A novel finite difference based numerical approach for Modified Atangana- Baleanu Caputo derivative

Reetika Chawla; Komal Deswal; Devendra Kumar; Dumitru Baleanu; Reetika Chawla; Komal Deswal; Devendra Kumar; Dumitru Baleanu

doi:10.3934/math.2022950

AIMS Mathematics

2022, Volume 7, Issue 9: 17252-17268. doi: 10.3934/math.2022950

Previous Article Next Article

Research article Special Issues

A novel finite difference based numerical approach for Modified Atangana- Baleanu Caputo derivative

1.
Department of Mathematics, Birla Institute of Technology and Science, Pilani, Rajasthan-333031, India
2.
Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Cankaya University, Ankara 06530, Turkey
3.
Institute of Space Science, Magurle-Bucharest 077125, Romania

Received: 23 May 2022 Revised: 29 June 2022 Accepted: 07 July 2022 Published: 25 July 2022
MSC : 26A33, 35R11, 65M06, 65M12, 65N22

In this paper, a new approach is presented to investigate the time-fractional advection-dispersion equation that is extensively used to study transport processes. The present modified fractional derivative operator based on Atangana-Baleanu's definition of a derivative in the Caputo sense involves singular and non-local kernels. A numerical approximation of this new modified fractional operator is provided and applied to an advection-dispersion equation. Through Fourier analysis, it has been proved that the proposed scheme is unconditionally stable. Numerical examples are solved that validate the theoretical results presented in this paper and ensure the proficiency of the numerical scheme.
- fractional derivative,
- advection dispersion equation,
- finite difference method
Citation: Reetika Chawla, Komal Deswal, Devendra Kumar, Dumitru Baleanu. A novel finite difference based numerical approach for Modified Atangana- Baleanu Caputo derivative[J]. AIMS Mathematics, 2022, 7(9): 17252-17268. doi: 10.3934/math.2022950

Related Papers:

Abstract

In this paper, a new approach is presented to investigate the time-fractional advection-dispersion equation that is extensively used to study transport processes. The present modified fractional derivative operator based on Atangana-Baleanu's definition of a derivative in the Caputo sense involves singular and non-local kernels. A numerical approximation of this new modified fractional operator is provided and applied to an advection-dispersion equation. Through Fourier analysis, it has been proved that the proposed scheme is unconditionally stable. Numerical examples are solved that validate the theoretical results presented in this paper and ensure the proficiency of the numerical scheme.

References

[1]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Amsterdam: Elsevier, 2006.
[2]	I. Podlubny, Fractional differential equations, San Diego: Academic Press, 1999.
[3]	H. Scher, E. W. Montroll, Anomalous transit-time dispersion in amorphous solids, Phys. Rev. B, 12 (1975), 2455–2477. https://doi.org/10.1103/PhysRevB.12.2455 doi: 10.1103/PhysRevB.12.2455
[4]	I. M. Sokolov, J. Klafter, A. Blumen, Ballistic versus diffusive pair-dispersion in the Richardson regime, Phys. Rev. E, 61 (2000), 2717–2722. https://doi.org/10.1103/PhysRevE.61.2717 doi: 10.1103/PhysRevE.61.2717
[5]	T. L. Szabo, J. Wu, A model for longitudinal and shear wave propagation in viscoelastic media, J. Acoust. Soc. Am., 107 (2000), 2437–2446. https://doi.org/10.1121/1.428630 doi: 10.1121/1.428630
[6]	R. Hilfer, Applications of fractional calculus in physics, Singapore: World Scientific, 2000. https://doi.org/10.1142/3779
[7]	Z. Liu, X. Li, A Crank-Nicolson difference scheme for the time variable fractional mobile-immobile advection-dispersion equation, J. Appl. Math. Comput., 56 (2018), 391–410. https://doi.org/10.1007/s12190-016-1079-7 doi: 10.1007/s12190-016-1079-7
[8]	M. M. Meerschaert, C. Tadjeran, Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math., 172 (2004), 65–77. https://doi.org/10.1016/j.cam.2004.01.033 doi: 10.1016/j.cam.2004.01.033
[9]	M. M. Meerschaert, C. Tadjeran, Finite difference approximations for two-sided space-fractional partial differential equations, Appl. Numer. Math., 56 (2006), 80–90. https://doi.org/10.1016/j.apnum.2005.02.008 doi: 10.1016/j.apnum.2005.02.008
[10]	M. Yaseen, M. Abbas, A. I. Ismail, T. Nazir, A cubic trigonometric B-spline collocation approach for the fractional sub-diffusion equations, Appl. Math. Comput., 293 (2017), 311–319. https://doi.org/10.1016/j.amc.2016.08.028 doi: 10.1016/j.amc.2016.08.028
[11]	M. Stynes, Singularities, In: Handbook of fractional calculus with applications, Volume 3, Walter de Gruyter GmbH, 2019,287–305. https://doi.org/10.1515/9783110571684-011
[12]	M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 73–85. http://doi.org/10.12785/pfda/010201 doi: 10.12785/pfda/010201
[13]	D. Baleanu, A. Mousalou, S. Rezapour, A new method for investigating approximate solutions of some fractional integro-differential equations involving the Caputo–Fabrizio derivative, Adv. Differ. Equ., 2017 (2017), 51. https://doi.org/10.1186/s13662-017-1088-3 doi: 10.1186/s13662-017-1088-3
[14]	Z. Liu, A. J. Cheng, X. Li, A second order finite difference scheme for quasilinear time fractional parabolic equation based on new fractional derivative, Int. J. Comput. Math., 95 (2018), 396–411. https://doi.org/10.1080/00207160.2017.1290434 doi: 10.1080/00207160.2017.1290434
[15]	M. Zhang, Y. Liu, H. Li, High-order local discontinuous Galerkin method for a fractal mobile/immobile transport equation with the Caputo-Fabrizio fractional derivative, Numer. Method. Part. Differ. Equ., 35 (2019), 1588–1612. https://doi.org/10.1002/num.22366 doi: 10.1002/num.22366
[16]	A. Atangana, D. Baleanu, New fractional derivatives with non-local and non-singular kernel: theory and application to heat transfer model, Therm. Sci., 20 (2016), 763–769. https://doi.org/10.2298/TSCI160111018A doi: 10.2298/TSCI160111018A
[17]	N. Sene, K. Abdelmalek, Analysis of the fractional diffusion equations described by Atangana-Baleanu-Caputo fractional derivative, Chaos Soliton. Fract., 127 (2019), 158–164. https://doi.org/10.1016/j.chaos.2019.06.036 doi: 10.1016/j.chaos.2019.06.036
[18]	M. Shafiq, M. Abbas, K. M. Abualnaja, M. J. Huntul, A. Majeed, T. Nazir, An efficient technique based on cubic B-spline functions for solving time-fractional advection diffusion equation involving Atangana-Baleanu derivative, Eng. Comput., 38 (2022), 901–917. https://doi.org/10.1007/s00366-021-01490-9 doi: 10.1007/s00366-021-01490-9
[19]	H. Tajadodi, A Numerical approach of fractional advection-diffusion equation with Atangana-Baleanu derivative, Chaos Soliton. Fract., 130 (2020), 109527. https://doi.org/10.1016/j.chaos.2019.109527 doi: 10.1016/j.chaos.2019.109527
[20]	M. A. Refai, D. Baleanu, On an extension of the operator with mittag-leffler kernel, Fractals, in press. https://doi.org/10.1142/S0218348X22401296
[21]	F. Liu, P. Zhuang, K. Burrage, Numerical methods and analysis for a class of fractional advection-dispersion models, Comput. Math. Appl., 64 (2012), 2990–3007. https://doi.org/10.1016/j.camwa.2012.01.020 doi: 10.1016/j.camwa.2012.01.020
[22]	M. K. Singh, A. Chatterjee, V. P. Singh, Solution of one-dimensional time fractional advection dispersion equation by homotopy analysis method, J. Eng. Mech., 143 (2017), 04017103. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001318 doi: 10.1061/(ASCE)EM.1943-7889.0001318
[23]	Y. Zhang, D. A. Benson, D. M. Reeves, Time and space nonlocalities underlying fractional-derivative models: Distinction and literature review of field applications, Adv. Water Resour., 32 (2009), 561–581. https://doi.org/10.1016/j.advwatres.2009.01.008 doi: 10.1016/j.advwatres.2009.01.008
[24]	K. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, New York: Wiley, 1993.
[25]	K. B. Oldham, J. Spanier, The fractional calculus, New York: Academic Press, 1974.
[26]	A. Jannelli, M. Ruggieri, M. P. Speciale, Analytical and numerical solutions of time and space fractional advection-diffusion-reaction equation, Commun. Nonlinear Sci. Numer. Simul., 70 (2019), 89–101. https://doi.org/10.1016/j.cnsns.2018.10.012 doi: 10.1016/j.cnsns.2018.10.012

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)