Single-step and multi-step methods for Caputo fractional-order differential equations with arbitrary kernels

Danuruj Songsanga; Parinya Sa Ngiamsunthorn; Danuruj Songsanga; Parinya Sa Ngiamsunthorn

doi:10.3934/math.2022822

AIMS Mathematics

2022, Volume 7, Issue 8: 15002-15028. doi: 10.3934/math.2022822

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Single-step and multi-step methods for Caputo fractional-order differential equations with arbitrary kernels

Danuruj Songsanga ¹,
Parinya Sa Ngiamsunthorn ^{1,2
,
,}

1.
Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi, Bangkok, 10140, Thailand
2.
Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Science Laboratory Building, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha Uthit Road, Bang Mod, Thung Khru, 10140, Bangkok, Thailand

Received: 21 April 2022 Revised: 05 June 2022 Accepted: 06 June 2022 Published: 14 June 2022
MSC : 26A33, 34A08, 65L05

We develop four numerical schemes to solve fractional differential equations involving the Caputo fractional derivative with arbitrary kernels. Firstly, we derive the four numerical schemes, namely, explicit product integration rectangular rule (forward Euler method), implicit product integration rectangular rule (backward Euler method), implicit product integration trapezoidal rule and Adam-type predictor-corrector method. In addition, the error estimation and stability for all four presented schemes are analyzed. To demonstrate the accuracy and effectiveness of the proposed methods, numerical examples are considered for various linear and nonlinear fractional differential equations with different kernels. The results show that theses numerical schemes are feasible in application.
- fractional differential equations,
- numerical methods,
- product integration,
- single-step method,
- multi-step method
Citation: Danuruj Songsanga, Parinya Sa Ngiamsunthorn. Single-step and multi-step methods for Caputo fractional-order differential equations with arbitrary kernels[J]. AIMS Mathematics, 2022, 7(8): 15002-15028. doi: 10.3934/math.2022822

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Abstract

We develop four numerical schemes to solve fractional differential equations involving the Caputo fractional derivative with arbitrary kernels. Firstly, we derive the four numerical schemes, namely, explicit product integration rectangular rule (forward Euler method), implicit product integration rectangular rule (backward Euler method), implicit product integration trapezoidal rule and Adam-type predictor-corrector method. In addition, the error estimation and stability for all four presented schemes are analyzed. To demonstrate the accuracy and effectiveness of the proposed methods, numerical examples are considered for various linear and nonlinear fractional differential equations with different kernels. The results show that theses numerical schemes are feasible in application.

References

[1]	O. P. Agrawal, Some generalized fractional calculus operators and their applications in integral equations, Fract. Calc. Appl. Anal., 15 (2012), 700–711. https://doi.org/10.2478/s13540-012-0047-7 doi: 10.2478/s13540-012-0047-7
[2]	R. Almeida, A caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Sci., 44 (2017), 460–481. https://doi.org/10.1016/j.cnsns.2016.09.006 doi: 10.1016/j.cnsns.2016.09.006
[3]	R. Almeida, M. Jleli, B. Samet, A numerical study of fractional relaxation-oscillation equations involving $\psi$-caputo fractional derivative, RACSAM, 113 (2019), 1873–1891. https://doi.org/10.1007/s13398-018-0590-0 doi: 10.1007/s13398-018-0590-0
[4]	R. Almeida, A. B. Malinowska, M. T. T. Monteiro, Fractional differential equations with a caputo derivative with respect to a kernel function and their applications, Math. Method. Appl. Sci., 41 (2018), 336–352. https://doi.org/10.1002/mma.4617 doi: 10.1002/mma.4617
[5]	H. Alrabaiah, I. Ahmad, R. Amin, K. Shah, A numerical method for fractional variable order pantograph differential equations based on haar wavelet, Eng. Comput., 38 (2022), 2655–2668. https://doi.org/10.1007/s00366-020-01227-0 doi: 10.1007/s00366-020-01227-0
[6]	R. Amin, H. Ahmad, K. Shah, M. B. Hafeez, W. Sumelka, Theoretical and computational analysis of nonlinear fractional integro-differential equations via collocation method, Chaos Soliton. Fract., 151 (2021), 111252. https://doi.org/10.1016/j.chaos.2021.111252 doi: 10.1016/j.chaos.2021.111252
[7]	R. Amin, B. Alshahrani, M. Mahmoud, A. H. Abdel-Aty, K. Shah, W. Deebani, Haar wavelet method for solution of distributed order time-fractional differential equations, Alex. Eng. J., 60 (2021), 3295–3303. https://doi.org/10.1016/j.aej.2021.01.039 doi: 10.1016/j.aej.2021.01.039
[8]	R. Amin, N. Senu, M. B. Hafeez, N. I. Arshad, A. Ahmadian, S. Salahshour, et al., A computational algorithm for the numerical solution of nonlinear fractional integral equations, Fractals, 30 (2021), 2240030. https://doi.org/10.1142/S0218348X22400308 doi: 10.1142/S0218348X22400308
[9]	R. Amin, K. Shah, H. Ahmad, A. H. Ganie, A. H. Abdel-Aty, T. Botmart, Haar wavelet method for solution of variable order linear fractional integro-differential equations, AIMS Mathematics, 7 (2022), 5431–5443. https://doi.org/10.3934/math.2022301 doi: 10.3934/math.2022301
[10]	R. Amin, K. Shah, M. Asif, I. Khan, A computational algorithm for the numerical solution of fractional order delay differential equations, Appl. Math. Comput., 402 (2021), 125863. https://doi.org/10.1016/j.amc.2020.125863 doi: 10.1016/j.amc.2020.125863
[11]	R. Amin, K. Shah, M. Asif, I. Khan, F. Ullah, An efficient algorithm for numerical solution of fractional integro-differential equations via haar wavelet, J. Comput. Appl. Math., 381 (2021), 113028. https://doi.org/10.1016/j.cam.2020.113028 doi: 10.1016/j.cam.2020.113028
[12]	D. Baleanu, G. C. Wu, Y. R. Bai, F. L. Chen, Stability analysis of caputo-like discrete fractional systems, Commun. Nonlinear Sci., 48 (2017), 520–530. https://doi.org/10.1016/j.cnsns.2017.01.002 doi: 10.1016/j.cnsns.2017.01.002
[13]	K. Burrage, N. Hale, D. Kay, An efficient implicit fem scheme for fractional-in-space reaction-diffusion equations, SIAM J. Sci. Comput., 34 (2012), A2145–A2172. https://doi.org/10.1137/110847007 doi: 10.1137/110847007
[14]	M. A. Chaudhry, A. Qadir, M. Rafique, S. Zubair, Extension of euler's beta function, J. Comput. Appl. Math., 78 (1997), 19–32. https://doi.org/10.1016/S0377-0427(96)00102-1 doi: 10.1016/S0377-0427(96)00102-1
[15]	W. Chen, J. Zhang, J. Zhang, A variable-order time-fractional derivative model for chloride ions sub-diffusion in concrete structures, Fract. Calc. Appl. Anal., 16 (2013), 76–92. https://doi.org/10.2478/s13540-013-0006-y doi: 10.2478/s13540-013-0006-y
[16]	C. Derbazi, Z. Baitiche, M. Benchohra, A. Cabada, Initial value problem for nonlinear fractional differential equations with $\psi$-caputo derivative via monotone iterative technique, Axioms, 9 (2020), 57. https://doi.org/10.3390/axioms9020057 doi: 10.3390/axioms9020057
[17]	K. Diethelm, An investigation of some nonclassical methods for the numerical approximation of caputo-type fractional derivatives, Numer. Algor., 47 (2008), 361–390. https://doi.org/10.1007/s11075-008-9193-8 doi: 10.1007/s11075-008-9193-8
[18]	K. Diethelm, The analysis of fractional differential equations: An application-oriented exposition using differential operators of Caputo type, Springer Science & Business Media, 2010.
[19]	K. Diethelm, An efficient parallel algorithm for the numerical solution of fractional differential equations, Fract. Calc. Appl. Anal., 14 (2011), 475–490. https://doi.org/10.2478/s13540-011-0029-1 doi: 10.2478/s13540-011-0029-1
[20]	K. Diethelm, N. J. Ford, A. D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynam., 29 (2002), 3–22. https://doi.org/10.1023/A:1016592219341 doi: 10.1023/A:1016592219341
[21]	Y. Y. Gambo, F. Jarad, D. Baleanu, T. Abdeljawad, On caputo modification of the hadamard fractional derivatives, Adv. Differ. Equ., 2014 (2014), 10. https://doi.org/10.1186/1687-1847-2014-10 doi: 10.1186/1687-1847-2014-10
[22]	R. Garrappa, Numerical solution of fractional differential equations: A survey and a software tutorial, Mathematics, 6 (2018), 16. https://doi.org/10.3390/math6020016 doi: 10.3390/math6020016
[23]	R. Garrappa, On linear stability of predictor-corrector algorithms for fractional differential equations, Int. J. Comput. Math., 87 (2010), 2281–2290. https://doi.org/10.1080/00207160802624331 doi: 10.1080/00207160802624331
[24]	R. Garrappa, M. Popolizio, On accurate product integration rules for linear fractional differential equations, J. Comput. Appl. Math., 235 (2011), 1085–1097. https://doi.org/10.1016/j.cam.2010.07.008 doi: 10.1016/j.cam.2010.07.008
[25]	R. Garrappa, M. Popolizio, Evaluation of generalized mittag-leffler functions on the real line, Adv. Comput. Math., 39 (2013), 205–225. https://doi.org/10.1007/s10444-012-9274-z doi: 10.1007/s10444-012-9274-z
[26]	M. Gohar, C. Li, Z. Li, Finite difference methods for caputo-hadamard fractional differential equations, Mediterr. J. Math., 17 (2020), 194. https://doi.org/10.1007/s00009-020-01605-4 doi: 10.1007/s00009-020-01605-4
[27]	F. Jarad, T. Abdeljawad, D. Baleanu, Caputo-type modification of the hadamard fractional derivatives, Adv. Differ. Equ., 2012 (2012), 142. https://doi.org/10.1186/1687-1847-2012-142 doi: 10.1186/1687-1847-2012-142
[28]	A. A. Kilbas, H. M. Srivastava, J. J. Trujill, Theory and applications of fractional differential equations, lsevier, 2006.
[29]	A. A. Kilbas, O. Marichev, S. Samko, Fractional integrals and derivatives: Theory and applications, Switzerland: Gordon and Breach, 1993.
[30]	J. D. Lambert, Numerical methods for ordinary differential systems: The initial value problem, John Wiley & Sons, 1991.
[31]	C. Li, A. Chen, Numerical methods for fractional partial differential equations, Int. J. Comput. Math., 95 (2018), 1048–1099. https://doi.org/10.1080/00207160.2017.1343941 doi: 10.1080/00207160.2017.1343941
[32]	H. L. Li, C. Hu, L. Zhang, H. Jiang, J. Cao, Complete and finite-time synchronization of fractional-order fuzzy neural networks via nonlinear feedback control, Fuzzy Set. Syst., 443 (2022), 50–69. https://doi.org/10.1016/j.fss.2021.11.004 doi: 10.1016/j.fss.2021.11.004
[33]	Y. Luchko, J. Trujillo, Caputo-type modification of the erdélyi-kober fractional derivative, Fract. Calc. Appl. Anal., 10 (2007), 249–267.
[34]	R. Metzler, J. H. Jeon, A. G. Cherstvy, E. Barkai, Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking, Phys. Chem. Chem. Phys., 16 (2014), 24128–24164. https://doi.org/10.1039/C4CP03465A doi: 10.1039/C4CP03465A
[35]	J. F. Reverey, J. H. Jeon, H. Bao, M. Leippe, R. Metzler, C. Selhuber-Unkel, Superdiffusion dominates intracellular particle motion in the supercrowded cytoplasm of pathogenic acanthamoeba castellanii, Sci. Rep., 5 (2015), 11690. https://doi.org/10.1038/srep11690 doi: 10.1038/srep11690
[36]	A. Suechoei, P. S. Ngiamsunthorn, Extremal solutions of $\varphi$-caputo fractional evolution equations involving integral kernels, AIMS Mathematics, 6 (2021), 4734–4757. https://doi.org/10.3934/math.2021278 doi: 10.3934/math.2021278
[37]	V. E. Tarasov, V. V. Tarasova, Long and short memory in economics: Fractional-order difference and differentiation, 2016, arXiv: 1612.07903V3.
[38]	G. C. Wu, D. Baleanu, H. P. Xie, F. L. Chen, Chaos synchronization of fractional chaotic maps based on the stability condition, Physica A, 460 (2016), 374–383. https://doi.org/10.1016/j.physa.2016.05.045 doi: 10.1016/j.physa.2016.05.045
[39]	A. Young, Approximate product-integration, Proc. Roy. Soc. A, 224 (1954), 552–561. https://doi.org/10.1098/rspa.1954.0179 doi: 10.1098/rspa.1954.0179
[40]	B. Yu, X. Jiang, C. Wang, Numerical algorithms to estimate relaxation parameters and caputo fractional derivative for a fractional thermal wave model in spherical composite medium, Appl. Math. Comput., 274 (2016), 106–118. https://doi.org/10.1016/j.amc.2015.10.081 doi: 10.1016/j.amc.2015.10.081

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