ORIGINAL PAPER
Analytical Solution of the Time Fractional Fokker-Planck Equation
More details
Hide details
1
Ranaghat Debnath Institution Mission Road, Ranaghat, PIN – 741 201 Nadia, West Bengal, INDIA
2
Department of Mathematics, Chakdaha College Chakdaha, PIN – 741 222 Nadia, West Bengal, INDIA
3
Heritage Institute of Technology BK-301, Sector-II, Chowbaga Road Anandapur, Kolkata – 700 107, INDIA
Online publication date: 2014-08-30
Publication date: 2014-05-01
International Journal of Applied Mechanics and Engineering 2014;19(2):435-440
KEYWORDS
ABSTRACT
A nonperturbative approximate analytic solution is derived for the time fractional Fokker-Planck (F-P) equation by using Adomian’s Decomposition Method (ADM). The solution is expressed in terms of Mittag- Leffler function. The present method performs extremely well in terms of accuracy, efficiency and simplicity.
FUNDING
This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.
CONFLICT OF INTEREST
The authors declare that they have no conflict of interest.
DATA AVAILABILITY
All data generated or analysed during this study are included in this published article.
REFERENCES (23)
1.
Abbaoui K. and Cherruault Y. (1994): Convergence of Adomian’s method applied to differential equations. - Computers Math. Applic., vol.28(5), pp.103-109.
2.
Abbaoui K. and Cherruault Y. (1995): New ideas for proving convergence of decomposition methods. - Computers Math. Applic., vol.29(7), pp.103-108.
3.
Adomian G. (1994): Solving Frontier Problems of Physics: The Decomposition Method. - Boston: Kluwer Academic Publishers.
4.
Adomian G. and Rach R. (1993): Analytic solution of nonlinear boundary-value problems in several dimensions by decomposition. - Jr. Math. Anal. Appl., vol.173(1), pp.118-137.
5.
Cherruault Y. (1989): Convergence of Adomian’s method. - Kybernetes, vol.18, pp.31-38.
6.
Datta B.K. (1993): The approximate evaluation of certain integrals. - Computers and Mathematics with Applications, vol.25(7), pp.47-49.
7.
Datta B.K. (2007): In analysis and estimation of stochastic physical and mechanical system. - UGC Minor Research Project No.F. PSW/061.
8.
Glockle W.G. and Nonnenmacher T.F. (1991): Fractional integral operators and Fox functions in the theory of viscoelasticity. - Macromolecules, vol.24, pp.6426-6434.
9.
Himoun N., Abbaoui K. and Cherruault Y. (1999): New results of convergence of Adomian’s method. - Kybernetes, vol.28(4-5), pp.423-429.
10.
Jiang K. (2005): Multiplicity of nonlinear thermal convection in a spherical shell. - Phys. Rev. E., Stat Nonlinear Soft Matter Phys., vol.71, No.1 Pt 2.
11.
Kaya D. (2006): The exact and numerical solitary-wave solutions for generalized modified Boussinesq equation. - Physics Letters A, vol.348(3-6), pp.244-250.
12.
Miller K.S. and Ross B. (1993): An introduction to the fractional calculus and fractional differential equations. - New York: John Wiley and Sons Inc.
13.
Ngarhasta N., Some B., Abbaoui K. and Cherruault Y. (2002): New numerical study of Adomian method applied to a diffusion model. - Kybernetes, vol.31(1), pp.61-75.
14.
Odibat Z. and Momani S. (2007): Numerical solution of Fokker-Planck equation with space and time-fractional derivatives. - Phys. Lett. A, vol.369, pp.349-358.
15.
Oksendal B. (2004): Stochastic Differential Equations. - Springer International.
16.
Oldham K.B. and Spanier J. (1974): The Fractional Calculus. - New York and London: Academic Press.
17.
Podlubny I. (1999): Fractional Differential Equations. - San Diego, California, USA: Academic Press.
18.
Rida, S.Z. and Sherbiny H.M. (2008): On the solution of the fractional nonlinear Schrödinger equation. - Phys. Lett. A, vol.372(5), pp.553-558.
19.
Saha R.S., Chaudhuri K.S. and Bera R.K. (2008): Application of modified decomposition method for the analytical solution of space fractional diffusion equation. - Applied Mathematics and Computation, vol.196(1), pp.294-302.
20.
Shawagfeh N.T. (2002): Analytical approximate solutions for nonlinear fractional differential equations. - Appl. Math. And Comp., vol.131, pp.517-529.
21.
Suarez L.E. and Shokooh A. (1997): An eigenvector expansion method for the solution of motion containing fractional derivatives. - Transaction ASME Journal of Applied Mechanics, vol.64(3), pp.629-635.
22.
Sutradhar T. (2009): Nonperturbative analytical solution of the time fractional nonlinear Burger’s equation. - Indian Journal of Physics, vol.83(12), pp.1681-1690.
23.
Wazwaz A.M. (2002): Partial Differential Equations. Methods and Applications. - Lisse, The Netherlands: A.A. Balkema Publishers.