Solution of the Liouville-Caputo time- and Riesz space-fractional Fokker-Planck equation via radial basis functions. (English) Zbl 1508.65146
Summary: In this paper, radial basis functions (RBFs) method is proposed for numerical solution of the Liouville-Caputo time- and Riesz space-fractional Fokker-Planck equation with a nonlinear source term. The left-sided and the right-sided Riemann-Liouville fractional derivatives of RBFs are computed and utilized to approximate the spatial fractional derivatives of the unknown function. Also, the time-fractional derivative is discretized by the high order formulas introduced in [J. Cao et al., Fract. Calc. Appl. Anal. 18, No. 3, 735–761 (2015; Zbl 1325.65121)]. In each time step, via a collocation method, the computations of fractional Fokker-Planck equation are reduced to a linear or nonlinear system of algebraic equations. Several numerical examples are included to demonstrate the applicability, accuracy and stability of the method. Some comparisons are made with the existing results.
MSC:
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |
| 65D12 | Numerical radial basis function approximation |
| 35G16 | Initial-boundary value problems for linear higher-order PDEs |
| 60J65 | Brownian motion |
| 26A33 | Fractional derivatives and integrals |
| 35R11 | Fractional partial differential equations |
| 35Q84 | Fokker-Planck equations |
Keywords:
collocation method; fractional Fokker-Planck equation; Liouville-Caputo derivative; radial basis functions; Riemann-Liouville derivativeCitations:
Zbl 1325.65121Software:
FODEReferences:
| [1] | Agarwal, R. P., Belmekki, M. and Benchohra, M., A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative, Adv. Difference Equ. (2009), Article ID: 981728, 47 pp. · Zbl 1182.34103 |
| [2] | Aminataei, A. and Vanani, S. K., Numerical solution of fractional Fokker-Planck equation using the operational collocation method, Appl. Comput. Math.12(1) (2013) 33-43. · Zbl 1290.26003 |
| [3] | Atanackovic, T. M. and Stankovic, B., On a system of differential equations with fractional derivatives arising in rod theory, J. Phys. A37 (2004) 1241-1250. · Zbl 1059.35011 |
| [4] | Atangana, A. and Baleanu, D., New fractional derivatives with nonlocal and nonsingular kernel: Theory and application to heat transfer model, Therm. Sci.20 (2016) 763-769. |
| [5] | Baeumer, B., Kov’acs, M., and Meerschaert, M. M., Numerical solutions for fractional reaction-diffusion equations, Comput. Math. Appl.55(10) (2008) 2212-2226. · Zbl 1142.65422 |
| [6] | Baleanu, D., Srivastava, H. M. and Yang, X. J., Local fractional variational iteration algorithm for the parabolic Fokker-Planck equation defined on cantor sets, Prog. Fract. Different. Appl.1(1) (2015) 1-11. |
| [7] | Benson, D. A., Wheatcraft, S. W. and Meerschaert, M. M., Application of a fractional advection-dispersion equation, Water Resour. Res.36(6) (2000) 1403-1412. |
| [8] | Buhman, M. D., Spectral convergence of multiquadric interpolation, Proc. Edinburg Math. Soc.36 (1993) 319-333. · Zbl 0791.41002 |
| [9] | Cao, X. N., Fu, J. L. and Huang, H., Numerical method for the time fractional Fokker-Planck equation, Adv. Appl. Math. Mech.4(6) (2012) 848-863. · Zbl 1262.65087 |
| [10] | Cao, J. X., Li, C. P. and Chen, Y. Q., High-order approximation to Caputo derivatives and Caputo-type advection-diffusion equations (II), Fract. Calculus Appl. Anal.18(3) (2015) 735-761. · Zbl 1325.65121 |
| [11] | Caputo, M., Linear model of dissipation whose q is almost frequency independent, J. Roy. Astr. Soc.13 (1967) 529-539. |
| [12] | Caputo, M. and Fabrizio, M., A new definition of fractional derivative without singular kernel, Prog. Fract. Different. Appl.1(2) (2015) 1-13. |
| [13] | Carlson, R. E. and Foley, T. A., The parameter \(r^2\) in multiquadric interpolation, Comput. Math. Appl.21 (1991) 29-42. · Zbl 0725.65009 |
| [14] | Chen, S., Liu, F., Zhuang, P. and Anh, V., Finite difference approximations for the fractional Fokker-Planck equation, Appl. Math. Model.33(1) (2009) 256-273. · Zbl 1167.65419 |
| [15] | Deng, W. H., Numerical algorithm for the time fractional Fokker-Planck equation, J. Comput. Phys.227(2) (2007) 1510-1522. · Zbl 1388.35095 |
| [16] | Deng, W. H., Finite element method for the space and time fractional Fokker-Planck equation, SIAM J. Numer. Anal.47(1), (2008) 204-226. · Zbl 1416.65344 |
| [17] | Diethelm, K., The Analysis of Fractional Differential Equations (Springer, New York, 2010). · Zbl 1215.34001 |
| [18] | Dubey, R. S., Alkahtani, B. S. T. and Atangana, A., Analytical solution of space-time fractional Fokker-Planck equation by homotopy perturbation Sumudu transform method, Math. Prob. Eng. (2015), https://doi.org/10.1155/2015/780929 · Zbl 1394.35075 |
| [19] | Freihet, A., Hasan, S., Alaroud, M., Al-Smadi, M., Ahmad, R. R. and Din, U. K. S., Toward computational algorithm for time-fractional Fokker-Planck models, Adv. Mech. Eng.11(10) (2019) 1-11. |
| [20] | Hartley, T. T., Chaos in a fractional order Chua system, IEEE Trans. circuits Syst.42 (1995) 485-490. |
| [21] | He, J. H., Some applications of nonlinear fractional differential equations and their approximations, Bull. Sci. Technol.15(2) (1999) 86-90. |
| [22] | Hilfer, R. (ed.), Applications of Fractional Calculus in Physics (World Scientific Publishing Company, Singapore, 2000). · Zbl 0998.26002 |
| [23] | Hilfer, R., Experimental evidence for fractional time evolution in glass forming materials, Chem. Phys.284(1-2) (2002) 399-408. |
| [24] | Huang, C., Le, K. N. and Stynes, M., A new analysis of a numerical method for the time-fractional Fokker-Planck equation with general forcing, IMA J. Numer. Anal. (2019), https://doi.org/10.1093/imanum/drz006 · Zbl 1455.65170 |
| [25] | Izadi, M. and Srivastava, H. M., A novel matrix technique for multi-order pantograph differential equations of fractional order, Proc. Roy. Soc. A: Math., Phys. Eng. Sci.477(2253) (2021) 1-21. |
| [26] | Izadi, M. and Srivastava, H. M., Numerical approximations to the nonlinear fractional-order Logistic population model with fractional-order Bessel and Legendre bases, Chaos Solitons Fractals145 (2021) 110779. · Zbl 1498.65117 |
| [27] | Jumarie, G., Modified Riemann-Liouville derivative and fractional Taylor series of non-differentiable functions further results, Comput. Math. Appl.51 (2006) 1367-1376. · Zbl 1137.65001 |
| [28] | Khalid, R., Al Horani, M., Yousef, A. and Sababheh, M., A new definition of fractional derivative, J. Comput. Appl. Math.264 (2014) 65-70. · Zbl 1297.26013 |
| [29] | Kilbas, A. A., Srivastava, H. M. and Trujillo, J. J., Theory and Application of Fractional Differential Equations (Elsevier, 2006). · Zbl 1092.45003 |
| [30] | Koca, I., Modeling the spread of Ebola virus with Atangana-Baleanu fractional operators, Eur. Phys. J. Plus133(3) (2018) 1-11. |
| [31] | Kumar, D., Singh, J. and Baleanu, D., A new analysis for fractional model of regularized long-wave equation arising in ion acoustic plasma waves, Math. Methods Appl. Sci.40(15) (2017) 5642-5653. · Zbl 1388.35212 |
| [32] | Le, K. N., McLean, W. and Mustapha, K., Numerical solution of the time-fractional Fokker-Planck equation with general forcing, SIAM J. Numer. Anal.54(3) (2016) 1763-1784. · Zbl 1404.65122 |
| [33] | Lenzy, E. K., Malacarne, L. C., Mendes, R. S. and Pedron, I. T., Anomalous diffusion, nonlinear fractional Fokker-Planck equation and solutions, Physica A319 (2003) 245-252. · Zbl 1008.82027 |
| [34] | Magin, R. L., Fractional calculus in bioengineering, Crit. Rev. Biomed. Eng.32 (2004) 1-104. |
| [35] | Mandelbrot, B., Some noises with 1/f spectrum, a bridge between direct current and white noise, IEEE Trans. Inform. Theory13 (1967) 289-298. · Zbl 0148.40507 |
| [36] | Metzler, R. and Klafter, J., Fractional Fokker-Planck equation: dispersive transport in an external forcefield, J. Molecul. Liq.86(1) (2000) 219-228. |
| [37] | Odibat, Z. and Momani, S., Numerical solution of Fokker-Planck equation with space- and time-fractional derivatives, Phys. Lett. A369(5) (2007) 349-358. · Zbl 1209.65114 |
| [38] | Oldham, K. B. and Spanier, J., The Fractional Calculus (Academic Press, London, 1974). · Zbl 0292.26011 |
| [39] | Piret, C. and Hanert, E., A radial basis functions method for fractional diffusion equations, J. Comput. Phys.238 (2013) 71-81. · Zbl 1286.65135 |
| [40] | Podlubny, I., Fractional Differential Equations (Academic Press, San Diego, 2006). |
| [41] | Quarteroni, A., Sacco, R. and Saleri, F., Numerical Mathematics (Springer, 2000). · Zbl 0943.65001 |
| [42] | Risken, H., The Fokker-Planck Equation (Springer, Berlin, 1989). · Zbl 0665.60084 |
| [43] | Samko, S. G., Kilbas, A. A. and Marichev, O. L., Fractional Integrals and Derivatives: Theory and Applications (Gordon and Breach, New York, 1993). · Zbl 0818.26003 |
| [44] | Saravanan, A. and Magesh, N., An efficient computational technique for solving the Fokker-Planck equation with space and time fractional derivatives, J. King Saud Univ.28 (2016) 160-166. |
| [45] | Schoenberg, I. J., Metric spaces and completely monotone functions, Ann. Math.39 (1938) 811-841. · JFM 64.0617.03 |
| [46] | Singh, H. and Srivastava, H. M., Jacobi collocation method for the approximate solution of some fractional-order Riccati differential equations with variable coefficients, Physica A: Statist. Mech. Appl.523 (2019) 1130-1149. · Zbl 07563443 |
| [47] | Srivastava, H. M., An introductory overview of fractional-calculus operators based upon the Fox-Wright and related higher transcendental functions, J. Adv. Eng. Comput.5(3) (2021) 135-166. |
| [48] | Srivastava, H. M., Fractional-Order derivatives and integrals: Introductory overview and recent developments, Kyungpook Math. J.60 (2020) 73-116. · Zbl 1453.26008 |
| [49] | Srivastava, H. M., Some parametric and argument variations of the operators of fractional calculus and related special functions and integral transformations, J. Nonlinear Convex Anal.22(8) (2021) 1501-1520. · Zbl 1515.26013 |
| [50] | Srivastava, H. M., Saad, K. M. and Khader, M. M., An efficient spectral collocation method for the dynamic simulation of the fractional epidemiological model of the Ebola virus, Chaos Solitons Fractals140 (2020) 110174. · Zbl 1495.92102 |
| [51] | Sun, H. H., Abdelvahab, A. A. and Onaral, B., Linear approximation of transfer function with a pole of fractional order, IEEE Trans. Automat. Control29 (1984) 441-444. · Zbl 0532.93025 |
| [52] | Sun, H. H., Onaral, B. and Tsao, Y., Application of positive reality principle to metal electrode linear polarization phenomena, IEEE Trans. Bimed. Eng.31 (1984) 664-674. |
| [53] | Sutradhar, T., Datta, B. K. and Bera, R. K., Analytical solution of the time fractional Fokker-Planck equation, Int. J. Appl. Mech. Eng.19(2) (2014) 435-440. |
| [54] | Yan, S. and Cui, M., Finite difference scheme for the time-fractional Fokker-Planck equation with time- and space-dependent forcing, Int. J. Comput. Math.96(2) (2019) 379-398. · Zbl 1499.65448 |
| [55] | Yang, Q., Liu, F. and Turner, I., Stability and convergence of an effective numerical method for the time-space fractional Fokker-Planck equation with a nonlinear source term, Int. J. Differential Equations2010 (2010), Article ID: 464321, 22 pp., https://doi.org/10.1155/2010/464321. · Zbl 1203.82068 |
| [56] | Zaslavsky, G. M., Chaos, fractional kinetics and anomalous transport, Phys. Rep.371(6) (2002) 461-580. · Zbl 0999.82053 |
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