Ozan ÖZKAN, Ali KURT

3415

A hybrid algorithm for solving fractional Fokker-Planck equations arising in physics and engineering

In this work, we proposed a hybrid algorithm to approximate the solution of Conformable Fractional Fokker-Planck Equation (CFFPE). This algorithm comprises of unification of two methods named Fractional Wave Transformation Method (FWTM) and Differential Transform Method (DTM). The method is based on two steps. The first step is to reduce the given CFPDEs to corresponding Partial Differential Equations (PDEs). Then, the second step is to solve obtained PDEs iteratively by using DTM. Moreover, the algorithm’s efficiency is shown by employing the method successfully to conformable time-fractional Fokker-Planck equation arising in surface physics, plasma physics, polymer physics, laser physics, biophysics, engineering, neurosciences, nonlinear hydrodynamics, population dynamics, pattern formation and marketing. As a result, the obtained data demonstrate that the algorithm is reliable and applicable.
Keywords:

Conformable fractional partial differential equations, Fractional wave transform method Differential transform method, Time fractional Fokker-Plank equation,

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  • S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, 1993.
  • I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
  • S. Saha Ray, R. K. Bera, An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method, Applied Mathematics and Computation, 167(1), (2005) 561–571.
  • Z. M. Odibat, S. Momani, Application of variational iteration method to nonlinear differential equations of fractional order, International Journal of Nonlinear Sciences and Numerical Simulation, 7(1), (2006) 27–34.
  • S. Momani, Z. Odibat, Homotopy perturbation method for nonlinear partial differential equations of fractional order, Physics Letters A, 365(5–6), (2007) 345–350.
  • L. Song, H. Zhang, Application of homotopy analysis method to fractional KdV-Burgers-Kuramoto equation, Physics Letters A, 367(1–2), (2007) 88–94.
  • J. K. Zhou, Differential Transformation and its Applications for Electric Circuits, Huazhong University Press, Wuhan, China, 1986, (in Chinese).
  • M. Casso, S. Wortmann, U. Rizza, Analytic modeling of two-dimensional transient atmospheric pollutant dispersion by double GITT and Laplace transform techniques, Environmental Modelling & Software, 24(1), (2009) 144–151.
  • E. Sejdic, I. Djurovic, L. Stankovic, Fractional Fourier transform as a signal processing tool: an overview of recent developments, Signal Processing, 91(6), (2011) 1351–1369.
  • R. M. Cotta, M. D. Mikhailov, Integral transform method, Applied Mathematical Modelling, 17(3), (1993) 156–161.
  • X. Yang, Local fractional integral transforms, Progress in Nonlinear Science, 4, (2011) 1–225.
  • Z. Odibat, S. Momani, Numerical solution of Fokker-Planck equation with space and time fractional derivatives, Physics Letters A, 369(5–6), (2007) 349–358.
  • M. Garg, P. Manohar, Analytical solution of space-time fractional Fokker-Planck equations by generalized differential transform method, Le Matematiche, 66, (2011) 91–101.
  • Ü. Cansu, O. Özkan, Solving Fokker-Planck equation by two-dimensional differential transform, Advances in Mathematical and Computational Methods, 73, (2011) 1368.
  • R. Khalil, M. A. Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, Journal of Computational and Applied Mathematics, 264, (2014) 65–70.
  • K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, 1993.
  • A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, San Diego, 2006.
  • H. Rezazadeh, J. Manaan, F. S. Khodadad, F. Nazari, Traveling wave solutions for density-dependent conformable fractional diffusion-reaction equation by the first integral method and the improved tan⁡〖1/2〗 φ(ξ)-expansion method, Optical and Quantum Electronics, 50(3), (2018) Article Number: 121, 1–15.
  • A. Korkmaz, O. E. Hepson, Hyperbolic tangent solution to the conformable time fractional Zakharov-Kuznetsov equation in 3D space, In AIP Conference Proceedings, 1926(1), (2018) 020023.
  • M. S. Hashemi, Some new exact solutions of (2+1)-dimensional nonlinear Heisenberg ferromagnetic spin chain with the conformable time-fractional derivative, Optical and Quantum Electronics, 50(2), (2018) Article Number: 79, 1–11.
  • C. Chen, Y. L. Jiang, Simplest equation method for some time-fractional partial differential equations with conformable derivative, Computers & Mathematics with Applications, 75(8), (2018) 2978–2988.
  • O. Özkan, A. Kurt, On conformable double Laplace transform, Optical and Quantum Electronics, 50(2), (2018) Article Number: 103, 1–9.
  • T. Abdeljawad, On conformable fractional calculus, Journal of Computational and Applied Mathematics, 279, (2015) 57–66.
  • M. J. Jang, C. L. Chen, Y. C. Liu, Two-dimensional differential transform for partial differential equations, Applied Mathematics and Computation 121(2–3), (2001) 261–270.
  • C. L. Chen, S. H. Ho, Solving partial differential equations by two-dimensional transform method, Applied Mathematics and Computation, 106(2–3), (1999) 171–179.
Journal of New Results in Science-Cover
  • ISSN: 1304-7981
  • Yayın Aralığı: Yılda 3 Sayı
  • Başlangıç: 2012
  • Yayıncı: Tokat Gaziosmanpaşa Üniversitesi
Arşiv
Sayıdaki Diğer Makaleler

Oğuz YAĞCI, Recep ŞAHİN, İ. Onur KIYMAZ, Ayşegül ÇETİNKAYA

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Ozan ÖZKAN, Ali KURT

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