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FODE

swMATH ID: 8377
Software Authors: Deng, Weihua
Description: Numerical algorithm for the time fractional Fokker-Planck equation. Anomalous diffusion is one of the most ubiquitous phenomena in nature, and it is present in a wide variety of physical situations, for instance, transport of fluid in porous media, diffusion of plasma, diffusion at liquid surfaces, etc. The fractional approach proved to be highly effective in a rich variety of scenarios such as continuous time random walk models, generalized Langevin equations, or the generalized master equation. To investigate the subdiffusion of anomalous diffusion, it would be useful to study a time fractional Fokker-Planck equation. In this paper, firstly the time fractional, the sense of Riemann-Liouville derivative, Fokker-Planck equation is transformed into a time fractional ordinary differential equation (FODE) in the sense of Caputo derivative by discretizing the spatial derivatives and using the properties of Riemann-Liouville derivative and Caputo derivative. Then combining the predictor-corrector approach with the method of lines, the algorithm is designed for numerically solving FODE with the numerical error \(O(k^{min{1+2alpha ,2}})+O(h^{2})\), and the corresponding stability condition is got. The effectiveness of this numerical algorithm is evaluated by comparing its numerical results for \(alpha =1.0\) with the ones of directly discretizing classical Fokker-Planck equation, some numerical results for time fractional Fokker-Planck equation with several different fractional orders are demonstrated and compared with each other, moreover for \(alpha =0.8\) the convergent order in space is confirmed and the numerical results with different time step sizes are shown.
Homepage: http://www.sciencedirect.com/science/article/pii/S0021999107004226
Keywords: Riemann-Liouville derivative; caputo derivative; Fokker-Planck equation; subdiffusion; predictor-corrector approach; method of lines; convergent order; numerical stability
Related Software: ma2dfc; Matlab; Fractional Order Chaotic Systems; FracPECE; Mittag-Leffler; Adaptative L1 TE and Predictive; ML; mftoolbox; IQPACK; BVPh; mlrnd; Regularization tools; Sprott's Software; Ninteger; Simulink; SpectralMeasures.jl; AGMG; FOTF Toolbox; FPINNs; iFEM
Referenced in: 329 Publications

Standard Articles

1 Publication describing the Software, including 1 Publication in zbMATH Year
Numerical algorithm for the time fractional Fokker-Planck equation. Zbl 1388.35095
Deng, Weihua
2007
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Referenced by 532 Authors

29 Deng, Weihua
18 Wang, Hong
9 Liu, Fawang
9 Wei, LeiLei
9 Wu, Yujiang
8 Dehghan Takht Fooladi, Mehdi
8 Huang, Chengming
8 Li, Meng
7 Abbaszadeh, Mostafa
7 Nie, Yufeng
7 Sun, Zhizhong
6 Cheng, Aijie
6 Jia, Jinhong
6 Jiang, Yingjun
6 Nie, Daxin
6 Sun, Jing
6 Turner, Ian William
6 Yuan, Zhanbin
6 Zhang, Xindong
6 Zheng, Xiangcheng
6 Zhu, Xiaogang
5 Chen, Minghua
5 Hesthaven, Jan S.
5 Huang, Yunqing
5 Li, Can
5 Li, Hong
5 Liu, Huan
5 Liu, Yang
5 Tang, Yifa
5 Xu, Chuanju
5 Yan, Yubin
5 Zhao, Lijing
4 Ahmad, Bashir
4 Anh, Vo V.
4 Bu, Weiping
4 Chen, Yanping
4 Jiang, Xiaoyun
4 Karniadakis, George Em
4 Li, Dongfang
4 Liao, HongLin
4 Ma, Jingtang
4 Petras, Ivo
4 Ren, Jincheng
4 Wang, Jungang
4 Xu, Da
4 Yang, Zongze
3 Atangana, Abdon
3 Bhrawy, Ali Hassan
3 Carrer, José Antonio Marques
3 Chen, Hu
3 Chen, Liping
3 Chen, Wenping
3 Deng, Jingwei
3 Fei, Mingfa
3 Fu, Hongfei
3 Fu, Yayun
3 He, Yinnian
3 Javidi, Mohammad
3 Li, Binjie
3 Li, Changpin
3 Lu, Shujuan
3 Seaïd, Mohammed
3 She, Zihang
3 Singh, Vineet Kumar
3 Sun, Haiwei
3 Trevelyan, Jon
3 Vong, Seakweng
3 Wang, Nan
3 Wang, Yuanming
3 Wang, Yushun
3 Wu, Ranchao
3 Xie, Xiaoping
3 Xu, Xuejun
3 Yang, Xuehua
3 Zaky, Mahmoud A.
3 Zhang, Haixiang
3 Zhang, Lu
3 Zhang, Yanan
3 Zheng, Minling
3 Zhou, Zhiqiang
2 Aboelenen, Tarek
2 Ali Akbar, M.
2 Angstmann, Christopher N.
2 Babolian, Esmail
2 Bai, Zhongzhi
2 Băleanu, Dumitru I.
2 Cai, Wenjun
2 Chai, Yi
2 Dai, Huiya
2 Devi, Vinita
2 Donnelly, I. C.
2 Duan, Jinqiao
2 Ford, Neville J.
2 Gu, Xian-Ming
2 He, Siriguleng
2 Henry, Bruce I.
2 Hou, Dianming
2 Huang, Jianfei
2 Jin, Xiaoqing
2 Khaliq, Abdul Q. M.
...and 432 more Authors
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Referenced in 78 Serials

29 Journal of Computational Physics
26 Journal of Scientific Computing
23 Computers & Mathematics with Applications
21 Numerical Algorithms
17 Applied Numerical Mathematics
14 Applied Mathematics and Computation
12 Journal of Computational and Applied Mathematics
9 International Journal of Computer Mathematics
9 Abstract and Applied Analysis
8 Applied Mathematical Modelling
7 Engineering Analysis with Boundary Elements
7 Nonlinear Dynamics
7 Advances in Difference Equations
6 Computational and Applied Mathematics
6 Advances in Applied Mathematics and Mechanics
5 BIT
5 Numerical Methods for Partial Differential Equations
5 Mathematical Problems in Engineering
5 East Asian Journal on Applied Mathematics
4 SIAM Journal on Scientific Computing
4 Advances in Computational Mathematics
4 Science China. Mathematics
3 Computer Methods in Applied Mechanics and Engineering
3 Physica A
3 Chaos, Solitons and Fractals
3 Mathematics and Computers in Simulation
3 SIAM Journal on Numerical Analysis
3 Applied Mathematics Letters
3 Discrete and Continuous Dynamical Systems. Series B
3 Computational Methods in Applied Mathematics
3 European Series in Applied and Industrial Mathematics (ESAIM): Mathematical Modelling and Numerical Analysis
3 Communications in Computational Physics
3 Advances in Mathematical Physics
3 Communications on Applied Mathematics and Computation
2 Journal of Statistical Physics
2 Mathematical Methods in the Applied Sciences
2 Calcolo
2 Computational Mechanics
2 Chaos
2 Fractional Calculus & Applied Analysis
2 Discrete Dynamics in Nature and Society
2 Communications in Nonlinear Science and Numerical Simulation
2 Journal of Applied Mathematics
2 Journal of Applied Mathematics and Computing
2 International Journal of Differential Equations
2 International Journal of Applied and Computational Mathematics
2 AIMS Mathematics
2 Nonlinear Physical Science
1 Computer Physics Communications
1 Mathematics of Computation
1 Journal of Optimization Theory and Applications
1 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods
1 Numerical Functional Analysis and Optimization
1 Numerische Mathematik
1 Acta Applicandae Mathematicae
1 SIAM Journal on Matrix Analysis and Applications
1 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering
1 International Journal of Numerical Methods for Heat & Fluid Flow
1 Fractals
1 Journal of Inverse and Ill-Posed Problems
1 Complexity
1 Arab Journal of Mathematical Sciences
1 Communications in Applied Analysis
1 Taiwanese Journal of Mathematics
1 Computational Geosciences
1 Bulletin of the Malaysian Mathematical Sciences Society. Second Series
1 Journal of Function Spaces and Applications
1 Fixed Point Theory and Applications
1 International Journal of Numerical Analysis and Modeling
1 Acta Mechanica Sinica
1 Applications and Applied Mathematics
1 Numerical Mathematics: Theory, Methods and Applications
1 Journal of Applied Analysis and Computation
1 Mathematical Sciences
1 Journal of Mathematics
1 Journal of Function Spaces
1 Computational Methods for Differential Equations
1 International Journal of Systems Science. Principles and Applications of Systems and Integration

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