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The Journal of Supercomputing
Erschienen in: The Journal of Supercomputing 12/2019

07.09.2019

A predictor–corrector scheme for the tempered fractional differential equations with uniform and non-uniform meshes

verfasst von: Mahdi Saedshoar Heris, Mohammad Javidi

Erschienen in: The Journal of Supercomputing | Ausgabe 12/2019

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Abstract

Tempered fractional derivatives and the corresponding tempered fractional differential equations have played a key role in physical science. In this paper, for solving the tempered fractional ordinary differential equation, the predictor–corrector (PC) methods with uniform and non-uniform meshes of Deng et al. (Numer Algorithms 74(3):717–754, 2017) are developed, by using the piecewise quadratic interpolation polynomial. The error bounds of proposed predictor–corrector schemes with uniform and equidistributing meshes are obtained. We proved that the presented numerical method has a higher-order convergence order \(O(h^3)\). Also, some numerical examples are constructed to demonstrate the efficacy and usefulness of the numerical methods. Finally, the results of PC schemes with uniform and non-uniform given in Deng et al. (2017) and presented schemes (improved PC with uniform and non-uniform meshes) are compared for different values of parameters.
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Literatur
1.
Butzer PL, Westphal U (2000) An introduction to fractional calculus. In: Hilfer R (ed) Applications of fractional calculus in physics. World Scientific, Singapore, pp 1–85MATH
2.
Heymans N, Podlubny I (2006) Physical interpretation of initial conditions for fractional differential equations with Riemann–Liouville fractional derivatives. Rheol Acta 45(5):765–771
3.
Miller KS, Ross B (1993) An introduction to the fractional calculus and fractional differential equations. Wiley, LondonMATH
4.
Ali Z, Kumam P, Shah K, Zada A (2019) Investigation of Ulam stability results of a coupled system of nonlinear implicit fractional differential equations. Mathematics 7(4):341
5.
Shah R, Khan H, Arif M, Kumam P (2019) Application of Laplace–Adomian decomposition method for the analytical solution of third-order dispersive fractional partial differential equations. Entropy 21(4):335MathSciNet
6.
Saoudi K, Agarwal P, Kumam P, Ghanmi A, Thounthong P (2018) The Nehari manifold for a boundary value problem involving Riemann–Liouville fractional derivative. Adv Differ Equ 2018(1):263MathSciNetMATH
7.
Chaipunya P, Kumam P (2015) Fixed point theorems for cyclic operators with application in fractional integral inclusions with delays. Conference Publications
8.
Podlubny I (1999) Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Mathematics in science and engineering, vol 198. Elsevier, AmsterdamMATH
9.
Kilbas AA, Marichev OI, Samko SG (1993) Fractional integral and derivatives (theory and applications), vol 1. Gordon and Breach, BaselMATH
10.
Debnath L (2003) Recent applications of fractional calculus to science and engineering. Int J Math Math Sci 54:3413–3442MathSciNetMATH
11.
Mainardi F (2010) Fractional calculus and waves in linear viscoelasticity: an introduction to mathematical models. World Scientific, SingaporeMATH
12.
Zaslavsky GM (2002) Chaos, fractional kinetics, and anomalous transport. Phys Rep 371(6):461–580MathSciNetMATH
13.
Matignon D (1996) Stability results for fractional differential equations with applications to control processing. In: Computational Engineering in Systems Applications, IMACS. IEEE-SMC Lille, France, vol 2, pp 963–968
14.
Matignon D, d’Andrea B (1997) Novel, observer-based controllers for fractional differential systems, In: Proceedings of the 36th IEEE Conference on Decision and Control, 1997, vol 5. IEEE, pp 4967–4972
15.
Diethelm K, Ford NJ, Freed AD (2004) Detailed error analysis for a fractional adams method. Numer Algorithms 36(1):31–52MathSciNetMATH
16.
Meerschaert MM, Sabzikar F, Phanikumar MS, Zeleke A (2014) Tempered fractional time series model for turbulence in geophysical flows. J Stat Mech Theory Exp 2014(9):P09023
17.
Sheng H, Chen Y, Qiu T (2011) Fractional processes and fractional-order signal processing: techniques and applications. Springer, BerlinMATH
18.
Reyes-Melo E, Martinez-Vega J, Guerrero-Salazar C, Ortiz-Mendez U (2005) Application of fractional calculus to the modeling of dielectric relaxation phenomena in polymeric materials. J Appl Polym Sci 98(2):923–935
19.
Schumer R, Benson DA, Meerschaert MM, Wheatcraft SW (2001) Eulerian derivation of the fractional advection–dispersion equation. J Contam Hydrol 48(1–2):69–88
20.
Heymans N, Bauwens JC (1994) Fractal rheological models and fractional differential equations for viscoelastic behavior. Rheologicaacta 33(3):210–219
21.
Raberto M, Scalas E, Mainardi F (2002) Waiting-times and returns in high-frequency financial data: an empirical study. Phys A Stat Mech Appl 314(1–4):749–755MATH
22.
Magin RL (2010) Fractional calculus models of complex dynamics in biological tissues. Comput Math Appl 59(5):1586–1593MathSciNetMATH
23.
Meral F, Royston T, Magin R (2010) Fractional calculus in viscoelasticity: an experimental study. Commun Nonlinear Sci Numer Simul 15(4):939–945MathSciNetMATH
24.
El-Sayed A, Gaber M (2006) The adomian decomposition method for solving partial differential equations of fractal order in finite domains. Phys Lett A 359(3):175–182MathSciNetMATH
25.
Golbabai A, Sayevand K (2011) Analytical treatment of differential equations with fractional coordinate derivatives. Comput Math Appl 62(3):1003–1012MathSciNetMATH
26.
Heris MS, Javidi M (2017) On fractional backward differential formulas for fractional delay differential equations with periodic and anti-periodic conditions. Appl Numer Math 118:203–220MathSciNetMATH
27.
Heris MS, Javidi M (2017) On fbdf5 method for delay differential equations of fractional order with periodic and anti-periodic conditions. Mediterr J Math 14(3):134MathSciNetMATH
28.
Heris MS, Javidi M (2018) On fractional backward differential formulas methods for fractional differential equations with delay. Int J Appl Comput Math 4(2):72MathSciNetMATH
29.
Heris MS, Javidi M (2019) Fractional backward differential formulas for the distributed-order differential equation with time delay. Bull Iran Math Soc 45:1159MathSciNetMATH
30.
Buhmann MD (2003) Radial basis functions: theory and implementations, vol 12. Cambridge University Press, CambridgeMATH
31.
Javidi M, Heris MS (2019) Analysis and numerical methods for the Riesz space distributed-order advection–diffusion equation with time delay. SeMA J 1–19
32.
Heris MS, Javidi M (2018) Second order difference approximation for a class of Riesz space fractional advection–dispersion equations with delay. arXiv preprint arXiv:​1811.​10513
33.
Diethelm K, Ford NJ, Freed AD (2002) A predictor–corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn 29(1–4):3–22MathSciNetMATH
34.
Deng W (2007) Short memory principle and a predictor–corrector approach for fractional differential equations. J Comput Appl Math 206(1):174–188MathSciNetMATH
35.
Deng W (2007) Numerical algorithm for the time fractional Fokker–Planck equation. J Comput Phys 227(2):1510–1522MathSciNetMATH
36.
Li C, Chen A, Ye J (2011) Numerical approaches to fractional calculus and fractional ordinary differential equation. J Comput Phys 230(9):3352–3368MathSciNetMATH
37.
Daftardar-Gejji V, Sukale Y, Bhalekar C (2014) A new predictor–corrector method for fractional differential equations. Appl Math Comput 244:158–182MathSciNetMATH
38.
Yan Y, Pal K, Ford NJ (2014) Higher order numerical methods for solving fractional differential equations. BIT Numer Math 54(2):555–584MathSciNetMATH
39.
Asl MS, Javidi M (2017) An improved PC scheme for nonlinear fractional differential equations: error and stability analysis. J Comput Appl Math 324:101–117MathSciNetMATH
40.
Asl MS, Javidi M (2018) Novel algorithms to estimate nonlinear fdes: applied to fractional order nutrient-phytoplankton–zooplankton system. J Comput Appl Math 339:193–207MathSciNetMATH
41.
Liu Y, Roberts J, Yan Y (2018) A note on finite difference methods for nonlinear fractional differential equations with non-uniform meshes. Int J Comput Math 95(6–7):1151–1169MathSciNet
42.
Liu Y, Roberts J, Yan Y (2018) Detailed error analysis for a fractional adams method with graded meshes. Numer Algorithms 78(4):1195–1216MathSciNetMATH
43.
Zhang YN, Sun ZZ, Liao HL (2014) Finite difference methods for the time fractional diffusion equation on non-uniform meshes. J Comput Phys 265:195–210MathSciNetMATH
44.
Metzler R, Klafter J (2000) The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys Rep 339(1):1–77MathSciNetMATH
45.
Cartea A, del Castillo-Negrete D (2007) Fractional diffusion models of option prices in markets with jumps. Phys A Stat Mech Appl 374(2):749–763
46.
Hanyga A (2001) Wave propagation in media with singular memory. Math Comput Model 34(12–13):1399–1421MathSciNetMATH
47.
Meerschaert MM, Zhang Y, Baeumer B (2008) Tempered anomalous diffusion in heterogeneous systems. Geophys Res Lett 35:L17403
48.
Deng J, Zhao L, Wu Y (2017) Fast predictor–corrector approach for the tempered fractional differential equations. Numer Algorithms 74(3):717–754MathSciNetMATH
49.
Kilbas AAA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations, vol 24. Elsevier Science Limited, AmsterdamMATH
50.
Hanyga A (2001) Wave propagation in media with singular memory. Math Comput Model 34:1399–1421MathSciNetMATH
51.
Metzler R, Joseph K (2000) The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys Rep 339(1):1–77MathSciNetMATH
52.
Moghaddam BP, Machado JT, Babaei A (2018) A computationally efficient method for tempered fractional differential equations with application. Comput Appl Math 37(3):3657–3671MathSciNetMATH
53.
Meerschaert MM, Sabzikar F (2013) Tempered fractional Brownian motion. Stat Probab Lett 83(10):2269–2275MathSciNetMATH
54.
Li C, Deng W, Zhao L (2019) Well-posedness and numerical algorithm for the tempered fractional differential equations. Discrete Contin Dyn Syst B 24(4):1989–2015MathSciNetMATH
55.
Kelley CT (2003) Solving nonlinear equations with Newton’s method, vol 1. SIAM, PhiladelphiaMATH
Metadaten
Titel
A predictor–corrector scheme for the tempered fractional differential equations with uniform and non-uniform meshes
verfasst von
Mahdi Saedshoar Heris
Mohammad Javidi
Publikationsdatum
07.09.2019
Verlag
Springer US
Erschienen in
The Journal of Supercomputing / Ausgabe 12/2019
Print ISSN: 0920-8542
Elektronische ISSN: 1573-0484
DOI
https://doi.org/10.1007/s11227-019-02979-3

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