Optimal solutions of the time-fractional Fokker–Planck equations

Muhammad Nawaz; Mehreen  Fiza; Hakeem Ullah; Syed Muhammad  Ghufran; Aasim Ullah  Jan

doi:10.5269/bspm.76004

Muhammad Nawaz Department of Mathematics, Abdul Wali Khan University, Mardan 23200, Pakistan.
Mehreen Fiza Department of Mathematics, Abdul Wali Khan University, Mardan 23200, Pakistan.
Hakeem Ullah Abdul Wali Khan University
Syed Muhammad Ghufran Department of Mathematics, Abdul Wali Khan University, Mardan 23200, Pakistan.
Aasim Ullah Jan Department of Mathematics and statistics, Bacha Khan University, Charsadda, Pakistan.

Résumé

This article investigates the comparative analysis of the time-fractional Fokker-plank equation (TFFPE), a mathematical model used in biological and physical sciences. In this work, we used two different methods for analytical solution of the model time-fractional Fokker-plank equation (TFFPE), namely optimal homotopy asymptotic method (OHAM) and optimal auxiliary fractional method (OAFM). The obtained results analyzed analytically and graphically to demonstrate the efficiency and applicability of the applied methods, as well as to investigate the effects of partial arrangement on the behavior of the solutions. Its results indicate that the used methods are effective and accurate for solving fractional order differential equations. These methods applied for the TFFPE model are simple and efficient, allowing us to fully recognize the analytical solutions to both linear and nonlinear fractional differential equations.

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Références

G. Baumann and F. Stenger, Fractional Fokker?Planck equation, Mathematics 5 (2017), 1?19.

X. N. Cao, J. L. Fu, and H. Huang, Numerical method for the time fractional Fokker?Planck equation, Adv. Appl. Math. Mech. 4 (2012), 848?863.

S. V. Dolgov, B. N. Khoromskij, and I. Oseledets, Fast solution of multi-dimensional parabolic problems in the TT/QTT formats with initial application to the Fokker?Planck equation, SIAM J. Sci. Comput. 34 (2012), A3016?A3038.

R. S. Dubey, B. S. Alkahtani, and A. Atangana, Analytical solution of space?time fractional Fokker?Planck equation by homotopy perturbation equation by Homotopy perturbation Sumudu transform method, Math. Probl. Eng. 2015 (2015), 780929.

H. Habenom and D. L. Suthar, Numerical solution for the time-fractional Fokker?Planck equation via shifted Chebyshev polynomials of the fourth kind, Adv. Differ. Eq. 2020 (2020), 1?16.

H. Habenom, D. L. Suthar, and M. Aychluh, Solution of fractional Fokker Planck equation using fractional power series method, J. Sci. Arts 48 (2019), 593?600.

S. Hesam, A. R. Nazemi, and A. Haghbin, Analytical solution for the Fokker?Planck equation by differential transform method, Sci. Iran. B 19 (2012), 1140 1145.

H. Risken, The Fokker?Planck equation methods of solution and applications,Springer, Berlin, Germany/New York, NY, 1989.

T. Sutradhar, B. K. Datta, and R. K. Bera, Analytical solution of the time fractional Fokker?Planck equation, Int. J. Appl. Mech. Eng. 19 (2014), 435?440.

Y. Yang, Y. Huang, and Y. Zhou, Numerical solutions for solving time fractional Fokker?Planck equations based on spectral collocation methods, J. Comput. Appl. Math. 339 (2018), 389?404.

L.T. Watson, Probabily one homotopies in computational sciences. Technical Report TR 00-03, comp. Sci. Virg. Tech. Vol. 1, 2000.

L.T. Watson, S.C. Billups, A.P. Morgan, HOMPACK: a suite of codes for globally convergent homotopy algorithms. ACM Trans. Math. Soft. 13 (1987) 281-310.

L.T. Watson, M. Sosonkina, R.C. Melvile, A.P.Morgan, H.F. Walker, Algorithm 777: HOMPACK 90: a suite of Fortran 90 codes for globally convergent homotopy algorithms. ACM Trans. Math. Soft. 23 (1997) 514-549.

M. Torrisi, R. Tracina, A. Valenti, A group analysis approach for a nonlinear differential system arising in diffusion phenomena. J. Math. Phys. 37 (9) (1996), 4758-4767.

P.G. Drzain, R.S. Johnson, An introduction discussion the theory of solution and its diverse applications. Camb. Uni. Pres. 1989.

B. Abdel-Hamid, Exact solutions of some nonlinear evolution equations using symbolic computations, Comp. Math. Appl. 40 (2000) 291-302.

G. Bluman, S. Kumei, on the remarkable nonlinear diffusion equations. J. Math. Phys. 21 (50 1019-1023.

LC. Chun, Fourier series based variational iteration method for a reliable treatment of heat equations with variable coefficients. Int. J. Non-linr. Sci. Num. Simu. 10 (2007) 1383-1388.

S.H. Chowdhury, A comparison between the modified homotopy perturbation method and adomian decomposition method for solving nonlinear heat transfer equations, J, Appl, Sci. 11 (8) (2011) 1416-1420.

H. Yaghoobi, M. Tirabi, The application of differential transformation method to nonlinear equation arising in heat transfer. Int. Comm. Heat Mass Transfer 38 (6) (2011) 815-820.

D.D. Ganji, The application of He?s homotopy perturbation method to nonlinear equation arising in heat transfer, Phy. Lett. A 355 (2006) 337-341.

R. Bellman, Perturbation techniques in Mathematics, Phys. and Engg. Holt, Rinehart and Winston, New York, 1964.

J.D. Cole, Perturbation Methods in Applied Mathematics, Blaisedell, Waltham, MA. 1968.

R.E. O?Malley, Introduction to Singular Perturbation. Acad. Pres. New York. 1974.

G.L. Liu, New research direction in singular perturbation theory: artificial Parameter approach and inverse perturbation technique, in: Conf. 7th Mod. Maths. Mech. 1997.

S.J. Liao, The proposed homotopy analysis technique for the solution of nonlinear Problems. PhD thesis, Shanghai Jiao Tong University; 1992.

V. Marinca, N.Herisanu, I.Nemes, Optimal homotopy asymptotic method with application to thin film flow. Central European Journal of Physics 2008; 6:64853.

N, Herisanu, V, Marinca, Explicit analytical approximation to large amplitude nonlinear osciallation of a uniform contilaver beam carrying an intermediate lumped mass and rotary inertia. Mecc, 45 847-855 2010.

V. Marinca, N.Herisanu, An optimal homotopy asymptotic method applied to the steady flow of a fourth-grade fluid past a porous plate. Applied Mathematics Letters 2009; 22:245?51.

V. Marinca, N.Herisanu, I.Nemes, A New Analytic Approach to Nonlinear Vibration of An Electrical Machine, Proceeding of the Romanian Academy 9(2008) 229-236.

H. Ullah, S. Islam, M. Idrees, R. Nawaz, Optimal Homotopy Asymptotic Method to doubly wave solutions of the coupled Drinfel?d-Sokolv-Wilson equations, Math. Prob. Engg. Arti. ID 362816, 8 pages.

H. Ullah, S. Islam, M. Idrees and M. Arif, Solution of Boundary Layer Problems with Heat Transfer by Optimal Homotopy Asymptotic Method. Abst. App. Anal.Volu. 2013 (2013), Arti. ID 324869, 10 pages.

H. Ullah, S. Islam, M. Idrees, M. Arif, Solution of boundary layer problems with heat treansfer by optimal homotopy asymptotic method. Abst. Appl. Anal. Volume (2013) Article ID 324869 10 pages.

H. Ullah, S. Islam, M. Idrees, R. Nawaz, Application of optimal homotopy asymptotic method to doubly wave solutions of the coupled Drinfel?d Sokolv-Wilson equations. Math. Prob. Engi. Volume (2013) Article ID 362816 8 pages.

H. Ullah, S. Islam, M. Idrees, R. Nawaz, M. Fiza, The flow past a rotating disk by optimal homotopy asymptotic method. World. Appl. Sci. J. 11 (2014) 1409-1414.

H. Ullah, S. Islam, M. Idrees, M. Fiza, An extenstion of the optimal homotopy asymptotic method to coupled Schrodinger-KdV equation. Int. J. Diff. Eqns. Volume 2014 Article ID 106934 12 pages.

H. Ullah, S. Islam, M. Idrees, M. Fiza, Solution of the differential-difference equation by optimal homotopy asymptotic method. Abst. Appl. Anal. Volume 2014 Aricle ID 520467 8 pages.

H. Ullah, S. Islam, M. Idrees, M. Fiza, The three dimensional flow past a stretching sheet by extended optimal homotopy asymptotic method. Sci. Int. 26 (2014) 567-576.

H. Ullah, S. Islam, I. Khan, S. Sharidna, M.Fiza, Formulation and implementation of optimal homotopy asymptotic method to coupled differential-difference equations. 10.1371/journal.pone.0120127. Plos One.

H. Ullah, S. Islam, I. Khan, S. Sharidan, M. Fiza, Approximate solution of the generalized coupled Hirota- Satsuma coupled KdV equation by extended optimal homotopy asymptotic method, Magnet research report Vol.2 (7). PP: 3022-3036.

H. Ullah, S. Islam, M. Fiza, The optimal homotopy asymptotic method with application to Inhomogeneous nonlinear wave equations. Sci.Int, 26(5), 1907-1913, 2014.

H. Ullah, S. Islam, I. Khan, L.C.C. Dennis, M. Fiza, Approximate solution of two dimensional nonlinear wave equation by optimal homotopy asymptotic method. Mathem. Prob. Engi. IF 1.082. In press.

B. Marinca, V. Marinca, Approximate analytical solutions for thin film flow of fourth grade fluid down a vertical cylinder, Proc. Roman. Acad. Ser. A. 19. (2018).

Mahdy, A. M. (2021). Numerical solutions for solving model time-fractional Fokker-Planck equation. Numerical Methods for Partial Differential Equations, 37(2), 1120-1135.