SPECIAL ISSUE PAPER
Comparison of two reliable methods to solve fractional Rosenau-Hyman equation
Mehmet Senol,
Orkun Tasbozan,
Ali Kurt,
Corresponding Author
Mehmet Senol
Department of Mathematics, Nevsehir Haci Bektas Veli University, Nevsehir, Turkey
Correspondence
Mehmet Senol, Department of Mathematics, Nevsehir Haci Bektas Veli University, Nevsehir 50300, Turkey.
Email: [email protected]
Communicated by: D. Zeidan
Search for more papers by this authorOrkun Tasbozan
Department of Mathematics, Mustafa Kemal University, Antakya, Turkey
Search for more papers by this authorAli Kurt
Department of Mathematics, Mustafa Kemal University, Antakya, Turkey
Search for more papers by this authorMehmet Senol,
Orkun Tasbozan,
Ali Kurt,
Corresponding Author
Mehmet Senol
Department of Mathematics, Nevsehir Haci Bektas Veli University, Nevsehir, Turkey
Correspondence
Mehmet Senol, Department of Mathematics, Nevsehir Haci Bektas Veli University, Nevsehir 50300, Turkey.
Email: [email protected]
Communicated by: D. Zeidan
Search for more papers by this authorOrkun Tasbozan
Department of Mathematics, Mustafa Kemal University, Antakya, Turkey
Search for more papers by this authorAli Kurt
Department of Mathematics, Mustafa Kemal University, Antakya, Turkey
Search for more papers by this authorAbstract
In this study, we examine the numerical solutions of the time-fractional Rosenau-Hyman equation, which is a KdV-like model. This model demonstrates the formation of patterns in liquid drops. For this purpose, two reliable methods, residual power series method (RPSM) and perturbation-iteration algorithm (PIA), are used to obtain approximate solutions of the model. The fractional derivative is taken in the Caputo sense. Obtained results are compared with each other and the exact solutions both numerically and graphically. The outcome shows that both methods are easy to implement, powerful, and reliable. So they are ready to implement for a variety of partial fractional differential equations.
REFERENCES
- 1Odibat Z, Momani S. Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order. Chaos Solitons Fractals. 2008; 36(1): 167-174.
- 2Wang Q. Homotopy perturbation method for fractional KdV-Burgers equation. Chaos Solitons Fractals. 2008; 35(5): 843-850.
- 3Ray SS, Bera RK. An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method. Appl Math Comput. 2005; 167(1): 561-571.
- 4Momani S, Odibat Z. Numerical approach to differential equations of fractional order. J Comput Appl Math. 2007; 207(1): 96-110.
- 5Yang XJ, Baleanu D, Khan Y, Mohyud-din ST. Local fractional variational iteration method for diffusion and wave equations on Cantor sets. Rom J Phys. 2014; 59(1-2): 36-48.
- 6Roul P. Numerical solutions of time fractional degenerate parabolic equations by variational iteration method with Jumarie-modified Riemann-Liouville derivative. Math Methods Appl Sci. 2011; 34(9): 1025-1035.
- 7Dehghan M, Jalil MH, Abbas S. Application of semi-analytic methods for the Fitzhugh-Nagumo equation, which models the transmission of nerve impulses. Math Methods Appl Sci. 2010; 3(11): 1384-1398.
- 8Yin XB, Kumar S, Kumar D. Modified homotopy analysis method for solution of fractional wave equations. Adv Mech Eng. 2015; 7(12): 1-8.
- 9Kurt A, Tasbozan O. Approximate analytical solution of the time fractional Whitham-Broer-Kaup equation using the homotopy analysis method. Int J Pure Appl Math. 2015; 98(4): 503-510.
10.12732/ijpam.v98i4.9 Google Scholar
- 10Diethelm K. An algorithm for the numerical solution of differential equations of fractional order. Electron Trans Numer Anal. 1997; 5: 1-6.
- 11Islam M, Zeidan D, Sekhar TR. On the wave interactions in the drift-flux equations of two-phase flows. Appl Math Comput. 2018; 327: 117-131.
- 12Zeidan D. Assessment of mixture two-phase flow equations for volcanic flows using Godunov-type methods. Appl Math Comput. 2016; 272: 707-719.
- 13Arqub OA. Fitted reproducing kernel Hilbert space method for the solutions of some certain classes of time-fractional partial differential equations subject to initial and Neumann boundary conditions. Comput Math Appl. 2017; 73: 1243-1261.
- 14Arqub OA. Numerical solutions for the Robin time-fractional partial differential equations of heat and fluid flows based on the reproducing kernel algorithm. Internat J Numer Methods Heat Fluid Flow. 2018; 28: 828-856.
- 15Arqub OA, Odibat Z, Al-Smadi M. Numerical solutions of time-fractional partial integrodifferential equations of Robin functions types in Hilbert space with error bounds and error estimates. Nonlinear Dyn. 2018; 94: 1819-1834.
- 16Al-Smadi M, Arqub OA. Computational algorithm for solving Fredholm time-fractional partial integrodifferential equations of Dirichlet functions type with error estimates. Appl Math Comput. 2019; 342: 280-294.
- 17Arqub OA, Maayah B. Numerical solutions of integrodifferential equations of Fredholm operator type in the sense of the Atangana-Baleanu fractional operator. Chaos Solitons Fractals. 2018; 117: 117-124.
- 18Arqub OA, Al-smadi M. Numerical algorithm for solving time-fractional partial integrodifferential equations subject to initial and Dirichlet boundary conditions. Numer Methods Partial Differential Equations. 2018; 34: 1577-1597.
- 19Arqub OA, Al-Smadi M. Atangana-Baleanu fractional approach to the solutions of Bagley-Torvik and Painlevé equations in Hilbert space. Chaos Solitons Fractals. 2018; 117: 161-167.
- 20Arqub OA. Solutions of time-fractional Tricomi and Keldysh equations of Dirichlet functions types in Hilbert space. Numer Methods Partial Differential Equations. 2018; 34: 1759-1780.
- 21Hemeda AA, Eladdad EE, Lairje IA. Local fractional analytical methods for solving wave equations with local fractional derivative. Math Methods Appl Sci. 2018; 41(6): 1-15.
- 22Aslan I. Exact solutions for fractional DDEs via auxiliary equation method coupled with the fractional complex transform. Math Methods Appl Sci. 2016; 39(18): 5619-5625.
- 23Rosenau P, Hyman JM. Compactons: solitons with finite wavelength. Phys Rev Lett. 1993; 70(5): 564.
- 24Senol M, Dolapci IT. On the perturbation-iteration algorithm for fractional differential equations. J King Saud Univ Sci. 2016; 28(1): 69-74.
- 25Senol M, Alquran M, Kasmaei HD. On the comparison of perturbation-iteration algorithm and residual power series method to solve fractional Zakharov-Kuznetsov equation. Results Phys. 2018; 9: 321-327.
- 26Arqub OA. Series solution of fuzzy differential equations under strongly generalized differentiability. J Adv Res Appl Math. 2013; 5: 31-52.
10.5373/jaram.1447.051912 Google Scholar
- 27Arqub OA, El-Ajou A, Bataineh A, Hashim I. A representation of the exact solution of generalized Lane-Emden equations using a new analytical method. Abstr Appl Anal. 2013; 2013: Article ID 378593, 10 pages.
- 28Alquran M, Al-Khaled K, Sarda T, Chattopadhyay J. Revisited fishers equation in a new outlook. A fractional derivative approach Phys A. 2015; 438: 81-93.
- 29Alquran M, Jaradat HM, Syam MI. Analytical solution of the time-fractional Phi-4 equation by using modified residual power series method. Nonlinear Dyn. 2017; 90(4): 2525-2529.
- 30Jaradat HM, Al-Shara S, Khan QJA, Alquran M, Al-Khaled K. Analytical solution of time-fractional Drinfeld-Sokolov-Wilson system using residual power series method. IAENG Int J Appl Math. 2016; 46(1): 64-70.
- 31Jaradat I, Al-Dolat M, Al-Zoubi K, Alquran M. Theory and applications of a more general form for fractional power series expansion. Chaos Solitons Fractals. 2018; 108: 107-110.
- 32Alquran M, Jaradat I. A novel scheme for solving Caputo time-fractional nonlinear equations: theory and application. Nonlinear Dyn. 2018; 91(4): 2389-2395.
- 33Jaradat HM, Jaradat I, Alquran M, et al. Approximate solutions to the generalized time-fractional Ito system. Ital J Pure Appl Math. 2017; 37: 699-710.
- 34Iyiola OS, Ojo GO, Mmaduabuchi O. The fractional Rosenau-Hyman model and its approximate solution. Alexandria Eng J. 2016; 55(2): 1655-1659.
- 35Molliq RY, Noorani MSM. Solving the fractional Rosenau-Hyman equation via variational iteration method and homotopy perturbation method. Int J Differ Equ. 2012; 2012: 472030.
- 36Singh J, Kumar D, Swroop R, Kumar S. An efficient computational approach for time-fractional Rosenau-Hyman equation. Neural Comput Appl. 2017: 1-8.
- 37El-Ajou A, Arqub OA, Momani S. Approximate analytical solution of the nonlinear fractional KdV-Burgers equation: a new iterative algorithm. J Comput Phys. 2015; 293: 81-95.
