×

zbMATH — the first resource for mathematics

A high-order scheme for fractional ordinary differential equations with the Caputo-Fabrizio derivative. (English) Zbl 07172818
Summary: In this paper, we consider numerical solutions of fractional ordinary differential equations with the Caputo-Fabrizio derivative, and construct and analyze a high-order time-stepping scheme for this equation. The proposed method makes use of quadratic interpolation function in sub-intervals, which allows to produce fourth-order convergence. A rigorous stability and convergence analysis of the proposed scheme is given. A series of numerical examples are presented to validate the theoretical claims. Traditionally a scheme having fourth-order convergence could only be obtained by using block-by-block technique. The advantage of our scheme is that the solution can be obtained step by step, which is cheaper than a block-by-block-based approach.
MSC:
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
34A08 Fractional ordinary differential equations and fractional differential inclusions
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
Software:
FODE
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Akman, T.; Yıldız, B.; Baleanu, D., New discretization of Caputo-Fabrizio derivative, Comput. Appl. Math., 37, 3, 3307-3333 (2018) · Zbl 1425.35207
[2] Alikhanov, Aa, A new difference scheme for the time fractional diffusion equation, J. Comput. Phys., 280, 424-438 (2015) · Zbl 1349.65261
[3] Atangana, A., On the new fractional derivative and application to nonlinear Fisher’s reaction-diffusion equation, Appl. Math. Comput., 273, 948-956 (2016) · Zbl 1410.35272
[4] Atangana, A.; Alqahtani, R., Numerical approximation of the space-time Caputo-Fabrizio fractional derivative and application to groundwater pollution equation, Adv. Differ. Equ., 2016, 1, 156 (2016) · Zbl 1422.65144
[5] Atangana, A.; Gómez-Aguilar, J., Numerical approximation of Riemann-Liouville definition of fractional derivative: from Riemann-Liouville to Atangana-Baleanu, Numer. Methods Partial Differ. Equ., 34, 5, 1502-1523 (2017) · Zbl 1417.65113
[6] Atangana, A.; Nieto, J., Numerical solution for the model of RLC circuit via the fractional derivative without singular kernel, Adv. Mech. Eng., 7, 10, 1-7 (2015)
[7] Baffet, D.; Hesthaven, J., A kernel compression scheme for fractional differential equations, SIAM J. Numer. Anal., 55, 2, 496-520 (2017) · Zbl 1359.65106
[8] Baffet, D.; Hesthaven, J., High-order accurate adaptive kernel compression time-stepping schemes for fractional differential equations, J. Sci. Comput., 72, 3, 1169-1195 (2017) · Zbl 1376.65104
[9] Cao, J.; Xu, C., A high order schema for the numerical solution of the fractional ordinary differential equations, J. Comput. Phys., 238, 154-168 (2013) · Zbl 1286.65092
[10] Caputo, M.; Fabrizio, M., A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1, 2, 73-85 (2015)
[11] Deng, Wh, Finite element method for the space and time fractional Fokker-Planck equation, SIAM J. Numer. Anal., 47, 1, 204-226 (2008)
[12] Djida, J.; Area, I.; Atangana, A., Numerical computation of a fractional derivative with non-local and non-singular kernel, Math. Model. Nat. Phenom., 12, 3, 4-13 (2017) · Zbl 1416.65190
[13] Firoozjaee, M.; Jafari, H.; Lia, A.; Baleanu, D., Numerical approach of Fokker-Planck equation with Caputo-Fabrizio fractional derivative using Ritz approximation, J. Comput. Appl. Math., 339, 367-373 (2018) · Zbl 1393.65029
[14] Gao, G.; Sun, Z.; Zhang, H., A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications, J. Comput. Phys., 259, 33-50 (2014) · Zbl 1349.65088
[15] Gómez-Aguilar, J.F., Yépez-Martínez, H., Escobar-Jiménez, R.F., et al.: Series solution for the time-fractional coupled mKdV equation using the homotopy analysis method. Math. Probl. Eng. 2016, (2016) · Zbl 1400.35066
[16] Gómez-Aguilar, J., Space-time fractional diffusion equation using a derivative with nonsingular and regular kernel, Phys. A, 465, 562-572 (2017) · Zbl 1400.82228
[17] Hou, D.; Xu, C., A fractional spectral method with applications to some singular problems, Adv. Comput. Math., 343, 5, 911-944 (2017) · Zbl 1382.65467
[18] Huang, J.; Tang, Y.; Vázquez, L., Convergence analysis of a block-by-block method for fractional differential equations, Numer. Math. Theor. Methods Appl., 5, 2, 229-241 (2012) · Zbl 1265.65141
[19] Jin, B.; Lazarov, R.; Zhou, Z., Two fully discrete schemes for fractional diffusion and diffusion-wave equations with nonsmooth data, SIAM J. Sci. Comput., 38, 1, A146-A170 (2016) · Zbl 1381.65082
[20] Ke, Rh; Ng, Mk; Sun, Hw, A fast direct method for block triangular Toeplitz-like with tri-diagonal block systems from time-fractional partial differential equations, J. Comput. Phys., 303, 203-211 (2015) · Zbl 1349.65404
[21] Lin, Y.; Xu, C., Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225, 2, 1533-1552 (2007) · Zbl 1126.65121
[22] Liu, F.; Shen, S.; Anh, V.; Turner, I., Analysis of a discrete non-Markovian random walk approximation for the time fractional diffusion equation, ANZIAM J., 46, E, C488-C504 (2005) · Zbl 1082.60511
[23] Liu, Z.; Cheng, A.; Li, X., A second order Crank-Nicolson scheme for fractional Cattaneo equation based on new fractional derivative, Appl. Math. Comput., 311, 361-374 (2017) · Zbl 1427.65173
[24] Liu, Z.; Cheng, A.; Li, X., A second-order finite difference scheme for quasilinear time fractional parabolic equation based on new fractional derivative, Int. J. Comput. Math., 95, 2, 396-411 (2018) · Zbl 1390.65075
[25] Mclean, W., Fast summation by interval clustering for an evolution equation with memory, J. Sci. Comput., 34, 6, A3039-A3056 (2012) · Zbl 1276.65059
[26] Podlubny, I., Fractional Differential Equations (1999), New York: Acad. Press, New York · Zbl 0918.34010
[27] Saad, Km; Khader, Mm; Gómez-Aguilar, Jf; Baleanu, Dumitru, Numerical solutions of the fractional Fisher’s type equations with Atangana-Baleanu fractional derivative by using spectral collocation methods, Chaos, 29, 2, 023116 (2019) · Zbl 1409.35225
[28] Shah, N.; Khan, I., Heat transfer analysis in a second grade fluid over and oscillating vertical plate using fractional Caputo-Fabrizio derivatives, Eur. Phys. J. C, 76, 362 (2016)
[29] Stynes, M.; O’Riordan, E.; Gracia, J., Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation, SIAM J. Numer. Anal., 55, 2, 1057-1079 (2017) · Zbl 1362.65089
[30] Sun, Z.; Wu, X., A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math., 56, 2, 193-209 (2006) · Zbl 1094.65083
[31] Yan, Y.; Sun, Z.; Zhang, J., Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations: a second-order scheme, Commun. Comput. Phys., 22, 4, 1028-1048 (2017)
[32] Yang, X.; Machado, J., A new fractional operator of variable order: application in the description of anomalous diffusion, Phys. A, 481, 276-283 (2017)
[33] Yang, J.; Huang, J.; Liang, D.; Tang, Y., Numerical solution of fractional diffusion-wave equation based on fractional multistep method, Appl. Math. Model., 38, 14, 3652-3661 (2014) · Zbl 1427.65196
[34] Zeng, F.; Turner, I.; Burrage, K., A stable fast time-stepping method for fractional integral and derivative operators, J. Sci. Comput., 77, 283-307 (2018) · Zbl 1406.65047
[35] Zhang, Q.; Zhang, J.; Jiang, S.; Zhang, Z., Numerical solution to a linearized time fractional KdV equation on unbounded domains, Math. Comp., 87, 310, 693-719 (2018) · Zbl 1381.65071
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.