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On a numerical investigation of the time fractional Fokker– Planck equation via local discontinuous Galerkin method

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Pages 1916-1942 | Received 01 Feb 2016, Accepted 07 Jul 2016, Published online: 31 Oct 2016

References

  • A. Aminataei and S.K. Vanani, Numerical solution of fractional Fokker–Planck equation using the operational collocation method, Appl. Comput. Math. 12 (2013), pp. 33–43.
  • R.L. Bagley and P.J. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity, J. Rheology 27 (1983), pp. 201–210. doi: 10.1122/1.549724
  • R.T. Baillie, Long memory processes and fractional integration in econometrics, J. Econometrics 73 (1996), pp. 5–59. doi: 10.1016/0304-4076(95)01732-1
  • B. Bonilla, M. Rivero, and J.J. Trujillo, On systems of linear fractional differential equations with constant coefficients, Appl. Math. Comput. 187 (2007), pp. 68–78.
  • B. Cockburn, Discontinuous Galerkin methods for convection-dominated problems, Lect. Notes Comput. Sci. Eng. 9 (1998), pp. 69–224. doi: 10.1007/978-3-662-03882-6_2
  • B. Cockburn and C.-W. Shu, TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws II: General framework, Math. Comput. 52 (1989), pp. 411–435.
  • B. Cockburn and C.-W. Shu, The local discontinuous Galerkin method for time-dependent convection–diffusion systems, SIAM J. Numer. Anal. 35 (1998), pp. 2440–2463. doi: 10.1137/S0036142997316712
  • B. Cockburn and C.-W. Shu, The Runge–Kutta discontinuous Galerkin method for conservation laws V: Multidimensional systems, J. Comput. Phys. 141 (1998), pp. 199–224. doi: 10.1006/jcph.1998.5892
  • V. Daftardar-Gejji and H. Jafari, Analysis of a system of nonautonomous fractional differential equations involving Caputo derivatives, J. Math. Anal. Appl. 328 (2007), pp. 1026–1033. doi: 10.1016/j.jmaa.2006.06.007
  • K. Diethelm and N.J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl. 265 (2002), pp. 229–248. doi: 10.1006/jmaa.2000.7194
  • K. Diethelm and A. Freed, The fracPECE subroutine for the numerical solution of differential equations of fractional order, in Forschung und wissenschaftliches Rechnen 1998, S. Heinzel and T. Plesser, eds., Gesellschaft für Wisseschaftliche Datenverarbeitung, pp. 57–71, Vandenhoeck & Ruprecht, Göttingen, 1999.
  • W. Deng, Finite element method for the space and time fractional Fokker–Planck equation, SIAM J. Numer. Anal. 47 (2007), pp. 204–226. doi: 10.1137/080714130
  • W. Deng, Numerical algorithm for the time fractional Fokker–Planck equation, J. Comput. Phys. 227 (2007), pp. 1510–1522. doi: 10.1016/j.jcp.2007.09.015
  • J. Eshaghi, H. Adibi, and S. Kazem, Solution of nonlinear weakly singular Volterra integral equations using the fractional-order Legendre functions and pseudospectral method, Math. Methods Appl. Sci. (2015), 10.1002/mma.3788.
  • G.-H. Gao and Z.-Z. Sun, A finite difference approach for the initial-boundary value problem of the fractional Klein–Kramers equation in phase space, Cent. Eur. J. Math. 10 (2012), pp. 101–115. doi: 10.2478/s11533-011-0105-0
  • R. Gorenflo and F. Mainardi, Fractional oscillation and Mittag–Leffler functions, Fachbereich Mathematik and Informatic, A14/96, Freie Universitaet, Berlin, 1996.
  • L. Guo, Z. Wang, and S. Vong, Fully discrete local discontinuous Galerkin methods for some time-fractional fourth-order problems, Int. J. Comput. Math. doi: 10.1080/00207160.2015.1070840.
  • J. He, Nonlinear oscillation with fractional derivative and its applications, in Proceedings of the International Conference on Vibrating Engineering, Dalian, Chaina, 1998, pp. 288–291.
  • J. He, Some applications of nonlinear fractional differential equations and their approximations, Bull. Sci. Technol. 15(2) (1999), pp. 86–90.
  • B. Jin, R. Lazarov, and Z. Zhou, Error estimates for a semidiscrete finite element method for fractional order parabolic equations, SIAM J. Numer. Anal. 51 (2013), pp. 445–466. doi: 10.1137/120873984
  • B. Jin, R. Lazarov, J. Pasciak, and Z. Zhou, Error analysis of semidiscrete finite element methods for inhomogeneous time-fractional diffusion, IMA J. Numer. Anal. 35 (2014), pp. 561–582. doi: 10.1093/imanum/dru018
  • B. Jin, R. Lazarov, J. Pasciak, and Z. Zhou, Error analysis of a finite element method for the space-fractional parabolic equation, SIAM J. Numer. Anal. 52 (2014), pp. 2272–2294. doi: 10.1137/13093933X
  • B. Jin, R. Lazarov, Y. Liu, and Z. Zhou, The Galerkin finite element method for a multi-term time-fractional diffusion equation, J. Comput. Phys. 281 (2015), pp. 825–843. doi: 10.1016/j.jcp.2014.10.051
  • S. Kazem, An integral operational matrix based on Jacobi polynomials for solving fractional-order differential equations, Appl. Math. Model. 37 (2013), pp. 1126–1136. doi: 10.1016/j.apm.2012.03.033
  • S. Kazem, S. Abbasbandy, and S. Kumar, Fractional-order Legendre functions for solving fractional-order differential equations, Appl. Math. Model. 37 (2013), pp. 5498–5510. doi: 10.1016/j.apm.2012.10.026
  • B.Q. Li, Discontinuous Finite Elements in Fluid Dynamics and Heat Transfer, Springer, London, 2006.
  • X. Liang, A.Q.M. Khaliq, and Y. Xing, Fourth order exponential time differencing method with local discontinuous Galerkin approximation for coupled nonlinear Schrodinger equations, Commun. Comput. Phys. 17 (2015), pp. 510–541. doi: 10.4208/cicp.060414.190914a
  • P. Linz, Analytical and Numerical Methods for Volterra Equations, SIAM, Philadelphia, PA, 1985.
  • F. Mainardi, Fractional calculus: ‘Some basic problems in continuum and statistical mechanics’, in Fractals and Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainardi, eds., Springer Verlag, New York, 1997, pp. 291–384.
  • D. Matignon, Stability results for fractional differential equations with applications to control processing, in Proceedings of IMACS-SMC, Lille, France, 1996, pp. 963–968.
  • R. Metzler and J. Klafter, The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A 37 (2004), pp. R161–R208. doi: 10.1088/0305-4470/37/31/R01
  • Z. Odibat and S. Momani, Numerical solution of Fokker–Planck equation with space- and time-fractional derivatives, Phys. Lett. A 369 (2007), pp. 349–358. doi: 10.1016/j.physleta.2007.05.002
  • K.B. Oldham and J. Spanier, Fractional Calculus: Theory and Applications, Differentiation and Integration to Arbitrary Order, Academic Press, Inc., New York, 1974.
  • I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, CA, 1999.
  • W. Reed and T. Hill, Triangular mesh methods for the neutrontransport equation, Tech. Rep. LA-UR-73-479, Los Alamos Scientific Laboratory, 1973.
  • Y.A. Rossikhin and M.V. Shitikova, Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids, Appl. Mech. Rev. 50 (1997), pp. 15–67. doi: 10.1115/1.3101682
  • A. Saadatmandi and M. Dehghan, A new operational matrix for solving fractional-order differential equations, Comput. Math. Appl. 59 (2010), pp. 1326–1336. doi: 10.1016/j.camwa.2009.07.006
  • G. Samko, A. Kilbas, and O. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Amsterdam, 1993.
  • L. Wei and Y. He, Analysis of a fully discrete local discontinuous Galerkin method for time-fractional fourth-order problems, Appl. Math. Model. 38 (2014), pp. 1511–1522. doi: 10.1016/j.apm.2013.07.040
  • L. Wei, Y. He, and Y. Zhang, Numerical analysis of the fractional seventh-order KdV equation using an implicit fully discrete local discontinuous Galerkin method, Int. J. Numer. Anal. Model. 10 (2013), pp. 430–444.
  • L. Wei, H. Dai, D. Zhang, and Z. Si, Fully discrete local discontinuous Galerkin method for solving the fractional telegraph equation, Calcolo 51 (2014), pp. 175–192. doi: 10.1007/s10092-013-0084-6
  • L. Wei, Y. He, X. Zhang, and S. Wang, Analysis of an implicit fully discrete local discontinuous Galerkin method for the time-fractional Schrodinger equation, Finite Elem. Anal. Des. 59 (2012), pp. 28–34. doi: 10.1016/j.finel.2012.03.008
  • Y. Xu and C.-W. Shu, Local discontinuous Galerkin methods for nonlinear Schrodinger equations, J. Comput. Phys. 205 (2005), pp. 72–97. doi: 10.1016/j.jcp.2004.11.001
  • Y. Xu and C.-W. Shu, Error estimates of the semi-discrete local discontinuous Galerkin method for nonlinear convection–diffusion and KdV equations, Comput. Methods Appl. Mech. Engrg. 196 (2007), pp. 3805–3822. doi: 10.1016/j.cma.2006.10.043
  • Y. Xu and C.-W. Shu, Local discontinuous Galerkin methods for high-order time-dependent partial differential equations, Commun. Comput. Phys. 7 (2010), pp. 1–46.
  • X. Zhang, Y. He, L. Wei, B. Tang, and S. Wang, A fully discrete local discontinuous Galerkin method for one-dimensional time-fractional Fisher's equation, Int. J. Comput. Math. 91 (2014), pp. 2021–2038. doi: 10.1080/00207160.2013.866233
  • Y. Zhao, W. Bu, J. Huang, D.-Y. Liu, and Y. Tang, Finite element method for two-dimensional space-fractional advection-dispersion equations, Appl. Math. Comput. 257 (2015), pp. 553–565.
  • Y. Zhao, Y. Zhang, F. Liu, I. Turner, Y. Tang, and V. Anh, Convergence and superconvergence of a fully-discrete scheme for multi-term time fractional diffusion equations, Comput. Math. Appl. doi: 10.1016/j.camwa.2016.05.005.

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