An efficient modified hybrid explicit group iterative method for the time-fractional diffusion equation in two space dimensions

Fouad Mohammad Salama; Nur Nadiah Abd Hamid; Norhashidah Hj. Mohd Ali; Umair Ali; Fouad Mohammad Salama; Nur Nadiah Abd Hamid; Norhashidah Hj. Mohd Ali; Umair Ali

doi:10.3934/math.2022134

AIMS Mathematics

2022, Volume 7, Issue 2: 2370-2392. doi: 10.3934/math.2022134

Previous Article Next Article

Research article

An efficient modified hybrid explicit group iterative method for the time-fractional diffusion equation in two space dimensions

1.
School of Mathematical Sciences, Universiti Sains Malaysia, Penang 11800, Malaysia
2.
Department of Applied Mathematics and Statistics, Institute of Space Technology, P. O. Box 2750, Islamabad 44000, Pakistan

Received: 31 July 2021 Accepted: 04 November 2021 Published: 11 November 2021
MSC : 35XX, 65N12

In this paper, a new modified hybrid explicit group (MHEG) iterative method is presented for the efficient and accurate numerical solution of a time-fractional diffusion equation in two space dimensions. The time fractional derivative is defined in the Caputo sense. In the proposed method, a Laplace transformation is used in the temporal domain, and, for the spatial discretization, a new finite difference scheme based on grouping strategy is considered. The unique solvability, unconditional stability and convergence are thoroughly proved by the matrix analysis method. Comparison of numerical results with analytical and other approximate solutions indicates the viability and efficiency of the proposed algorithm.
- Caputo fractional derivative,
- fractional diffusion equation,
- Laplace transform,
- finite difference scheme,
- grouping strategy,
- stability and convergence
Citation: Fouad Mohammad Salama, Nur Nadiah Abd Hamid, Norhashidah Hj. Mohd Ali, Umair Ali. An efficient modified hybrid explicit group iterative method for the time-fractional diffusion equation in two space dimensions[J]. AIMS Mathematics, 2022, 7(2): 2370-2392. doi: 10.3934/math.2022134

Related Papers:

Abstract

In this paper, a new modified hybrid explicit group (MHEG) iterative method is presented for the efficient and accurate numerical solution of a time-fractional diffusion equation in two space dimensions. The time fractional derivative is defined in the Caputo sense. In the proposed method, a Laplace transformation is used in the temporal domain, and, for the spatial discretization, a new finite difference scheme based on grouping strategy is considered. The unique solvability, unconditional stability and convergence are thoroughly proved by the matrix analysis method. Comparison of numerical results with analytical and other approximate solutions indicates the viability and efficiency of the proposed algorithm.

References

[1]	H. R. Ghehsareh, A. Zaghian, S. M. Zabetzadeh, The use of local radial point interpolation method for solving two-dimensional linear fractional cable equation, Neural Comput. Appl., 29 (2018), 745–754. doi: 10.1007/s00521-016-2595-y. doi: 10.1007/s00521-016-2595-y
[2]	Y. L. Zhao, T. Z. Huang, X. M. Gu, W. H. Luo, A fast second-order implicit difference method for time-space fractional advection-diffusion equation, Numer. Func. Anal. Opt., 41 (2020), 257–293. doi: 10.1080/01630563.2019.1627369. doi: 10.1080/01630563.2019.1627369
[3]	M. Hussain, S. Haq, Weighted meshless spectral method for the solutions of multi-term time fractional advection-diffusion problems arising in heat and mass transfer, Int. J. Heat Mass Tran., 129 (2019), 1305–1316. doi: 10.1016/j.ijheatmasstransfer.2018.10.039. doi: 10.1016/j.ijheatmasstransfer.2018.10.039
[4]	M. Abbaszadeh, A. Mohebbi, A fourth-order compact solution of the two dimensional modified anomalous fractional sub-diffusion equation with a nonlinear source term, Comput. Math. Appl., 66 (2013), 1345–1359. doi: 10.1016/j.camwa.2013.08.010. doi: 10.1016/j.camwa.2013.08.010
[5]	G. M. Zaslavsky, Chaos, fractional kinetics, and anomalous transport, Phys. Rep., 371 (2002), 461–580. doi: 10.1016/S0370-1573(02)00331-9. doi: 10.1016/S0370-1573(02)00331-9
[6]	S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives: Theory and applications, Yverdon: Gordon and Breach, 1993.
[7]	I. Podlubny, Fractional differential equations, mathematics in science and engineering, New York: Academic Press, 1999.
[8]	B. Guo, X. Pu, F. Huang, Fractional partial differential equations and their numerical solutions, Singapore: World Scientific, 2015.
[9]	A. Kochubei, Y. Luchko, V. E. Tarasov, I. Petra, Handbook of fractional calculus with applications, Berlin: De Gruyter Grand Forks, 2019.
[10]	J. Y. Shen, Z. Z. Sun, R. Du, Fast finite difference schemes for time fractional diffusion equations with a weak singularity at initial time, E. Asian J. Appl. Math., 8 (2018), 834–858. doi: 10.4208/eajam.010418.020718. doi: 10.4208/eajam.010418.020718
[11]	A. Chen, C. Li, A novel compact adi scheme for the time-fractional subdiffusion equation in two space dimensions, Int. J. Comput. Math., 93 (2016), 889–914. doi: 10.1080/00207160.2015.1009905. doi: 10.1080/00207160.2015.1009905
[12]	G. H. Gao, Z. Z. Sun, Y. N. Zhang, A finite difference scheme for fractional sub-diffusion equations on an unbounded domain using artificial boundary conditions, J. Comput. Phys., 231 (2012), 2865–2879. doi: 10.1016/j.jcp.2011.12.028. doi: 10.1016/j.jcp.2011.12.028
[13]	M. Tamsir, N. Dhiman, D. Nigam, A. Chauhan, Approximation of Caputo time-fractional diffusion equation using redefined cubic exponential B-spline collocation technique, AIMS Mathematics, 6 (2021), 3805–3820. doi: 10.3934/math.2021226. doi: 10.3934/math.2021226
[14]	J. Shen, X. M. Gu, Two finite difference methods based on an H2N2 interpolation for two-dimensional time fractional mixed diffusion and diffusion-wave equations, Discrete Cont. Dyn. B, 2021. doi: 10.3934/dcdsb.2021086.
[15]	X. M. Gu, Y. L. Zhao, X. L. Zhao, B. Carpentieri, Y. Y. Huang, A note on parallel preconditioning for the all-at-once solution of Riesz fractional diffusion equations, Numer. Math. Theor. Meth. Appl., 14 (2021), 893–919. doi: 10.4208/nmtma.OA-2020-0020. doi: 10.4208/nmtma.OA-2020-0020
[16]	X. M. Gu, H. W. Sun, Y. L. Zhao, X. Zheng, An implicit difference scheme for time-fractional diffusion equations with a time-invariant type variable order, Appl. Math. Lett., 120 (2021), 107270. doi: 10.1016/j.aml.2021.107270. doi: 10.1016/j.aml.2021.107270
[17]	Y. Xu, Y. Zhang, J. Zhao, Backward difference formulae and spectral galerkin methods for the riesz space fractional diffusion equation, Math. Comput. Simulat., 166 (2019), 494–507. doi: 10.1016/j.matcom.2019.07.007. doi: 10.1016/j.matcom.2019.07.007
[18]	X. Gao, B. Yin, H. Li, Y. Liu, Tt-m FE method for a 2D nonlinear time distributed-order and space fractional diffusion equation, Math. Comput. Simulat., 181 (2021), 117–137. doi: 10.1016/j.matcom.2020.09.021. doi: 10.1016/j.matcom.2020.09.021
[19]	X. Zheng, H. Wang, An error estimate of a numerical approximation to a hidden-memory variable-order space-time fractional diffusion equation, SIAM J. Numer. Anal., 58 (2020), 2492–2514. doi: 10.1137/20M132420X. doi: 10.1137/20M132420X
[20]	X. Zheng, H. Wang, A Hidden-memory variable-order time-fractional optimal control model: Analysis and approximation, SIAM J. Control Optim., 59 (2021), 1851–1880. doi: 10.1137/20M1344962. doi: 10.1137/20M1344962
[21]	X. Zheng, H. Wang, Optimal-order error estimates of finite element approximations to variable-order time-fractional diffusion equations without regularity assumptions of the true solutions, IMA J. Numer. Anal., 41 (2021), 1522–1545. doi: 10.1093/imanum/draa013. doi: 10.1093/imanum/draa013
[22]	H. Wang, X. Zheng, Wellposedness and regularity of the variable-order time-fractional diffusion equations, J. Math. Anal. Appl., 475 (2019), 1778–1802. doi: 10.1016/j.jmaa.2019.03.052. doi: 10.1016/j.jmaa.2019.03.052
[23]	Y. Luchko, Initial-boundary-value problems for the one-dimensional time-fractional diffusion equation, Fract. Calc. Appl. Anal., 15 (2012), 141–160. doi: 10.2478/s13540-012-0010-7. doi: 10.2478/s13540-012-0010-7
[24]	B. Jin, B. Li, Z. Zhou, Numerical analysis of nonlinear subdiffusion equations, SIAM J. Numer. Anal., 56 (2018), 1–23. doi: 10.1137/16M1089320. doi: 10.1137/16M1089320
[25]	X. Zheng, H. Wang, Wellposedness and regularity of a variable-order space-time fractional diffusion equation, Anal. Appl., 18 (2020), 615–638. doi: 10.1142/S0219530520500013. doi: 10.1142/S0219530520500013
[26]	H. Fu, M. K. Ng, H. Wang, A divide-and-conquer fast finite difference method for space-time fractional partial differential equation, Comput. Math. Appl., 73 (2017), 1233–1242. doi: 10.1016/j.camwa.2016.11.023. doi: 10.1016/j.camwa.2016.11.023
[27]	F. M. Salama, N. H. M. Ali, Fast O(N) hybrid method for the solution of two dimensional time fractional cable equation, Compusoft, 8 (2019), 3453–3461.
[28]	F. M. Salama, N. H. M. Ali, Computationally efficient hybrid method for the numerical solution of the 2D time fractional advection-diffusion equation, Int. J. Math. Eng. Manag., 5 (2020), 432–446. doi: 10.33889/IJMEMS.2020.5.3.036. doi: 10.33889/IJMEMS.2020.5.3.036
[29]	X. M. Gu, S. L. Wu, A parallel-in-time iterative algorithm for Volterra partial integro-differential problems with weakly singular kernel, J. Comput. Phys., 417 (2020), 109576. doi: 10.1016/j.jcp.2020.109576. doi: 10.1016/j.jcp.2020.109576
[30]	X. L. Lin, M. K. Ng, H. W. Sun, A multigrid method for linear systems arising from time-dependent two-dimensional space-fractional diffusion equations, J. Comput. Phys., 336 (2017), 69–86. doi: 10.1016/j.jcp.2017.02.008. doi: 10.1016/j.jcp.2017.02.008
[31]	J. Ren, Z. Z. Sun, W. Dai, New approximations for solving the caputo-type fractional partial differential equations, Appl. Math. Model., 40 (2016), 2625–2636. doi: 10.1016/j.apm.2015.10.011. doi: 10.1016/j.apm.2015.10.011
[32]	N. A. Khan, S. Ahmed, Finite difference method with metaheuristic orientation for exploration of time fractional partial differential equations, Int. J. Appl. Comput. Math., 7 (2021), 1–22. doi: 10.1007/s40819-021-01061-y. doi: 10.1007/s40819-021-01061-y
[33]	A. Ahmadian, S. Salahshour, M. Ali-Akbari, F. Ismail, D. Baleanu, A novel approach to approximate fractional derivative with uncertain conditions, Chaos Soliton. Fract., 104 (2017), 68–76. doi: 10.1016/j.chaos.2017.07.026. doi: 10.1016/j.chaos.2017.07.026
[34]	F. M. Salama, N. H. M. Ali, N. N. Abd Hamid, Fast O(N) hybrid Laplace transform-finite difference method in solving 2D time fractional diffusion equation, J. Math. Comput. Sci., 23 (2021), 110–123. doi: 10.22436/jmcs.023.02.04. doi: 10.22436/jmcs.023.02.04
[35]	N. H. M. Ali, L. M. Kew, New explicit group iterative methods in the solution of two dimensional hyperbolic equations, J. Comput. Phys., 231 (2012), 6953–6968. doi: 10.1016/j.jcp.2012.06.025. doi: 10.1016/j.jcp.2012.06.025
[36]	N. H. Mohd Ali, A. Mohammed Saeed, Convergence analysis of the preconditioned group splitting methods in boundary value problems, Abstr. Appl. Anal., 2012 (2012), 867598. doi: 10.1155/2012/867598. doi: 10.1155/2012/867598
[37]	L. M. Kew, N. H. M. Ali, New explicit group iterative methods in the solution of three dimensional hyperbolic telegraph equations, J. Comput. Phys., 294 (2015), 382–404. doi: 10.1016/j.jcp.2015.03.052. doi: 10.1016/j.jcp.2015.03.052
[38]	A. Saudi, J. Sulaiman, Robot path planning using four point-explicit group via nine-point laplacian (4EG9L) iterative method, Procedia Engineering, 41 (2012), 182–188. doi: 10.1016/j.proeng.2012.07.160. doi: 10.1016/j.proeng.2012.07.160
[39]	N. H. M. Ali, A. M. Saeed, Preconditioned modified explicit decoupled group for the solution of steady state navier-stokes equation, Appl. Math. Inform. Sci., 7 (2013), 1837–1844. doi: 10.12785/amis/070522. doi: 10.12785/amis/070522
[40]	M. A. Khan, N. H. M. Ali, N. N. Abd Hamid, A new fourth-order explicit 270 group method in the solution of two-dimensional fractional rayleigh–stokes problem for a heated generalized second-grade fluid, Adv. Diff. Equ., 2020 (2020), 598. doi: 10.1186/s13662-020-03061-6. doi: 10.1186/s13662-020-03061-6
[41]	N. Abdi, H. Aminikhah, A. H. Sheikhani, J. Alavi, M. Taghipour, An efficient explicit decoupled group method for solving two–dimensional fractional Burgers' equation and its convergence analysis, Adv. Math. Phys., 2021 (2021), 6669287. doi: 10.1155/2021/6669287. doi: 10.1155/2021/6669287
[42]	N. Abdi, H. Aminikhah, A. R. Sheikhani, High-order rotated grid point iterative method for solving 2D time fractional telegraph equation and its convergence analysis, Comp. Appl. Math., 40 (2021), 54. doi: 10.1007/s40314-021-01451-4. doi: 10.1007/s40314-021-01451-4
[43]	F. M. Salama, N. H. M. Ali, N. N. Abd Hamid, Efficient hybrid group iterative methods in the solution of two-dimensional time fractional cable equation, Adv. Differ. Equ., 2020 (2020), 257. doi: 10.1186/s13662-020-02717-7. doi: 10.1186/s13662-020-02717-7
[44]	N. Moraca, Bounds for norms of the matrix inverse and the smallest singular value, Linear Algebra Appl., 429 (2008), 2589–2601. doi: 10.1016/j.laa.2007.12.026. doi: 10.1016/j.laa.2007.12.026
[45]	A. Ali, N. H. M. Ali, On skewed grid point iterative method for solving 2d hyperbolic telegraph fractional differential equation, Adv. Differ. Equ., 2019 (2019), 303. doi: 10.1186/s13662-019-2238-6. doi: 10.1186/s13662-019-2238-6

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)