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Higher order conformally invariant equations in $ {\mathbb R}^3 $ with prescribed volume
Existence, uniqueness and regularity of the solution of the time-fractional Fokker–Planck equation with general forcing
| 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 |
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.
References:
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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. |
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K. Diethelm, The Analysis of Fractional Differential Equations, vol. 2004 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2010
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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. |
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L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, 2nd edition, American Mathematical Society, Providence, RI, 2010.
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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). |
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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. |
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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. |
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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. |
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W. McLean,
Regularity of solutions to a time-fractional diffusion equation, ANZIAM J., 52 (2010), 123-138.
doi: 10.1017/S1446181111000617. |
| [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. |
| [11] |
W. McLean, K. Mustapha, R. Ali and O. Knio, Well-posedness of time-fractional advection-diffusion-reaction equations, arXiv e-prints. |
| [12] |
W. McLean, K. Mustapha, R. Ali and O. Knio, Regularity theory for time-fractional, advection-diffusion-reaction equations, arXiv e-prints. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
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. |
| [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. |
| [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. |
| [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. |
| [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). |
| [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. |
| [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. |
| [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. |
| [9] |
W. McLean,
Regularity of solutions to a time-fractional diffusion equation, ANZIAM J., 52 (2010), 123-138.
doi: 10.1017/S1446181111000617. |
| [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. |
| [11] |
W. McLean, K. Mustapha, R. Ali and O. Knio, Well-posedness of time-fractional advection-diffusion-reaction equations, arXiv e-prints. |
| [12] |
W. McLean, K. Mustapha, R. Ali and O. Knio, Regularity theory for time-fractional, advection-diffusion-reaction equations, arXiv e-prints. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
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