev, “Higher-order approximation difference scheme for the generalized aller equation of fractional order”, Vladikavkaz. Mat. Zh., 23:3 (2021), 5–15 | MR | Zbl

[42] A.A. Samarskii, A.V. Gulin, Stability of difference schemes, Nauka, M., 1973 | MR | Zbl

[43] A.A. Samarskii, Theory of difference schemes, Nauka, M., 1977 | MR | Zbl

[44] A.A. Samarskiy, P.N. Vabishchevich, Vychislitel'naya teploperedacha, Editorial URSS, M., 2003