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Singh, Brajesh Kumar, Kumar, Anil, Gupta, Mukesh. (1402). New approximations of space-time fractional Fokker-Planck equations. سامانه مدیریت نشریات علمی, 11(3), 495-521. doi: 10.22034/cmde.2022.51295.2134
Brajesh Kumar Singh; Anil Kumar; Mukesh Gupta. "New approximations of space-time fractional Fokker-Planck equations". سامانه مدیریت نشریات علمی, 11, 3, 1402, 495-521. doi: 10.22034/cmde.2022.51295.2134
Singh, Brajesh Kumar, Kumar, Anil, Gupta, Mukesh. (1402). 'New approximations of space-time fractional Fokker-Planck equations', سامانه مدیریت نشریات علمی, 11(3), pp. 495-521. doi: 10.22034/cmde.2022.51295.2134
Singh, Brajesh Kumar, Kumar, Anil, Gupta, Mukesh. New approximations of space-time fractional Fokker-Planck equations. سامانه مدیریت نشریات علمی, 1402; 11(3): 495-521. doi: 10.22034/cmde.2022.51295.2134

New approximations of space-time fractional Fokker-Planck equations

Computational Methods for Differential Equations
مقاله 5، دوره 11، شماره 3، شهریور 2023، صفحه 495-521    اصل مقاله (2.21 M)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22034/cmde.2022.51295.2134
نویسندگان
Brajesh Kumar Singh* ؛ Anil Kumar؛ Mukesh Gupta
School of Physical and Decision Sciences, Department of Mathematics, Babasaheb Bhimrao Ambedkar University Lucknow-226025 (UP), India.
چکیده
The present study focuses on the two new hybrid methods: variational iteration J-transform technique (J-VIT) and J-transform method with optimal homotopy analysis (OHAJTM) for analytical assessment of space-time fractional Fokker-Planck equations (STF-FPE), appearing in many realistic physical situations, e.g., in ultra-slow kinetics, Brownian motion of particles, anomalous diffusion, polymerases of Ribonucleic acid, deoxyribonucleic acid, continuous random movement, and formation of wave patterns. OHAJTM is developed via optimal homotopy analysis after implementing the properties of J-transform while (J-VIT) is produced by implementing properties of the J-transform and the theory of variational iteration. Banach approach is utilized to analyze the convergence of these methods. In addition, it is demonstrated that J-VIT is T-stable. Computed new approximations are reported as a closed form expression of the Mittag-Leffler function, and in addition, the effectiveness/validity of the proposed new approximations is demonstrated via three test problems of STF-FPE by computing the error norms: L2 and absolute errors. The numerical assessment demonstrates that the developed techniques perform better for STF-FPE and for a given iteration, and OHAJTM produces new approximations with better accuracy as compared to J-VIT as well as the techniques developed recently.
کلیدواژه‌ها
Fractional Fokker-Planck equations(STF-FPE)‎؛ ‎$mathbb{J}$‎- ‎transform‎؛ ‎Optimal homotopy analysis $mathbb{J}$‎- ‎transform method~(${}_O$HA$mathbb{J}$TM)‎؛ ‎Variational calculus‎؛ ‎Variational iteration technique‎؛ ‎$mathbb{J}$-VIT
مراجع
  • [1] T. A. Abassy, M. A. El-Tawil, and H. El-Zoheiry, Modified variational iteration method for Boussinesq equation, Computers and Mathematics with Applications, 54(7-8) (2007), 955965.
  • [2] A. Ali and N. H. M. Ali, On numerical solution of fractional order delay differential equation using Chebyshev collocation method, New Trends in Mathematical sciences, 6(1) (2018), 817.
  • [3] M. Brics, J. Kaupu˘zs, and R. Mahnke, How to solve Fokker-Planck equation treating mixed eigenvalue spectrum?, Condensed Matter Physics, 16 (2013), 113.
  • [4] A. V. Chechkin, J. Klafter, and I.M. Sokolov, Fractional Fokker-Planck equation for ultra-slow kinetics, Europhys Letters, 63(3) (2003), 326-32.
  • [5] L. Cooke, D. Driessche, and X. Zou, Interaction of maturation delay and  nonlinear  birth  in  population  and  epidemic models, Journal of Mathematical Biology, 39 (1999), 332352.
  • [6] A. Y. Esmaeelzade, F. Behnaz, and J. Hosein, Numerical approach to simulate diffusion model of a fluid-flow in a porous media, Thermal Science, 25 (2021), 255-261.
  • [7] A. Y. Esmaeelzade, H. Mesgarani, G. M. Moremedi, and M. Khoshkhahtinat, High accuracy numerical scheme for solving the space-time fractional advection-diffusion equation with convergence analysis, Alexandria Engineering Journal, 61 (2022), 217-225.
  • [8] B. A. Finlayson, The Method of Weighted Residuals and Variational Principles. Academic Press, New York, 1972.
  • [9] P. Goswami and R. Alqahtani, Solutions of fractional differential equations by sumudu transform and variational iteration method. Journal of Nonlinear Science and Application, 9(4) (2016), 19441951.
  • [10] S. Gupta,Numerical simulation of time-fractional black-scholes equation using fractional variational iteration method, Journal of Computer and Mathematical Sciences,9(9) (2019), 11011110.
  • [11] J. H. He, Approximate analytical solution for seepage flow with fractional derivatives in porous media, Computer Methods in Applied Mechanics and Engineering, 167(1-2) (1998), 57-68.
  • [12] J. H. He, Variational iteration methoda kind of non-linear analytical technique: Some examples,  International  Journal of Non-Linear Mechanics, 34(4) (1999), 699-708.
  • [13] J. H. He, Variational iteration method-Some  recent  results  and  new  interpretations,  Journal  of  Computational and Applied Mathematics, 207(1) (2007), 3 17.
  • [14] S. Hesama, A. R. Nazemia, and A. Haghbinb, Analytical solution for the FokkerPlanck equation by differential transform method, Scientia Iranica B, 19 (2012), 11401145.
  • [15] H. Jafari and A. Alipoor, A new method for calculating general Lagrange multiplier in the variational iteration method, Numerical Methods for Partial Differential Equations, 27 (2011), 996 1001.
  • [16] H. Jafari and V. Daftardar-Gejji, Solving linear and nonlinear fractional diffusion and wave equations by adomian decomposition, Journal of Applied Mathematics and Computing, 180 (2006), 488497.
  • [17] H. Jafari and S. Momani, Solving fractional diffusion and wave equations by modified  homotopy  perturbation  method, Physics Letters A, 370(56) (2007), 388-396.
  • [18] H. Jafari and S. Seifi, Homotopy analysis method for solving linear and  nonlinear fractional diffusion-wave  equa-  tion, Commu. Nonli. Sci. Numer. Simul., 14(5) (2009), 20062012.
  • [19]I. Jaradat, M. Alquran, and R. Abdel-Muhsen, An Analytical framework of 2D diffusion, wave-like, telegraph, and Burgers models with twofold Caputo derivatives ordering, Nonlinear Dyn., 93 (2018), 1911-1922.
  • [20] H. Khan, A. Khan, W. Chen, and K. Shah, Stability analysis and a numerical scheme for fractional Klein-Gordon equations, Methods in the Applied Sciences, 42(2) (2019), 723 732.
  • [21] K. Kim and Y.S. Kong, Anomalous behaviours in fractional Fokker-Planck equation, The Journal of the Korean Physical Society, 40(6) (2002), 979-82.
  • [22] E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley and Sons, New York, 1978.
  • [23] S. Kumar, Numerical computation of time-fractional Fokker - Planck equation arising in solid state physics and circuit theory, Zeitschrift fur Naturforschung, 68 (2013), 1-8.
  • [24] S. Kumar, A. Kumar, D. Baleanu, Two analytical methods for time-fractional nonlinear coupled BoussinesqBurgers equations arise in propagation of shallow water waves, Nonlinear Dyn, 85 (2016), 699715.
  • [25] S. J. Liao, Beyond perturbation: introduction to homotopy analysis method, Eur. P, khys. J. Plus., (2003).
  • [26] F. Liua, V. Anh, I. Turnerb, Numerical solution of the space fractional Fokker-Planck equation, J. Comput. Appl. Math. 166 (2004), 209219.
  • [27] M. Magdziarz, A. Weron, and K. Weron, Fractional Fokker-Planck dynamics: stochastic representation and computer simulation, Physical Review E, 75 (2007), 1-6.
  • [28] H. Mesgarani, M. Bakhshandeh, and Y. Esmaeelzade, The Stability and Convergence of The Numerical Compu- tation for the Temporal Fractional Black-Scholes Equation, J. Math. Ext., 15 (2021), 1-18.
  • [29] H. Mesgarani, A. Y. Esmaeelzade, and H. Tavakoli, Numerical Simulation to Solve Two-Dimensional Temporal- Space Fractional BlochTorrey Equation Taken of the Spin Magnetic Moment Diffusion, Int. J. Appl. Comput. Math., 7(3) (2021), 1-14.
  • [30] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, (1993).
  • [31] S. Momani and Z. M. Odibat, The variational iteration method: an efficient scheme for handling fractional partial differential equation in fluid mechanics, Computers & Mathematics with Applications, 58 (2009), 21992208.
  • [32] Nadeem, Muhammad, F. Li, and H. Ahmad, Modified laplace variational iteration method for solving fourth-order parabolic partial differential equation with variable coefficients, Comput. Math. Appli., 78(6) (2019), 2052-2062.
  • [33] Z. M. Odibat, A study on the convergence of variational iteration method, Math. Comput. Model., 51(9-10) (2010), 11811192.
  • [34] Z. M. Odibat and S. Momani, Numerical solution of Fokker-Planck equation with space- and time-fractional derivatives, Phy. Letters A, 369 (2007), 349-358.
  • [35] M. D. Ortigueira, Fractional Calculus for Scientists and Engineers, Springer Dordrecht Heidelberg London New York, (2011).
  • [36] I. Podlubny, Fractional Differential Equations, Academic Press, New York, (1999).
  • [37] A. Prakash and H. Kaur, Numerical solution for fractional model of Fokker-Planck equation by using q-HATM, Chaos, Solitons and Fractals, 105 (2017), 99110.
  • [38] Y. Qing and B. E. Rhoades, Stability of Picard iteration in metric spaces, Fixed Point Theory and Applications, 2008 (2008), 1-4.
  • [39] S. S. Ray, Fractional Calculus with Applications for Nuclear Reactor Dynamics, CRC Press Taylor and Francis Group, New York, (2016).
  • [40] H. Safdari, H. Mesgarani, M. Javidi, and A. Y. Esmaeelzade, Convergence analysis of the space fractional-order diffusion equation based on the compact finite difference scheme, Computational and Applied Mathematics, 39(2) (2020), 1-15.
  • [41] A. Saravanan and N. Magesh, An efficient computational technique for solving the Fokker-Planck equation with space and time fractional derivatives, Journal of King Saud University - Science 28 (2016), 160-166.
  • [42] A. Sayfy and S. A. Khuri, A laplace variational iteration strategy for the solution of differential equations, Appl. Math. Letters, 25 (2012), 22982305.
  • [43] B. K. Singh, A novel approach for numeric study of 2D biological population model, Cogent Math, 3 (2016), 1261527.
  • [44] B. K. Singh,  Fractional  reduced  differential transform method for numerical computation of a system of linear  and nonlinear fractional partial differential equations, Int. J. Open Problems Compt. Math., 9(3) (2016), 2038.
  • [45] B. K. Singh, Homotopy perturbation new integral transform method for numeric study of space and time fractional (n+1)-dimensional heat and wave-like equations, Waves, Wavelets Frac., 4 (2018), 1936.
  • [46] B. K. Singh and S. Agrawal, A new approximation of conformable time fractional partial differential equations with proportional delay, Appl. Numer. Math., 157 (2020), 419433.
  • [47] B. K. Singh and S. Agrawal, Study of time fractional proportional delayed multi-pantograph system and integro- differential equations, Math. Meth. Appl. Sci., 45 (2022), 8305-8328.
  • [48] B. K. Singh and P. Kumar, A novel approach for numerical computation of Burgers equation in (1 +1) and (2 +  1) dimensions, Alex. Eng. J., 55(4) (2016), 33313344.
  • [49] B. K. Singh and P. Kumar, Numerical computation for time - fractional gas dynamics equations by fractional reduced differential transforms method. J. Math. Sys. Sci., 6 (2016), 248259.
  • [50] B. K. Singh and P. Kumar, Extended Fractional Reduced Differential Transform for Solving Fractional Partial Differential Equations with Proportional Delay, Int. J. Appl. Comput. Math., 3(1) (2017), 631649.
  • [51] B. K. Singh and P. Kumar, Fractional variational iteration method for solving fractional partial differential equa- tions with proportional delay, Int. J. Differ. Eqns., 88(8) (2017), 111.
  • [52] B. K. Singh and P. Kumar, FRDTM for numerical simulation of multi-dimensional, time-fractional model of NavierStokes equation, Ain Shams Eng. J., 9(4) (2018), 827-834.
  • [53] B. K. Singh and P. Kumar, An algorithm based on a new DQM with modified extended cubic B-splines for numerical study of two dimensional hyperbolic telegraph equation, Alex. Eng. J., 57(1) (2018), 175191.
  • [54] B. K. Singh and P. Kumar, Homotopy perturbation transform method for solving fractional partial differential equations with proportional delay, SeMA J., em 75 (2018), 111125.
  • [55] B. K. Singh and A. Kumar, Numerical Study of Conformable Space and Time Fractional FokkerPlanck Equation via CFDT Method,In:N. Deo , V. Gupta V., A. Acu , P. Agrawal, (eds) Mathematical Analysis II: Optimisation, Differential Equations and Graph Theory, ICRAPAM 2018 Springer Proceedings in Mathematics and Statistics, Springer, 307 (2020).
  • [56] B. K. Singh and P. Kumar, and V. Kumar, Homotopy perturbation method for solving time fractional coupled viscous Burgers equation in (2+1) and (3+1) dimensions, Int. J. Appl. Comput. Math., 4(38) (2018).
  • [57] B. K. Singh, A. Kumar, and M. Gupta, Efficient New Approximations for Space-Time Fractional Multi- dimensional Telegraph Equation, Int. J. Appl. Comput. Math., 8 (2022), 218.
  • [58] B. K. Singh and M. Gupta, A comparative study of analytical solutions of space-time fractional hyperbolic-like equations with two reliable methods, Arab J. Basic Appl. Sci., 26(1) (2019), 4157.
  • [59] B. K. Singh and M. Gupta, A new efficient fourth order collocation scheme for solving Burgers’ equation, Appl. Math. Comput., 399(15)(2021), 126011.
  • [60] B. K. Singh and M. Gupta, Trigonometric tension B-spline collocation approximations for time fractional Burgers equation, J. Ocean Eng. Sci., https://doi.org/10.1016/j.joes.2022.03.023.
  • [61] B. K. Singh, J. P. Shukla, and M. Gupta, Study of one dimensional hyperbolic telegraph equation via a hybrid cubic B-spline differential quadrature method, Int. J. Appl. Comput. Math., 7(1), 14 (2021)
  • [62] B. K. Singh and V. K. Srivastava, Approximate series solution of multi-dimensional, time fractional-order (heat- like) diffusion equations using frdtm, Royal Society Open Science, 2(5) (2015), 140511.
  • [63] L. Song, S. Xu, and J. Yang, Dynamical models of happiness with fractional order. Commu. Nonli. Sci. Numer. Simul., 15(3) (2010), 616628.
  • [64] I. M. Sokolov, Thermodynamics and fractional Fokker-Planck equations, Physical Review E, 63(5) (2001), 561111- 18.
  • [65] A. A. Stanislavsky, Subordinated Brownian motion and its fractional Fokker Planck equation, Physica Scripta, 67(4) (2003), 265-268.
  • [66] V. E. Tarasov, Fokker-Planck equation for fractional systems, International Journal of Modern Physics, 21(6) (2007), 955967.
  • [67] M. Tataria, M. Dehghana, and M. Razzaghib, Application of the Adomian decomposition method for the Fokker- Planck equation, Mathematical and Computer Modelling, 45 (2007), 639650.
  • [68] L. Yan, Numerical solutions of fractional Fokker Planck equations using iterative Laplace transform method, Abstract and applied analysis, 2013 (2013) 465160, 7 pages.
  • [69] Q. Yang, F. Liu, and I. Turner, Computationally efficient numerical methods for time and space-fractional Fokker- Planck equations, Physica Scripta, 36(2009) Article ID 014026, 7 pages.
  • [70] J. J. Yao, A. Kumar, and S. Kumar, A fractional model to describe the Brownian motion of  particles  and  its analytical solution, Advances in Mechanical Engineering, 7(12) (2015), 111.
  • [71] A. Yildirim, Analytical approach to Fokker-Planck equation with space- and time fractional derivatives by means  of the homotopy perturbation method, Journal of King Saud University-Science, 22(4) (2010), 257264.
  • [72] S. E. Wakil and M. A. Zahran, Fractional Fokker-Planck equation. Chaos, Solitons Fractals, 11(5) (2000), 791-8.
  • [73] W. Zhao and S. Maitama,Beyond sumudu transform and natural  transform:  j-transform  properties  and  applica- tions, Journal of Applied Analysis and Computation, 10(4) (2020), 12231241.
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