Finite difference scheme for the time-fractional Fokker–Planck equation with time- and space-dependent forcing

References

• T.S. Aleroev, H.T. Aleroeva, J. Huang, M.V. Tamm, Y. Tang, and Y. Zhao, Boundary value problems of fractional Fokker–Planck equations, Comput. Math. Appl. 73(6) (2017), pp. 959–969.
   Web of Science ®Google Scholar
• C.N. Angstmann, I.C. Donnelly, B.I. Henry, T.A.M. Langlands, and P. Straka, Generalised continuous time random walks, master equations, and fractional Fokker–Planck equations, SIAM J. Appl. Math. 75 (2015), pp. 1445–1468.
   Web of Science ®Google Scholar
• G. Baumann and F. Stenger, Fractional Fokker–Planck equation, Mathematics 5(1) (2017), p. 12.
   Web of Science ®Google Scholar
• J.P. Bouchaud and A. Georges, Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications, Phys. Rep. 195(4) (1990), pp. 127–293.
   Web of Science ®Google Scholar
• X.N. Cao, J.L. Fu, and H. Huang, Numerical method for the time fractional Fokker–Planck equation, Adv. Appl. Math. Mech. 4 (2012), pp. 848–863.
   Web of Science ®Google Scholar
• S. Chen, F. Liu, P. Zhuang, and V. Anh, Finite difference approximations for the fractional Fokker–Planck equation, Appl. Math. Model. 33 (2009), pp. 256–273.
   Web of Science ®Google Scholar
• M.R. Cui, Compact exponential scheme for the time fractional convection–diffusion reaction equation with variable coefficients, J. Comput. Phys. 280 (2015), pp. 143–163.
   Web of Science ®Google Scholar
• W.H. Deng, Numerical algorithm for the time fractional Fokker–Planck equation, J. Comput. Phys. 227 (2007), pp. 1510–1522.
   Web of Science ®Google Scholar
• K.Y. Deng and W.H. Deng, Finite difference/predictor-corrector approximations for the space and time fractional Fokker–Planck equation, Appl. Math. Lett. 25(11) (2012), pp. 1815–1821.
   Web of Science ®Google Scholar
• J. Dixon and S. Mckee, Weakly singular discrete Gronwall inequalities, ZAMM Z. Angew. Math. Mech. 66 (1986), pp. 535–544.
   Web of Science ®Google Scholar
• G. Fairweather, H.X. Zhang, X.H. Yang, and D. Xu, A backward euler orthogonal spline collocation method for the time-fractional Fokker–Planck equation, Numer. Methods Partial Differential Equations 31(5) (2015), pp. 1534–1550.
   Web of Science ®Google Scholar
• B.B. Guo, W. Jiang, and C.P. Zhang, A new numerical method for solving nonlinear fractional Fokker–Planck differential equations, J. Comput. Nonlinear Dynam. 12(5) (2017), 051004.
   Web of Science ®Google Scholar
• E. Heinsalu, M. Patriarca, I. Goychuk, and P. Hänggi, Use and abuse of a fractional Fokker–Planck dynamics for time-dependent driving, Phys. Rev. Lett. 99 (2007), 120602.
   PubMed Web of Science ®Google Scholar
• B.I. Henry, T.A.M. Langlands, and P. Straka, Fractional Fokker–Planck equations for subdiffusion with space- and time- dependent forces, Phys. Rev. Lett. 105 (2010), 170602.
   PubMed Web of Science ®Google Scholar
• C. Huang, K.N. Le, and M. Stynes, A new analysis of a numerical method for the time-fractional Fokker–Planck equation with general forcing. Available at https://www.researchgate.net/publication/318883074.
   Google Scholar
• Y.J. Jiang, A new analysis of stability and convergence for finite difference schemes solving the time fractional Fokker–Planck equation, Appl. Math. Model. 39 (2015), pp. 1163–1171.
   Web of Science ®Google Scholar
• A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Application of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
   Google Scholar
• R.C. Koeller, Applications of fractional calculus to the theory of viscoelasticity, J. Appl. Mech. 51(2) (1984), pp. 299–307.
   Web of Science ®Google Scholar
• D. Kusnezov, A. Bulgac, and G.D. Dang, Quantum levy processes and fractional kinetics, Phys. Rev. Lett. 82 (1999), pp. 1136–1139.
   Web of Science ®Google Scholar
• K.N. Le, W. Mclean, and K. Mustapha, Numerical solution of the time-fractional Fokker–Planck equation with general forcing, SIAM J. Numer. Anal. 54 (2016), pp. 1763–1784.
   Web of Science ®Google Scholar
• F. Liu, P. Zhuang, and Q. Liu, Numerical Methods of Fractional Partial Differential Equations and Applications, Science Press, China, 2015. ISBN 978-7-03-046335-7.
   Google Scholar
• M.M. Meerschaert and E. Scalas, Coupled continuous time random walks in finance, Phys. A 370 (2006), pp. 114–118.
   Web of Science ®Google Scholar
• R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep. 339(1) (2000), pp. 1–77.
   Web of Science ®Google Scholar
• R. Metzler, E. Barkai, and J. Klafter, Deriving fractional Fokker–Planck equations from a generalised master equation, Europhys. Lett. 46 (1999), pp. 431–436.
   Google Scholar
• 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 (1999), pp. 3563–3567.
   Web of Science ®Google Scholar
• K. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.
   Google Scholar
• Z. Odibat, Rectangular decomposition method for fractional diffusion-wave equations, Appl. Math. Comput. 179 (2006), pp. 92–97.
   Web of Science ®Google Scholar
• G.F. Pang, W. Chen, and Z.J. Chen, Space-fractional advection-dispersion equations by the Kansa method, J. Comput. Phys. 293 (2015), pp. 280–296.
   Web of Science ®Google Scholar
• L. Pinto and E. Sousa, Numerical solution of a time-space fractional Fokker–Planck equation with variable force field and diffusion, Commun. Nonlinear Sci. Numer. Simul. 50 (2017), pp. 211–228.
   Web of Science ®Google Scholar
• I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
   Google Scholar
• M. Stynes, E. O'Riordan, and J.L. Gracia, Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation, SIAM J. Numer. Anal. 55 (2017), pp. A1057–A1079.
   Web of Science ®Google Scholar
• S.W. Vong and Z.B. Wang, A high order compact finite difference scheme for time fractional Fokker–Planck equations, Appl. Math. Lett. 43 (2015), pp. 38–43.
   Web of Science ®Google Scholar
• M. Zheng, F. Liu, I. Turner, and V. Anh, A novel high order space-time spectral method for the time fractional Fookker-Planck equation, SIAM J. Sci. Comput. 37(2) (2015), pp. 701–724.
   Web of Science ®Google Scholar
• M.L. Zuparic and A.C. Kalloniatis, Analytic solution to space-fractional Fokker–Planck equations for tempered-stable Levy distributions with spatially linear, time-dependent drift, J. Phys. A 51(3) (2018), 035101.
   Google Scholar