Geodesic
Crank-Nicolson scheme for two-dimensional in space fractional diffusion equations with functional delay
Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 57 (2021), pp. 128-141.

Voir la notice de l'article provenant de la source Math-Net.Ru

A two-dimensional in space fractional diffusion equation with functional delay of a general form is considered. For this problem, the Crank-Nicolson method is constructed, based on shifted Grunwald-Letnikov formulas for approximating fractional derivatives with respect to each spatial variable and using piecewise linear interpolation of discrete history with continuation extrapolation to take into account the delay effect. The Douglas scheme is used to reduce the emerging high-dimensional system to tridiagonal systems. The residual of the method is investigated. To obtain the order of the method, we reduce the systems to constructions of the general difference scheme with heredity. A theorem on the second order of convergence of the method in time and space steps is proved. The results of numerical experiments are presented.
Keywords: two spatial coordinates, functional delay, Grunwald-Letnikov approximation, Crank-Nicolson method, factorization, order of convergence.
Mots-clés : diffusion equation 
@article{IIMI_2021_57_a4,
     author = {M. Ibrahim and V. G. Pimenov},
     title = {Crank-Nicolson scheme for two-dimensional in space fractional diffusion equations with functional delay},
     journal = {Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta},
     pages = {128--141},
     publisher = {mathdoc},
     volume = {57},
     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IIMI_2021_57_a4/}
}
TY  - JOUR
AU  - M. Ibrahim
AU  - V. G. Pimenov
TI  - Crank-Nicolson scheme for two-dimensional in space fractional diffusion equations with functional delay
JO  - Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta
PY  - 2021
SP  - 128
EP  - 141
VL  - 57
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IIMI_2021_57_a4/
LA  - en
ID  - IIMI_2021_57_a4
ER  - 
%0 Journal Article
%A M. Ibrahim
%A V. G. Pimenov
%T Crank-Nicolson scheme for two-dimensional in space fractional diffusion equations with functional delay
%J Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta
%D 2021
%P 128-141
%V 57
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IIMI_2021_57_a4/
%G en
%F IIMI_2021_57_a4
M. Ibrahim; V. G. Pimenov. Crank-Nicolson scheme for two-dimensional in space fractional diffusion equations with functional delay. Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 57 (2021), pp. 128-141. http://geodesic.mathdoc.fr/item/IIMI_2021_57_a4/

[1] V. G. Pimenov, Raznostnye metody resheniya uravnenii v chastnykh proizvodnykh s nasledstvennostyu, Izd-vo Ural. un-ta, Ekaterinburg, 2014

[2] M. M. Meerschaert, C. Tadjeran, “Finite difference approximations for two-sided space-fractional partial differential equations”, Applied Numerical Mathematics, 56:1 (2006), 80–90 | DOI | MR | Zbl

[3] C. Tadjeran, M. M. Meerschaert, H. P. Scheffler, “A second-order accurate numerical approximation for the fractional diffusion equation”, Journal of Computational Physics, 213:1 (2006), 205–213 | DOI | MR | Zbl

[4] V. G. Pimenov, A. S. Hendy, “A fractional analog of Crank-Nicolson method for the two sided space fractional partial equation with functional delay”, Ural Mathematical Journal, 2:1 (2016), 48–57 | DOI | Zbl

[5] M. M. Meerschaert, H. P. Scheffler, C. Tadjeran, “Finite difference methods for two-dimensional fractional dispersion equation”, Journal of Computational Physics, 211:1 (2006), 249–261 | DOI | MR | Zbl

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

[7] X. Q. Jin, F. R. Lin, Z. Zhao, “Preconditioned iterative methods for two-dimensional space-fractional diffusion equations”, Communications in Computational Physics, 18:2 (2015), 469–488 | DOI | MR | Zbl

[8] 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”, Journal of Scientific Computing, 75:2 (2018), 1102–1127 | DOI | MR | Zbl

[9] X. L. Lin, M. K. Ng, “A fast solver for multidimensional time-space fractional diffusion equation with variable coefficients”, Computers and Mathematics with Applications, 78:5 (2019), 1477–1489 | DOI | MR | Zbl

[10] A. S. Hendy, J. E. Macías-Díaz, “A conservative scheme with optimal error estimates for a multidimension al space-fractional Gross-Pitaevskii equation”, International Journal of Applied Mathematics and Computer Science, 29:4 (2019), 713–723 | DOI | MR | Zbl

[11] X. Yue, S. Shu, X. Xu, W. Bu, K. Pan, “Parallel-in-time multigrid for space-time finite element approximations of two-dimensional space-fractional diffusion equations”, Computers and Mathematics with Applications, 78:11 (2019), 3471–3484 | DOI | MR | Zbl

[12] R. Du, A. A. Alikhanov, Z. Z. Sun, “Temporal second order difference schemes for the multidimensional variable-order time fractional sub-diffusion equations”, Computers and Mathematics with Applications, 79:10 (2020), 2952–2972 | DOI | MR | Zbl

[13] M. A. Zaky, J. T. Machado, “Multi-dimensional spectral tau methods for distributed-order fractional diffusion equations”, Computers and Mathematics with Applications, 79:2 (2020), 476–488 | DOI | MR | Zbl

[14] A. V. Kim, V. G. Pimenov, i-Gladkii analiz i chislennye metody resheniya funktsionalno-differentsialnykh uravnenii, Regulyarnaya i khaoticheskaya dinamika, M.–Izhevsk, 2004

[15] A. V. Lekomtsev, V. G. Pimenov, “Skhodimost metoda peremennykh napravlenii chislennogo resheniya uravneniya teploprovodnosti s zapazdyvaniem”, Trudy Instituta matematiki i mekhaniki UrO RAN, 16, no. 1, 2010, 102–118

[16] V. Pimenov, A. Hendy, M. Ibrahim, “Numerical method for two-dimensional space fractional equations with functional delay”, AIP Conference Proceedings, 2312 (2020), 050017, 1–6 | DOI