Abstract
In this paper, we design and develop some algorithms by using the piecewise linear interpolation polynomial for solving the partial fractional differential equations involving Caputo derivative, with uniform and non-uniform meshes. For designing new methods, we select the mesh points based on the two equal-height and equal-area distribution. Furthermore, the error bounds of proposed methods with uniform and equidistributing meshes are obtained. We also show that our numerical method is stable and convergent with the accuracy of \(O(\kappa ^2 + h)\). Also, some numerical examples are constructed to demonstrate the efficacy and usefulness of the numerical methods. Finally, a comparative study for different values of parameters is also presented.
Similar content being viewed by others
Availability data and material
Not applicable.
References
Bagley RL, Calico R (1991) Fractional order state equations for the control of viscoelasticallydamped structures. J Guid Control Dyn 14(2):304–311
Magin RL (2006) Fractional calculus in bioengineering. Begell House Redding
Marks R, Hall M (1981) Differintegral interpolation from a bandlimited signal’s samples. IEEE Trans Acous Speech Signal Process 29(4):872–877
Wang Z, Huang X, Shi G (2011) Analysis of nonlinear dynamics and chaos in a fractional order financial system with time delay. Comput Math Appl 62(3):1531–1539
Gaul L, Klein P, Kemple S (1991) Damping description involving fractional operators. Mech Syst Signal Process 5(2):81–88
Gorenflo R (1997) Fractional calculus: some numerical methods. Courses and lectures-international centre for mechanical sciences. pp 277–290
Sebaa N, Fellah ZEA, Lauriks W, Depollier C (2006) Application of fractional calculus to ultrasonic wave propagation in human cancellous bone. Signal Process 86(10):2668–2677
Assaleh K, Ahmad WM (2007) Modeling of speech signals using fractional calculus. In: 2007 9th international symposium on signal processing and its applications, IEEE. 1–4
Magin R, Ovadia M (2008) Modeling the cardiac tissue electrode interface using fractional calculus. J Vibr Control 14(9–10):1431–1442
Fellah Z, Depollier C, Fellah M (2002) Application of fractional calculus to the sound waves propagation in rigid porous materials: validation via ultrasonic measurements. Acta Acust United Acust 88(1):34–39
Suárez JI, Vinagre BM, Calderón A, Monje C, Chen Y (2003) Using fractional calculus for lateral and longitudinal control of autonomous vehicles. In: International conference on computer aided systems theory. Springer, 337–348
Soczkiewicz E (2002) Application of fractional calculus in the theory of viscoelasticity. Mol Quantum Acoust 23:397–404
Mathieu B, Melchior P, Oustaloup A, Ceyral C (2003) Fractional differentiation for edge detection. Signal Process 83(11):2421–2432
Kulish VV, Lage JL (2002) Application of fractional calculus to fluid mechanics. J Fluids Eng 124(3):803–806
Ciuchi F, Mazzulla A, Scaramuzza N, Lenzi E, Evangelista L (2012) Fractional diffusion equation and the electrical impedance: experimental evidence in liquid-crystalline cells. J Phys Chem C 116(15):8773–8777
Chen W, Hu S, Cai W (2016) A causal fractional derivative model for acoustic wave propagation in lossy media. Arch Appl Mech 86(3):529–539
Momani S, Odibat Z (2007) Homotopy perturbation method for nonlinear partial differential equations of fractional order. Phys Lett A 365(5–6):345–350
Jafari H, Daftardar-Gejji V (2006) Solving a system of nonlinear fractional differential equations using adomian decomposition. J Comput Appl Math 196(2):644–651
Lesnic D (2006) The decomposition method for initial value problems. Appl Math Comput 181(1):206–213
Daftardar-Gejji V, Jafari H (2005) Adomian decomposition: a tool for solving a system of fractional differential equations. J Math Anal Appl 301(2):508–518
Zurigat M, Momani S, Odibat Z, Alawneh A (2010) The homotopy analysis method for handling systems of fractional differential equations. Appl Math Modell 34(1):24–35
Podlubny I (1998) Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Vol 198. Elsevier
Heris MS, Javidi M (2017) On fractional backward differential formulas for fractional delay differential equations with periodic and anti-periodic conditions. Appl Numer Math 118:203–220
Heris MS, Javidi M (2017) On fbdf5 method for delay differential equations of fractional order with periodic and anti-periodic conditions. Mediterranean J Math 14(3):134
Heris MS, Javidi M, Ahmad B (2019) Analytical and numerical solutions of Riesz space fractional advection-dispersion equations with delay. Comput Model Eng Sci 121(1):249–272
Diethelm K (1997) An algorithm for the numerical solution of differential equations of fractional order. Electron Trans Numer Anal 5(1):1–6
Diethelm K, Ford NJ, Freed AD (2004) Detailed error analysis for a fractional Adams method. Numer Algorithms 36(1):31–52
Diethelm K, Luchko Y (2004) Numerical solution of linear multi-term initial value problems of fractional order. J Comput Anal Appl 6(3):243–263
Blank L (1997) Numerical treatment of differential equations of fractional order. Nonlinear World 4:473–492
Garrappa R, Popolizio M (2011) On accurate product integration rules for linear fractional differential equations. J Comput Appl Math 235(5):1085–1097
Galeone L, Garrappa R (2006) On multistep methods for differential equations of fractional order. Mediterranean J Math 3(3–4):565–580
Garrappa R, Moret I, Popolizio M (2015) Solving the time-fractional Schrödinger equation by Krylov projection methods. J Comput Phys 293:115–134
Li C, Chen A, Ye J (2011) Numerical approaches to fractional calculus and fractional ordinary differential equation. J Comput Phys 230(9):3352–3368
Diethelm K, Ford NJ (2004) Multi-order fractional differential equations and their numerical solution. Appl Math Comput 154(3):621–640
Diethelm K (2010) The analysis of fractional differential equations: an application-oriented exposition using differential operators of Caputo type, vol 2004. Springer, Berlin
Heris MS, Javidi M (2019) A predictor-corrector scheme for the tempered fractional differential equations with uniform and non-uniform meshes. J Supercomput 75:8168–8206
Chen J, Liu F, Anh V (2008) Analytical solution for the time-fractional telegraph equation by the method of separating variables. J Math Anal Appl 338(2):1364–1377
Al-Khaled K, Momani S (2005) An approximate solution for a fractional diffusion-wave equation using the decomposition method. Appl Math Comput 165(2):473–483
Odibat Z, Momani S (2009) The variational iteration method: an efficient scheme for handling fractional partial differential equations in fluid mechanics. Comput Math Appl 58(11–12):2199–2208
Ganji D, Sadighi A (2007) Application of homotopy-perturbation and variational iteration methods to nonlinear heat transfer and porous media equations. J Comput Appl Math 207(1):24–34
Momani S, Odibat Z, Erturk VS (2007) Generalized differential transform method for solving a space-and time-fractional diffusion-wave equation. Phys Lett A 370(5–6):379–387
Momani S, Odibat Z (2008) Numerical solutions of the space-time fractional advection-dispersion equation. Numer Methods Partial Diff Equ Int J 24(6):1416–1429
Meerschaert MM, Tadjeran C (2006) Finite difference approximations for two-sided space-fractional partial differential equations. Appl Numer Math 56(1):80–90
Tadjeran C, Meerschaert MM, Scheffler H-P (2006) A second-order accurate numerical approximation for the fractional diffusion equation. J Comput Phys 213(1):205–213
Liu Q, Zeng F, Li C (2015) Finite difference method for time-space-fractional Schrödinger equation. Int J Comput Math 92(7):1439–1451
Ding H, Li C (2013) Numerical algorithms for the fractional diffusion-wave equation with reaction term. In: Abstract and applied analysis. Vol 2013. Hindawi
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
Heris MS, Javidi M (2018) On fractional backward differential formulas methods for fractional differential equations with delay. Int J Appl Comput Math 4(2):72
Heris MS, Javidi M (2019) Fractional backward differential formulas for the distributed-order differential equation with time delay. Bullet Iran Math Soc 45(4):1159–1176
Javidi M, Heris MS (2019) Analysis and numerical methods for the Riesz space distributed-order advection-diffusion equation with time delay. SeMA J 76:533–551
Deng J, Zhao L, Wu Y (2017) Fast predictor-corrector approach for the tempered fractional differential equations. Numer Algorithms 74(3):717–754
Javidi M, Heris MS, Ahmad B (2019) A predictor-corrector scheme for solving nonlinear fractional differential equations with uniform and nonuniform meshes. Int J Model Simul Sci Comput 10:1950033
Kozyakin V (2009) On accuracy of approximation of the spectral radius by the Gelfand formula. Linear Algebra Appl 431(11):2134–2141
Thomas JW (2013) Numerical partial differential equations: finite difference methods, vol 22. Springer, New York
Acknowledgements
The authors would like to express special thanks to the referees for carefully reading, constructive comments, and valuable remarks which significantly improved the quality from this paper. This research is supported by a research grant of the University of Tabriz (Number 940).
Funding
Not applicable.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors have no conicts of interest and there is no fnancial interest to report.
Ethical Approval
Not applicable.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary Information
Below is the link to the electronic supplementary material.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Javidi, M., Saedshoar Heris, M. New numerical methods for solving the partial fractional differential equations with uniform and non-uniform meshes. J Supercomput 79, 14457–14488 (2023). https://doi.org/10.1007/s11227-023-05198-z
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11227-023-05198-z