American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Optimal indirect stability of a weakly damped elastic abstract system of second order equations coupled by velocities
  • CPAA Home
  • This Issue
  • Next Article
    Higher order conformally invariant equations in $ {\mathbb R}^3 $ with prescribed volume
September  2019, 18(5): 2765-2787. doi: 10.3934/cpaa.2019124

Existence, uniqueness and regularity of the solution of the time-fractional Fokker–Planck equation with general forcing

Kim-Ngan Le 1, , William McLean 2, and  Martin Stynes 3,

1. 

School of Mathematical Sciences, Monash University, VIC 3800, Australia
2. 

School of Mathematics and Statistics, The University of New South Wales, Sydney 2052, Australia
3. 

Applied and Computational Mathematics Division, Beijing Computational Science Research Center, Beijing 100193, China

Received  November 2018 Revised  November 2018 Published  April 2019

Fund Project: The research of the first author is supported in part by the Australian Government through the Australian Research Council Discovery Projects funding scheme (project number DP170100605). The research of the third author is supported in part by the National Natural Science Foundation of China under grants 91430216 and NSAF-U1530401.

A time-fractional Fokker–Planck initial-boundary value problem is considered, with differential operator $ u_t-\nabla\cdot(\partial_t^{1-\alpha}\kappa_\alpha\nabla u -{\bf{F}}\partial_t^{1-\alpha}u) $, where $ 0<\alpha <1 $. The forcing function $ {\bf{F}} = {\bf{F}}(t,x) $, which is more difficult to analyse than the case $ {\bf{F}} = {\bf{F}}(x) $ investigated previously by other authors. The spatial domain $ \Omega \subset\mathbb{R}^d $, where $ d\ge 1 $, has a smooth boundary. Existence, uniqueness and regularity of a mild solution $ u $ is proved under the hypothesis that the initial data $ u_0 $ lies in $ L^2(\Omega) $. For $ 1/2<\alpha<1 $ and $ u_0\in H^2(\Omega)\cap H_0^1(\Omega) $, it is shown that $ u $ becomes a classical solution of the problem. Estimates of time derivatives of the classical solution are derived—these are known to be needed in numerical analyses of this problem.

Keywords: Fokker–Planck equation, Riemann–Liouville derivative, time-fractional, regularity of solution.
Mathematics Subject Classification: 35R11.
Citation: Kim-Ngan Le, William McLean, Martin Stynes. Existence, uniqueness and regularity of the solution of the time-fractional Fokker–Planck equation with general forcing. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2765-2787. doi: 10.3934/cpaa.2019124
References:
[1]

F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, vol. 183, Springer-Verlag, New York, 2013. doi: 10.1007/978-1-4614-5975-0.  Google Scholar

[2]

K. Diethelm, The Analysis of Fractional Differential Equations, vol. 2004 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2010 doi: 10.1007/978-3-642-14574-2.  Google Scholar

[3]

S. D. Eidelman and A. N. Kochubei, Cauchy problem for fractional diffusion equations, Journal of Differential Equations, 199 (2004), 211-255.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[4]

L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, 2nd edition, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.  Google Scholar

[5]

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, IMA J. Numer. Anal. (to appear). Google Scholar

[6]

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), 1763-1784.  doi: 10.1137/15M1031734.  Google Scholar

[7]

Y. Luchko, Maximum principle for the generalized time-fractional diffusion equation, Journal of Mathematical Analysis and Applications, 351 (2009), 218-223.  doi: 10.1016/j.jmaa.2008.10.018.  Google Scholar

[8]

Y. Luchko, Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation, Computers & Mathematics with Applications, 59 (2010), 1766-1772.  doi: 10.1016/j.camwa.2009.08.015.  Google Scholar

[9]

W. McLean, Regularity of solutions to a time-fractional diffusion equation, ANZIAM J., 52 (2010), 123-138.  doi: 10.1017/S1446181111000617.  Google Scholar

[10]

W. McLean, Fast summation by interval clustering for an evolution equation with memory, SIAM J. Sci. Comput., 34 (2012), A3039-A3056.  doi: 10.1137/120870505.  Google Scholar

[11]

W. McLean, K. Mustapha, R. Ali and O. Knio, Well-posedness of time-fractional advection-diffusion-reaction equations, arXiv e-prints. Google Scholar

[12]

W. McLean, K. Mustapha, R. Ali and O. Knio, Regularity theory for time-fractional, advection-diffusion-reaction equations, arXiv e-prints. Google Scholar

[13]

J. Mu, B. Ahmad and S. Huang, Existence and regularity of solutions to time-fractional diffusion equations, Computers & Mathematics with Applications, 73 (2017), 985-996.  doi: 10.1016/j.camwa.2016.04.039.  Google Scholar

[14]

K. Mustapha and D. Schötzau, Well-posedness of hp-version discontinuous Galerkin methods for fractional diffusion wave equations, IMA Journal of Numerical Analysis, 34 (2014), 1426-1446.  doi: 10.1093/imanum/drt048.  Google Scholar

[15]

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), 211-228.  doi: 10.1016/j.cnsns.2017.03.004.  Google Scholar

[16]

R.-N. Wang, D.-H. Chen and T.-J. Xiao, Abstract fractional Cauchy problems with almost sectorial operators, Journal of Differential Equations, 252 (2012), 202-235.  doi: 10.1016/j.jde.2011.08.048.  Google Scholar

[17]

H. Ye, J. Gao and Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328 (2007), 1075-1081.  doi: 10.1016/j.jmaa.2006.05.061.  Google Scholar

show all references

References:
[1]

F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, vol. 183, Springer-Verlag, New York, 2013. doi: 10.1007/978-1-4614-5975-0.  Google Scholar

[2]

K. Diethelm, The Analysis of Fractional Differential Equations, vol. 2004 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2010 doi: 10.1007/978-3-642-14574-2.  Google Scholar

[3]

S. D. Eidelman and A. N. Kochubei, Cauchy problem for fractional diffusion equations, Journal of Differential Equations, 199 (2004), 211-255.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[4]

L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, 2nd edition, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.  Google Scholar

[5]

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, IMA J. Numer. Anal. (to appear). Google Scholar

[6]

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), 1763-1784.  doi: 10.1137/15M1031734.  Google Scholar

[7]

Y. Luchko, Maximum principle for the generalized time-fractional diffusion equation, Journal of Mathematical Analysis and Applications, 351 (2009), 218-223.  doi: 10.1016/j.jmaa.2008.10.018.  Google Scholar

[8]

Y. Luchko, Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation, Computers & Mathematics with Applications, 59 (2010), 1766-1772.  doi: 10.1016/j.camwa.2009.08.015.  Google Scholar

[9]

W. McLean, Regularity of solutions to a time-fractional diffusion equation, ANZIAM J., 52 (2010), 123-138.  doi: 10.1017/S1446181111000617.  Google Scholar

[10]

W. McLean, Fast summation by interval clustering for an evolution equation with memory, SIAM J. Sci. Comput., 34 (2012), A3039-A3056.  doi: 10.1137/120870505.  Google Scholar

[11]

W. McLean, K. Mustapha, R. Ali and O. Knio, Well-posedness of time-fractional advection-diffusion-reaction equations, arXiv e-prints. Google Scholar

[12]

W. McLean, K. Mustapha, R. Ali and O. Knio, Regularity theory for time-fractional, advection-diffusion-reaction equations, arXiv e-prints. Google Scholar

[13]

J. Mu, B. Ahmad and S. Huang, Existence and regularity of solutions to time-fractional diffusion equations, Computers & Mathematics with Applications, 73 (2017), 985-996.  doi: 10.1016/j.camwa.2016.04.039.  Google Scholar

[14]

K. Mustapha and D. Schötzau, Well-posedness of hp-version discontinuous Galerkin methods for fractional diffusion wave equations, IMA Journal of Numerical Analysis, 34 (2014), 1426-1446.  doi: 10.1093/imanum/drt048.  Google Scholar

[15]

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), 211-228.  doi: 10.1016/j.cnsns.2017.03.004.  Google Scholar

[16]

R.-N. Wang, D.-H. Chen and T.-J. Xiao, Abstract fractional Cauchy problems with almost sectorial operators, Journal of Differential Equations, 252 (2012), 202-235.  doi: 10.1016/j.jde.2011.08.048.  Google Scholar

[17]

H. Ye, J. Gao and Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328 (2007), 1075-1081.  doi: 10.1016/j.jmaa.2006.05.061.  Google Scholar

[1]

Alessandro Carbotti, Giovanni E. Comi. A note on Riemann-Liouville fractional Sobolev spaces. Communications on Pure & Applied Analysis, 2021, 20 (1) : 17-54. doi: 10.3934/cpaa.2020255
[2]

Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137
[3]

Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020318
[4]

S. Sadeghi, H. Jafari, S. Nemati. Solving fractional Advection-diffusion equation using Genocchi operational matrix based on Atangana-Baleanu derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020435
[5]

Teresa D'Aprile. Bubbling solutions for the Liouville equation around a quantized singularity in symmetric domains. Communications on Pure & Applied Analysis, 2021, 20 (1) : 159-191. doi: 10.3934/cpaa.2020262
[6]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247
[7]

Anh Tuan Duong, Phuong Le, Nhu Thang Nguyen. Symmetry and nonexistence results for a fractional Choquard equation with weights. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 489-505. doi: 10.3934/dcds.2020265
[8]

Sihem Guerarra. Maximum and minimum ranks and inertias of the Hermitian parts of the least rank solution of the matrix equation AXB = C. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 75-86. doi: 10.3934/naco.2020016
[9]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432
[10]

Min Xi, Wenyu Sun, Jun Chen. Survey of derivative-free optimization. Numerical Algebra, Control & Optimization, 2020, 10 (4) : 537-555. doi: 10.3934/naco.2020050
[11]

Oussama Landoulsi. Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the $ L^2 $-supercritical case. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 701-746. doi: 10.3934/dcds.2020298
[12]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319
[13]

Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020445
[14]

Abdollah Borhanifar, Maria Alessandra Ragusa, Sohrab Valizadeh. High-order numerical method for two-dimensional Riesz space fractional advection-dispersion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020355
[15]

Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, 2021, 20 (1) : 301-317. doi: 10.3934/cpaa.2020267
[16]

Sören Bartels, Jakob Keck. Adaptive time stepping in elastoplasticity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 71-88. doi: 10.3934/dcdss.2020323
[17]

Yi An, Bo Li, Lei Wang, Chao Zhang, Xiaoli Zhou. Calibration of a 3D laser rangefinder and a camera based on optimization solution. Journal of Industrial & Management Optimization, 2021, 17 (1) : 427-445. doi: 10.3934/jimo.2019119
[18]

Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442
[19]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348
[20]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (87)
  • HTML views (175)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences