International Journal of Differential Equations

Application of Residual Power Series Method to Fractional Coupled Physical Equations Arising in Fluids Flow

Anas Arafa and Ghada Elmahdy

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

The approximate analytical solution of the fractional Cahn-Hilliard and Gardner equations has been acquired successfully via residual power series method (RPSM). The approximate solutions obtained by RPSM are compared with the exact solutions as well as the solutions obtained by homotopy perturbation method (HPM) and q-homotopy analysis method (q-HAM). Numerical results are known through different graphs and tables. The fractional derivatives are described in the Caputo sense. The results light the power, efficiency, simplicity, and reliability of the proposed method.

Article information

Source
Int. J. Differ. Equ., Volume 2018 (2018), Article ID 7692849, 10 pages.

Dates
Received: 28 March 2018
Accepted: 3 June 2018
First available in Project Euclid: 19 September 2018

Permanent link to this document
https://projecteuclid.org/euclid.ijde/1537322436

Digital Object Identifier
doi:10.1155/2018/7692849

Mathematical Reviews number (MathSciNet)
MR3827849

Zentralblatt MATH identifier
06915961

Citation

Arafa, Anas; Elmahdy, Ghada. Application of Residual Power Series Method to Fractional Coupled Physical Equations Arising in Fluids Flow. Int. J. Differ. Equ. 2018 (2018), Article ID 7692849, 10 pages. doi:10.1155/2018/7692849. https://projecteuclid.org/euclid.ijde/1537322436


Export citation

References

  • L. Debnath, “Recent applications of fractional calculus to science and engineering,” International Journal of Mathematics and Mathematical Sciences, no. 54, pp. 3413–3442, 2003.MR2025566
  • M. Rahimy, “Applications of fractional differential equations,” Applied Mathematical Sciences, vol. 4, no. 49-52, pp. 2453–2461, 2010.MR2726272
  • A. A. Arafa and S. Z. Rida, “Numerical solutions for some generalized coupled nonlinear evolution equations,” Mathematical and Computer Modelling, vol. 56, no. 11-12, pp. 268–277, 2012.MR2974546
  • A. A. Arafa, “Series solutions of time-fractional host-parasitoid systems,” Journal of Statistical Physics, vol. 145, no. 5, pp. 1357–1367, 2011.MR2863738
  • N. H. Sweilam, M. M. Khader, and R. F. Al-Bar, “Numerical studies for a multi-order fractional differential equation,” Physics Letters A, vol. 371, no. 1-2, pp. 26–33, 2007.MR2419274
  • A. Golbabai and K. Sayevand, “Fractional calculus- a new approach to the analysis of generalized fourth-order diffusion-wave equations,” Computers & Mathematics with Applications. An International Journal, vol. 61, no. 8, pp. 2227–2231, 2011.MR2785590
  • K. A. Gepreel, “The homotopy perturbation method applied to the nonlinear fractional Kolmogorov-Petrovskii-PISkunov equations,” Applied Mathematics Letters, vol. 24, no. 8, pp. 1428–1434, 2011.MR2793646
  • A. A. Arafa, S. Z. Rida, and H. Mohamed, “Approximate analytical solutions of Schnakenberg systems by homotopy analysis method,” Applied Mathematical Modelling: Simulation and Computation for Engineering and Environmental Systems, vol. 36, no. 10, pp. 4789–4796, 2012.MR2930371
  • A. A. Arafa, S. Z. Rida, and M. Khalil, “The effect of anti-viral drug treatment of human immunodeficiency virus type 1 (HIV-1) described by a fractional order model,” Applied Mathematical Modelling: Simulation and Computation for Engineering and Environmental Systems, vol. 37, no. 4, pp. 2189–2196, 2013.MR3002310
  • M. Javidi, “A numerical solution of the generalized Burgers-Huxley equation by spectral collocation method,” Applied Mathematics and Computation, vol. 178, no. 2, pp. 338–344, 2006.MR2248493
  • A.-M. Wazwaz, “Analytic study on Burgers, Fisher, Huxley equations and combined forms of these equations,” Applied Mathematics and Computation, vol. 195, no. 2, pp. 754–761, 2008.MR2381254
  • X.-W. Zhou, “Exp-function method for solving Huxley equation,” Mathematical Problems in Engineering, Art. ID 538489, 7 pages, 2008.MR2402979
  • A. A. Arafa, S. Z. Rida, A. A. Mohammadein, and H. M. Ali, “Solving nonlinear fractional differential equation by generalized Mittag-Leffler function method,” Communications in Theoretical Physics, vol. 59, no. 6, pp. 661–663, 2013.MR3155428
  • M. Sari and G. Gürarslan, “Numerical solutions of the generalized Burgers-Huxley equation by a differential quadrature method,” Mathematical Problems in Engineering, Art. ID 370765, 11 pages, 2009.MR2491525
  • O. Abu Arqub, “An iterative method for solving fourth-order boundary value problems of mixed type integro-differential equations,” Journal of Computational Analysis and Applications, vol. 18, no. 5, pp. 857–874, 2015.MR3308505
  • O. Abu Arqub, “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,” Computers & Mathematics with Applications, vol. 73, no. 6, pp. 1243–1261, 2017.
  • A.-M. Wazwaz, “Solitons and singular solitons for the Gardner-KP equation,” Applied Mathematics and Computation, vol. 204, no. 1, pp. 162–169, 2008.MR2458351
  • C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, “Method for solving the Korteweg-deVries equation,” Physical Review Letters, vol. 19, no. 19, pp. 1095–1097, 1967.
  • A.-M. Wazwaz, “New solitons and kink solutions for the Gardner equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 12, no. 8, pp. 1395–1404, 2007.MR2332640
  • Z. Fu, S. Liu, and S. Liu, “New kinds of solutions to Gardner equation,” Chaos, Solitons & Fractals, vol. 20, no. 2, pp. 301–309, 2004.MR2025581
  • G.-q. Xu, Z.-b. Li, and Y.-p. Liu, “Exact solutions to a large class of nonlinear evolution equations,” Chinese Journal of Physics, vol. 41, no. 3, pp. 232–241, 2003.MR2010807
  • D. Baldwin, Ü. Göktas, W. Hereman, L. Hong, R. S. Martino, and J. C. Miller, “Symbolic computation of exact solutions expressible in hyperbolic and elliptic functions for nonlinear PDEs,” Journal of Symbolic Computation, vol. 37, no. 6, pp. 669–705, 2004.MR2095366
  • W. Hereman and A. Nuseir, “Symbolic methods to construct exact solutions of nonlinear partial differential equations,” Mathematics and Computers in Simulation, vol. 43, no. 1, pp. 13–27, 1997.MR1438817
  • Y. Ugurlu and D. g. Kaya, “Solutions of the Cahn-Hilliard equation,” Computers & Mathematics with Applications. An International Journal, vol. 56, no. 12, pp. 3038–3045, 2008.MR2474558
  • J. W. Cahn and J. E. Hilliard, “Free energy of a nonuniform system. I. Interfacial free energy,” The Journal of Chemical Physics, vol. 28, no. 2, pp. 258–267, 1958.
  • S. M. Choo, S. K. Chung, and Y. J. Lee, “A conservative difference scheme for the viscous Cahn-Hilliard equation with a nonconstant gradient energy coefficient,” Applied Numerical Mathematics, vol. 51, no. 2-3, pp. 207–219, 2004.MR2091400
  • M. E. Gurtin, “Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance,” Physica D: Nonlinear Phenomena, vol. 92, no. 3-4, pp. 178–192, 1996.MR1387065
  • O. S. Iyiola and O. G. Olayinka, “Analytical solutions of time-fractional models for homogeneous Gardner equation and non-homogeneous differential equations,” Ain Shams Engineering Journal, vol. 5, no. 3, pp. 999–1004, 2014.
  • Y. Pandir and H. H. Duzgun, “New Exact Solutions of Time Fractional Gardner Equation by Using New Version of F-Expansion Method,” Communications in Theoretical Physics, vol. 67, no. 1, pp. 9–14, 2017.
  • J. Ahmad and S. T. Mohyud-Din, “An efficient algorithm for some highly nonlinear fractional PDEs in mathematical physics,” PLoS ONE, vol. 9, no. 12, Article ID e109127, 2014.
  • W. Li, H. Yang, and B. He, “Exact solutions of fractional Burgers and Cahn-Hilliard equations using extended fractional Riccati expansion method,” Mathematical Problems in Engineering, Art. ID 104069, 9 pages, 2014.MR3224383
  • H. Jafari, H. Tajadodi, N. Kadkhoda, and D. Baleanu, “Fractional subequation method for Cahn-Hilliard and Klein-Gordon equations,” Abstract and Applied Analysis, vol. 2013, Article ID 587179, 5 pages, 2013.MR3034884
  • M. S. Mohamed and K. S. Mekheimer, “Analytical approximate solution for nonlinear space-time fractional Cahn-Hilliard equation,” International Electronic Journal of Pure and Applied Mathematics, vol. 7, no. 4, pp. 145–159, 2014.MR3259465
  • Z. Dahmani and M. Benbachir, “Solutions of the Cahn-Hilliard equation with time- and space-fractional derivatives,” International Journal of Nonlinear Science, vol. 8, no. 1, pp. 19–26, 2009.MR2557974
  • D. Baleanu, Y. Ugurlu, M. Inc, and B. Kilic, “Improved ( G ' / G) -Expansion Method for the Time-Fractional Biological Population Model and Cahn–Hilliard Equation,” Journal of Computational and Nonlinear Dynamics, vol. 10, no. 5, p. 051016, 2015.
  • A. Bouhassoun and M. Hamdi Cherif, “Homotopy Perturbation Method For Solving The Fractional Cahn-Hilliard Equation,” Journal of Interdisciplinary Mathematics, vol. 18, no. 5, pp. 513–524, 2015.
  • J. Ahmad and S. T. Mohyud-Din, “An efficient algorithm for nonlinear fractional partial differential equations,” Proceedings of the Pakistan Academy of Sciences, vol. 52, no. 4, pp. 381–388, 2015.MR3452511
  • J. Manafian and M. Lakestani, “A new analytical approach to solve some of the fractional-order partial differential equations,” Indian Journal of Physics, vol. 91, no. 3, pp. 243–258, 2017.
  • S. Tuluce Demiray, Y. Pandir, and H. Bulut, “Generalized Kudryashov method for time-fractional differential equations,” Abstract and Applied Analysis, Art. ID 901540, 13 pages, 2014.MR3240569
  • O. Abu Arqub, “Series solution of fuzzy differential equations under strongly generalized differentiability,” Journal of Advanced Research in Applied Mathematics, vol. 5, no. 1, pp. 31–52, 2013.
  • O. Abu Arqub, A. El-Ajou, A. S. Bataineh, and I. Hashim, “A representation of the exact solution of generalized Lane-Emden equations using a new analytical method,” Abstract and Applied Analysis, Art. ID 378593, 10 pages, 2013.MR3073508
  • O. Abu Arqub, Z. Abo-Hammour, R. Al-Badarneh, and S. Momani, “A reliable analytical method for solving higher-order initial value problems,” Discrete Dynamics in Nature and Society, vol. 2013, Article ID 673829, 12 pages, 2013.MR3145428
  • A. El-Ajou, O. Abu Arqub, and S. Momani, “Approximate analytical solution of the nonlinear fractional KdV-Burgers equation: a new iterative algorithm,” Journal of Computational Physics, vol. 293, pp. 81–95, 2015.MR3342458
  • O. A. Arqub, A. El-Ajou, and S. Momani, “Constructing and predicting solitary pattern solutions for nonlinear time-fractional dispersive partial differential equations,” Journal of Computational Physics, vol. 293, pp. 385–399, 2015.MR3342478
  • O. A. Arqub, A. El-Ajou, Z. A. Zhour, and S. Momani, “Multiple solutions of nonlinear boundary value problems of fractional order: A new analytic iterative technique,” Entropy, vol. 16, no. 1, pp. 471–493, 2014.
  • A. El-Ajou, O. Abu Arqub, S. Momani, D. Baleanu, and A. Alsaedi, “A novel expansion iterative method for solving linear partial differential equations of fractional order,” Applied Mathematics and Computation, vol. 257, pp. 119–133, 2015.MR3320653
  • A. El-Ajou, O. Abu Arqub, Z. Al Zhour, and S. Momani, “New results on fractional power series: theories and applications,” Entropy. An International and Interdisciplinary Journal of Entropy and Information Studies, vol. 15, no. 12, pp. 5305–5323, 2013.MR3147055
  • A. El-Ajou, O. Abu Arqub, and M. Al-Smadi, “A general form of the generalized Taylor's formula with some applications,” Applied Mathematics and Computation, vol. 256, pp. 851–859, 2015.MR3316113 \endinput