A. PRAKASH, M. KUMAR

4017

NUMERICAL SOLUTION OF TIME-FRACTIONAL ORDER FOKKER-PLANCK EQUATION

In this article, new iterative method NIM is employed to nd the numerical solution of linear and nonlinear time-fractional order Fokker-Planck equation FPE , which is applied in many elds of engineering and applied science. The introduced technique renders an analytical solution in the form of a convergent series with easily computable components without using any restrictive assumptions. Three numerical examples are tested using this method. Plotted graph illustrate the eciency and accuracy of the proposed method.
Keywords:

Fokker-Planck equation, New iterative method NIM Analytic solution, Caputo fractional derivative.,

PDF

___

  • He, J. H., (1997), Variational iteration method for delay differential equations, Commun. Nonlinear Sci. Numer. Simul., 2(4), pp. 235-236.
  • He, J. H., (1998), Approximate solution of nonlinear differential equations with convolution product nonlinearities, Comput. Methods Appl. Mech. Engrg., 167, pp. 69-73.
  • He, J. H., (1999), Variational iteration method a kind of non-linear analytical technique: some examples, Int. J. Nonlinear Mech., 34, pp. 699-708.
  • Prakash, A., Kumar. M and Sharma, K. K., (2015), Numerical method for solving fractional coupled Burgers equations, Applied Mathematics and Computation, 260, pp. 314-320.
  • Prakash, A. and Kumar. M., (2016), He’s Variational iteration method for the solution of nonlinear Newell-Whitehead-Segel equation, Journal of Applied Analysis and Computation, 6(3), pp. 738-748.
  • Prakash, A. and Kumar, M., (2016), Numerical solution of two dimensional time fractional-order biological population model, Open Physics, 14, pp. 177186.
  • Prakash, A. and Kumar, M., (2017), Numerical method for solving time fractional multi-dimensional diffusion equations, Int. J. Computing Science and Mathematics, 8(3), pp. 257-267.
  • Aruna. K and Ravi Kanth, A. S. V., (2013), Approximate solutions of nonlinear fractional Schrodinger equation via differential transform method and modified differential transform method, National Academy Science Letters, 36(2), pp. 201-213
  • Prakash, A., (2016), Analytical method for space-fractional telegraph equation by homotopy petur bation transform method, Nonlinear Engineering, 5(2), pp. 123-128.
  • Kumar, S., Kocak, H. and Yildirim, A.,(2012), A fractional model of gas dynamics equations and its analytical approximate solution using Laplace transform, Z. Naturforsch, 67a, pp. 389-396.
  • Kumar. S, Yildirim, A., Khan, Y. and Wei, L., (2012), A fractional model of the diffusion equation and its analytical solution using Laplace transform, Sci. Iran. B, 19 (4), pp. 1117-1123.
  • Kumar, D., Singh, J. and Baleanu, D., (2016), Numerical computation of a fractional model of differential-difference equation, Journal of Computational and Nonlinear Dynamics, 11(6), pp. 0610041-0610044.
  • Kumar, D., Singh, J. and Baleanu, D., (2017), A new analysis for fractional model of regularized long- wave equation arising in ion acoustic plasma waves, Math. Meth. Appl. Sci., DOI: 10.1002/mma.4414.
  • Singh, J., Kumar, D., Swroop, R. and Kumar, S., (2017), An efficient computational approach for time-fractional Rosenau-Hyman equation, Neural Computing and Applications, DOI 10.1007/s00521- 017-2909-8.
  • Srivastava, H. M., Kumar, D. and Singh, J., (2017), An efficient analytical technique for fractional model of vibration equation, Applied Mathematical Modelling, 45, pp. 192-204.
  • Kumar, D., Singh, J. and Baleanu, D., (2017), A hybrid computational approach for Klein-Gordon equations on Cantor sets, Nonlinear Dynamics, 87, pp. 511-517.
  • Daftardar-Gejji, V. and Jafari, H.,(2006), An iterative method for solving nonlinear functional equa- tions, J. Math. Anal. Appl., 316, pp. 753-763.
  • Daftardar-Gejji, V. and Bhalekar, S., (2010), Solving fractional boundary value problems with Dirich- let boundary conditions, Comput. Math. Appl., 59, pp. 1801-1809.
  • Bhalekar, S. and Daftardar-Gejji,V., (2008), New iterative method: application to partial differential equations, Appl. Math. Comput. 203, pp. 778-783.
  • Daftardar-Gejji, V. and Bhalekar, S.,(2008), Solving fractional diffusion-wave equations using the new iterative method, Frac. Calc. Appl. Anal., 11 , pp. 193-202.
  • Bhalekar, S. and Daftardar-Gejji, V., (2012), Numeric-Analytic solutions of dynamical systems using a new iterative method, Journal of Applied Nonlinear Dynamics, 1(2), pp. 141-158.
  • Al-luhaibi, M. S., (2017), An analytical treatment to fractional Fornberg-Whitham equation, Math. Sci., 11(1), pp. 1-6.
  • Bhalekar, S. and Daftardar-Gejji, V., (2010), Solving evolution equations using a new iterative method, Numer. Methods Partial Differential Equations, 26 (4), pp. 906-916.
  • Risken, H., (1988), The FokkerPlanck Equation, Springer, Berlin.
  • Liu, F., Anh, V. and Turner, I., (2004), Numerical solution of the space fractional Fokker-Planck equation, J. Comp. Appl. Math., 166 (1), pp. 209-219.
  • Yan, L., (2013), Numerical solutions of fractional Fokker- Planck equations using iterative Laplace transform method, Abstract and Applied Analysis, Article ID 465160.
  • Yildirim, A., (2010), Analytical approach to Fokker-Planck equation with space-and time-fractional derivatives by homotopy perturbation method, J. King Saud University (Science), 22, pp. 257-264.
  • Liu, F., Anh, V. and Turner, I., (2004), Numerical solution of the space fractional Fokker-Planck equation, Journal of Computational and Applied Mathematics, 166, pp. 209-219.
  • Metzler, R. and Nonnenmacher, T. F., (2002), Space- and time fractional diffusion and wave equations, fractional Fokker-Planck equations, and physical motivation, Chemical Physics, 284(1-2), pp. 67- 90.
  • Metzler, R., Barkai, E. and Klafter, J., (1999), Deriving fractional Fokker-Planck equations from generalised master equation, Europhysics Letter, 46(4), pp. 431-436.
  • El-Wakil, S. A. and Zahran, M. A., (2000), Fractional Fokker-Planck equation, Chaos Solitons and Fractals, 11(5), pp. 791-798.
  • Chechkin, A. V., Klafter, J. and Sokolov, I. M., (2003), Fractional Fokker-Planck equation for ultraslow kinetics, Eurphysics Letters, 63(3), pp. 326-332.
  • Stanislavsky, A. A., (2003), Subordinated Brownian motion and its fractional Fokker-Planck equation, Physica Scripta, 67(4), pp. 265-268.
  • Kim, K. and Kong, Y. S., (2002), Anomalous behaviours in fractional Fokker-Planck equation, Journal of the Korean Physical Society, 40(6), pp. 979-982.
  • Sokolov, I. M., (2001), Thermodynamics and fractional Fokker-Planck equations, Physical Review, Statistical, Nonlinear and Soft Matter Physics, 63(5), pp. 561111-561118.
  • Tarasov, V. E.,(2007), Fokker-Planck equation for fractional systems, International Journal of Modern Physics B, 21(6), pp. 955-967.
  • Liu, F., Anh, V. and Turner, I., (2004), Numerical solution of the space fractional Fokker-Planck equation, Journal of Computational and Applied Mathematics, 166(1), pp. 209-219.
  • Deng, W., (2004), Finite element method for the space and time-fractional Fokker-Planck equation, SIAM Journal on Numerical Analysis, 47(1), pp. 204-226.
  • Odibat, Z. and Momani, S., (2007), Numerical solution of Fokker-Planck equation with space and time-fractional derivatives, Physics Letters A: General, Atomic and Solid State Physics, 369(5-6), pp. 349-358.
  • Kumar, S., (2013), Numerical Computation of Time-Fractional Fokker- Planck equation Arising in Solid State Physics and Circuit theory, Zeitschrift fur Naturforschung, 68a , pp. 1-8.
  • Harrison, G.,(1988), Numerical solution of the Fokker-Planck equation using moving finite elements, Numer. Methods Partial Differential Equations, 4, pp. 219-232.
  • Yao, J. J., Kumar, A. and Kumar, S., (2015), A fractional model to describe the Brownian motion of particles and its analytical solution, Advances in Mechanical Engineering, 7(12), pp. 1-11.
  • Podlubny, I., (1999), Fractional Differential Equations, Academic Press, San Diego.
  • Kilbas, A. A., Srivastava, H. M. and Trujillo, J. J., (2006), Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204.
  • Liao, S., (2012), Homotopy Analysis Method in Nonlinear Differential Equations, Springer.
TWMS Journal of Applied and Engineering Mathematics-Cover
  • ISSN: 2146-1147
  • Başlangıç: 2010
  • Yayıncı: Turkic World Mathematical Society
Arşiv
Sayıdaki Diğer Makaleler

M. DÜZ, T. M. ELZAKİ

F. ÇELİK, I. N. CANGÜL

C. JANA, K. HAYAT, M. PAL

H. DEMİRAY

F. A. ALİEV, V. B. LARİN

E. KHOJASTEHNEZHAD, M. BİDKHAM

H. NOSRATIPOUR, K. AMİNİ

C. ADİGA, B. R. RAKSHİTH, - SUMİTHRA

V. LOKESHA, K. Zeba YASMEEN

I. YESİLCE