دانشگاه تبریز
فهرست نشریات دارای اعتبار وزارت علوم، تحقیقات و فناوری
پایگاه استنادی علوم جهان اسلام (ISC)

 آمار

تعداد نشریات43
تعداد شماره‌ها1,222
تعداد مقالات15,019
تعداد مشاهده مقاله50,482,884
تعداد دریافت فایل اصل مقاله13,571,452
Sepehrian, Behnam, Shamohammadi, Zahra. (1400). Radial basis functions method for nonlinear time- and space-fractional Fokker-Planck equation. سامانه مدیریت نشریات علمی, 9(4), 1128-1147. doi: 10.22034/cmde.2020.36633.1633
Behnam Sepehrian; Zahra Shamohammadi. "Radial basis functions method for nonlinear time- and space-fractional Fokker-Planck equation". سامانه مدیریت نشریات علمی, 9, 4, 1400, 1128-1147. doi: 10.22034/cmde.2020.36633.1633
Sepehrian, Behnam, Shamohammadi, Zahra. (1400). 'Radial basis functions method for nonlinear time- and space-fractional Fokker-Planck equation', سامانه مدیریت نشریات علمی, 9(4), pp. 1128-1147. doi: 10.22034/cmde.2020.36633.1633
Sepehrian, Behnam, Shamohammadi, Zahra. Radial basis functions method for nonlinear time- and space-fractional Fokker-Planck equation. سامانه مدیریت نشریات علمی, 1400; 9(4): 1128-1147. doi: 10.22034/cmde.2020.36633.1633

Radial basis functions method for nonlinear time- and space-fractional Fokker-Planck equation

Computational Methods for Differential Equations
مقاله 14، دوره 9، شماره 4، دی 2021، صفحه 1128-1147    اصل مقاله (497.89 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22034/cmde.2020.36633.1633
نویسندگان
Behnam Sepehrian* ؛ Zahra Shamohammadi
Department of Mathematics, Faculty of Science, Arak University, Arak 38156-8-8349, Iran.
چکیده
In this study, a radial basis functions (RBFs) method for solving nonlinear timeand space-fractional Fokker-Planck equation is presented. The time-fractional derivative is of the Caputo type, and the space-fractional derivatives are considered in the sense of Caputo or Riemann-Liouville. The Caputo and Riemann-Liouville fractional derivatives of RBFs are computed and utilized for approximating the spatial fractional derivatives of the unknown function. Also, in each time step, the time-fractional derivative is approximated by the high order formulas introduced in [6], and then a collocation method is applied. The centers of RBFs are chosen as suitable collocation points. Thus, in each time step, the computations of fractional Fokker-Planck equation are reduced to a nonlinear system of algebraic equations. Several numerical examples are included to demonstrate the applicability, accuracy, and stability of the method. Numerical experiments show that the experimental order of convergence is 4 − α where α is the order of time derivative.
کلیدواژه‌ها
Fokker-Planck equation؛ Fractional derivative؛ Newton method؛ Radial basis functions
مراجع
  • [1]          M. H. Al-Towaiq and Y. S. Abu hour, Two improved classes of Broyden’s methods for solving nonlinear systems of equations, J. Math. Computer Sci., 17 (2017), 22–31.
  • [2]          A. Aminataei and S. K. Vanani, Numerical Solution of fractional Fokker-Planck equation using the operational collocation method, Appl. Comput. Math., 12(1) (2013), 33–43.
  • [3]          L. Beilina, Adaptive finite element/difference method for inverse elastic scattering waves, Appl. comput. Math., 2(2) (2003), 119–134.
  • [4]          M. Bologna, C. Tsallis, and P. Grigolini, Anomalous diffusion associated with nonlinear frac- tional derivative Fokker-Planck-Like equation: Exact time dependent solutions, Phys. Rev. E, 62(2) (2000), 2213–2218.
  • [5]          M. D. Buhman, Spectral convergence of multiquadratic interpolation, Proc. Edinburg Math. Soc., 36 (1993), 319–333.
  • [6]          J. X. Cao, C. P. Li, and Y. Q. Chen, High-order approximation to Caputo derivatives and Caputo-type advection-diffusion equations (II), Fract. Calc. Appl. Anal., 18(3) (2015), 735–761.
  • [7]          R. E. Carlson and T. A. Foley, The parameter r2 in multiquadratic interpolation, Comput. Math. Appl., 21 (1991), 29–42.
  • [8]          S. Chen, F. Liu, P. Zhuang, and V. Anh, Finite difference approximations for the fractional Fokker-Planck equation, Appl. Math. Model., 33(1) (2009), 256–273.
  • [9]          W. Deng, Numerical algorithm for the time fractional Fokker-Planck equation, J. Comput. Phys., 227(2) (2007), 1510–1522.
  • [10]        M. Duarte and J. T. Oden, An h − p adaptive method using clouds, Comput. Methods Appl. Mech. Engrg., 139 (1996), 237–262.
  • [11]        A. Golbabai, E. Mohebianfar, and H. Rabiei, On the new variable shape parameter strategies for radial basis functions, Comp. Appl. Math., 34 (2015), 691–704.
  • [12]        A. Golbabai and H. Rabiei, A meshfree method based on radial basis functions for the eigen- values of transient Stokes equations, Eng. Anal. Bound. Elem., 36(11) (2012), 1555–1559.
  • [13]        A. Golbabai and A. Saeedi, An investigation of radial basis function approximation methods with application in dynamic investment model, IJST, 39A2 (2015), 221–231.
  • [14]        E. J. Kansa, Multiquadrics a scattered data approximation scheme with applications to com- putational fluid-dynamics-I, Comput. Math. Appl., 19 (1990), 127–145.
  • [15]        F. Liu, V. Anh, and I. Turner, Numerical solution of the space fractional Fokker-Planck equa- tion, J. Comput. Appl. Math., 166(1) (2004), 209–219.
  • [16]        W. K. Liu and W. M. Han, Reproducing kernel element method. Part I: Theoritical information, Comput. Methods Appl. Mech. Engrg., 193 (2004), 933–951.
  • [17]        J. Ma and Y. Liu, Exact solutions for a generalized nonlinear fractional Fokker-Planck equation, Nonlinear Analysis: Real World Applications, 11 (1) (2010), 515–521.
  • [18]        J. M. Melenk and I. Babuska, The partition of unity method: basic theory and applications, Comput. Methods Appl. Mech. Engrg., 139 (1996), 289–314.
  • [19]        R. Metzler, E. Barkai and, J. Klafter, Anomalous diffusion and relaxation close to thermal equilibrium: A fractional Fokker-Planck equation approach, Phys. Rev. Lett., 82(1) (1999), 35–63.
  • [20]        R. Metzler and J. Klafter, The fractional Fokker-Planck equation: dispersive transport in an external force field, J. of Molec. Liq., 86(2) (2000), 219–228.
  • [21]        Z. Odibat and S. Momani, Numerical solution of Fokker-Planck equation with space- and time- fractional derivatives, Phys. Lett. A, 369(5) (2007), 349–358
  • [22]        C. Piret and E. Hanert, A radial basis functions method for fractional diffusion equations, J. Comput. Phys., 238 (2013), 71–81.
  • [23]        I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
  • [24]        A. Prakash and H. Kaur, Numerical solution for fractional model of Fokker-Planck equation by using q-HATM, Chaos, Solitons and Fractals, 105 (2017), 99–110.
  • [25]        A. Saravanan and N. Magesh, An efficient computational technique for solving the Fokker- Planck equation with space and time fractional derivatives, J. King Saud Univer., 28 (2016), 160–166.
  • [26]        S. A. Sarra , A random variable shape parameter strategy for radial basis function approximation methods, Eng. Anal. Bound. Elem., 33 (2009), 1239–1245.
  • [27]        I. J. Schoenberg, Metric spaces and completely monotone functions, Ann. Math., 39 (1938), 811–841.
  • [28]        C. Tsallis and E. K. Lenzi, Anomalous diffusion: Nonlinear fractional Fokker-Planck equation, Chem. Phys., 284(1) (2002), 341–347.
  • [29]        L. Yan, Numerical solutions of fractional Fokker-Planck equations using iterative Laplace trans- form method, Abstr. Appl. Anal., Article ID 465160, (2013), DOI: 10.1155/2013/465160, 7 pages.
  • [30]        A. Yildirim, Analytical approach to Fokker-Planck equation with space- and time-fractional derivatives by means of the homotopy perturbation method, J. King Saud Univ., 22(4) (2010), 257–264.
  • [31]        M. P. Zorzano, H. Mais, and L. Vazquez, Numerical solution of two dimensional Fokker-Planck equations, Appl. Math. Comput., 98 (1999), 109–117.
آمار
تعداد مشاهده مقاله: 482
تعداد دریافت فایل اصل مقاله: 369


سامانه مدیریت نشریات علمی. قدرت گرفته از سیناوب