GENERAL |
|
|
|
|
Solution to the Fokker–Planck Equation with Piecewise-Constant Drift |
Bin Cheng1, Ya-Ming Chen2**, Xiao-Gang Deng2,3 |
1College of Computer, National University of Defense Technology, Changsha 410073, China 2College of Aerospace Science and Engineering, National University of Defense Technology, Changsha 410073, China 3Chinese Academy of Military Science, Beijing 100071, China
|
|
Cite this article: |
Bin Cheng, Ya-Ming Chen, Xiao-Gang Deng 2020 Chin. Phys. Lett. 37 060201 |
|
|
Abstract We study the solution to the Fokker–Planck equation with piecewise-constant drift, taking the case with two jumps in the drift as an example. The solution in Laplace space can be expressed in closed analytic form, and its inverse can be obtained conveniently using some numerical inversion methods. The results obtained by numerical inversion can be regarded as exact solutions, enabling us to demonstrate the validity of some numerical methods for solving the Fokker–Planck equation. In particular, we use the solved problem as a benchmark example for demonstrating the fifth-order convergence rate of the finite difference scheme proposed previously [Chen Y and Deng X Phys. Rev. E 100 (2019) 053303].
|
|
Received: 04 February 2020
Published: 26 May 2020
|
|
PACS: |
02.60.Cb
|
(Numerical simulation; solution of equations)
|
|
52.65.Ff
|
(Fokker-Planck and Vlasov equation)
|
|
02.30.Uu
|
(Integral transforms)
|
|
|
Fund: *Supported by the National Natural Science Foundation of China (Grant Nos. 11972370 and 61772542). |
|
|
[1] | Crandall S H, Lee S S and Williams Jr J H 1974 J. Appl. Mech. 41 1094 |
[2] | Crandall S H and Lee S S 1976 Ingenieur-Arch. 45 361 |
[3] | Reimann P 2002 Phys. Rep. 361 57 |
[4] | De Gennes P G 2005 J. Stat. Phys. 119 953 |
[5] | Gnoli A, Puglisi A and Touchette H 2013 Europhys. Lett. 102 14002 |
[6] | Baule A, Touchette H and Cohen E G D 2011 Nonlinearity 24 351 |
[7] | Menzel A M and Goldenfeld N 2011 Phys. Rev. E 84 011122 |
[8] | Baule A and Sollich P 2012 Europhys. Lett. 97 20001 |
[9] | Baule A and Sollich P 2013 Phys. Rev. E 87 032112 |
[10] | Chen Y and Just W 2014 Phys. Rev. E 89 022103 |
[11] | Chen Y and Just W 2014 Phys. Rev. E 90 042102 |
[12] | Geffert P M and Just W 2017 Phys. Rev. E 95 062111 |
[13] | Xu W, Wang L, Feng J, Qiao Y and Han P 2018 Chin. Phys. B 27 110503 |
[14] | Hayakawa H 2005 Physica D 205 48 |
[15] | Risken H 1989 The Fokker–Planck Equation: Methods of Solution and Applications (Berlin: Springer) |
[16] | Touchette H, Der Straeten E V and Just W 2010 J. Phys. A 43 445002 |
[17] | Caughey T K and Dienes J K 1961 J. Appl. Phys. 32 2476 |
[18] | Touchette H, Prellberg T and Just W 2012 J. Phys. A 45 395002 |
[19] | Karatzas I and Shreve S E 1984 Ann. Probab. 12 819 |
[20] | Simpson D J W and Kuske R 2014 Disc. Contin. Dyn. Syst. B 19 2889 |
[21] | Baule A, Cohen E G D and Touchette H 2010 J. Phys. A 43 025003 |
[22] | Chen Y, Baule A, Touchette H and Just W 2013 Phys. Rev. E 88 052103 |
[23] | Leobacher G and Szölgyenyi M 2016 BIT Numer. Math. 56 151 |
[24] | Ngo H L and Taguchi D 2017 Stat. Probab. Lett. 125 55 |
[25] | Papaspiliopoulos O, Roberts G O and Taylor K B 2016 Adv. Appl. Probab. 48 249 |
[26] | Dereudre D, Mazzonetto S and Roelly S 2017 SIAM J. Sci. Comput. 39 A711 |
[27] | Chen Y and Deng X 2018 Phys. Rev. E 98 033302 |
[28] | Chen Y and Deng X 2019 Phys. Rev. E 100 053303 |
[29] | Atkinson J D and Caughey T K 1968 Int. J. Non-Linear Mech. 3 137 |
[30] | Atkinson J D and Caughey T K 1968 Int. J. Non-Linear Mech. 3 399 |
[31] | Talbot A 1979 IMA J. Appl. Math. 23 97 |
[32] | Abate J and Valkó P P 2004 Int. J. Numer. Methods Eng. 60 979 |
[33] | Abate J and Whitt W 2006 INFORMS J. Comput. 18 408 |
[34] | Zhang M and Shu C W 2003 Math. Mod. Meth. Appl. Sci. 13 395 |
[35] | Tian X and Du Q 2013 SIAM J. Numer. Anal. 51 3458 |
|
|
Viewed |
|
|
|
Full text
|
|
|
|
|
Abstract
|
|
|
|
|