Analytical Approach to Fractional Fokkerplanck Equations by New Homotopy Perturbation Method

Volume 9, Issue 4, pp 426 - 437 http://dx.doi.org/10.22436/jmcs.09.04.18

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 2068 Downloads
• 2921 Views

Authors

Z. Ayati - Department of Engineering sciences, Faculty of Technology and Engineering East of Guilan, University of Guilan, P.C. 44891-63157, Rudsar-Vajargah, Iran. J. Biazar - Department of Applied Mathematics, Faculty of Mathematical Science, University of Guilan, P.O. Box 41635-19141, P.C. 4193833697, Rasht, Iran. S. Ebrahimi - Department of Applied Mathematics, Faculty of Mathematical Science, University of Guilan, P.O. Box 41635-19141, P.C. 4193833697, Rasht, Iran.

Abstract

In this paper, a new form of homotopy perturbation method has been adopted for solving the spacetime dependent fractional Fokker-Planck equation. The fractional derivatives are described in the Caputo sense. The method gives an analytic solution in the form of a convergent series with easily computable components, requiring no linearization or small perturbation. The numerical results show that the approaches are easy to implement and accurate when applied to the space-time dependent fractional Fokker-Planck equations. The method introduces a promising tool for solving many spacetime fractional partial differential equations.

Share and Cite

Keywords

• New Homotopy perturbation method
• Fokker-Plank equation
• functional equation.

MSC

•  65H20
•  34E10
•  82C31
•  58J35
•  34A34
•  94A17

References

• [1] H. Risken, The Fokker–Planck Equation, Springer, Berlin (1988)
  • View Article
  • Google Scholar


• [2] F. Liu,V. Anh, I. Turner, Numerical solution of spacefractional Fokker-Planck equation, J. Comp. and Appl. Math., 166 (2004), 209-219
  • View Article
  • Google Scholar


• [3] T. D. Frank, Stochastic feedback, nonlinear families of Markov processes, and nonlinear Fokker– Planck equations, Physical A , 331 (2004), 391–408
  • View Article
  • MathSciNet
  • Google Scholar


• [4] J. A. Acebron, A Concise Introduction to the Statistical Physics of Complex Systems, et al., Rev. Mod. Phys. , 77 137 (2005),


• [5] T. D. Frank, Nonlinear Fokker–Planck Equations, Fundamentals and Applications, Springer, Berlin (2005)
  • View Article
  • Google Scholar
  • MATH


• [6] J. H. He, Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering, 178 (1999), 257–262
  • Google Scholar


• [7] J. H. He, New interpretation of homotopy perturbation method, International Journal of Modern Physics B, 20 (2006), 2561–2568
  • View Article
  • MathSciNet
  • Google Scholar


• [8] J. H. He, Recent development of homotopy perturbation method, Topological Methods in Nonlinear Analysis, 31 (2008), 205–209
  • MathSciNet
  • Google Scholar
  • MATH


• [9] A. Rajabi, D. D. Ganji, Application of homotopy perturbation method in nonlinear heat conduction and convection equations, Physics Letters A, 360 (2007), 570–573
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [10] D. D. Ganji, A. Sadighi , Application of homotopy perturbation and variational iteration methods to nonlinear heat transfer and porous media equations, Journal of Computational and Applied Mathematics, 207 (2007), 24–34
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [11] D. D. Ganji, The application of He’s homotopy perturbation method to nonlinear equations arising in heat transfer, Physics Letters A, 355 (2006), 337–341
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [12] S. Abbasbandy, A numerical solution of Blasius equation by Adomian’s decomposition method and comparison with homotopy perturbation method, Chaos, Solitons and Fractals, 31 (2007), 257–260
  • View Article
  • Google Scholar


• [13] J. Biazar, H. Ghazvini, Exact solutions for nonlinear Schrödinger equations by He’s homotopy perturbation method, Physics Letters A, 366 (2007), 79–84
  • View Article
  • Google Scholar
  • MATH


• [14] J. H. He, Homotopy perturbation method for solving boundary value problems, Physics Letters A, 350 (2006), 87–88
  • View Article
  • Google Scholar


• [15] Q. Wang, Homotopy perturbation method for fractional KdV-Burgers equation, Chaos, Solitons and Fractals, 35 (2008), 843–850
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [16] E. Yusufoglu, Homotopy perturbation method for solving a nonlinear system of second order boundary value problems, International Journal of Nonlinear Sciences and Numerical Simulation, 8 (2007), 353–358
  • View Article
  • Google Scholar


• [17] K. B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York (1974)


• [18] K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations , John Wiley and Sons, New York (1993)
  • MathSciNet
  • Google Scholar


• [19] Y. Luchko, R. Gorenflo, The initial value problem for some fractional differential equations with the Caputo derivative, Preprint series A08–98, Fachbreich Mathematik und Informatik, Freic Universitat Berlin (1998)
  • Google Scholar


• [20] M. Caputo, J. R. Astron, Vibrations of an infinite plate with a frequency independent Q, Soc. , 13 (1967), 529.
  • View Article
  • Google Scholar


• [21] M. Rabbani, New Homotopy Perturbation Method to Solve Non-Linear Problems, The journal of mathematics and computer science, 7 (2013), 272-275.
  • Google Scholar


• [22] Hadi Kashefi, Maryam Ghorbani, Solutions Exact to Fredholm Fuzzy Integral Equations with Optimal Homotopy Asymptotic Method, The journal of mathematics and computer science, 8 (2014), 153-162.
  • Google Scholar


• [23] Z. Odibat, S. Momani, Numerical solution of Fokker–Planck equationwith space- and time-fractional derivatives, Physics Letters A, 369 (2007), 349–358
  • View Article
  • Google Scholar