## 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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### Referenced in 28 Fields

 285 Numerical analysis (65-XX) 243 Partial differential equations (35-XX) 70 Real functions (26-XX) 49 Ordinary differential equations (34-XX) 20 Statistical mechanics, structure of matter (82-XX) 17 Fluid mechanics (76-XX) 8 Probability theory and stochastic processes (60-XX) 7 Integral equations (45-XX) 6 Approximations and expansions (41-XX) 5 Linear and multilinear algebra; matrix theory (15-XX) 5 Dynamical systems and ergodic theory (37-XX) 5 Biology and other natural sciences (92-XX) 4 Operator theory (47-XX) 3 Special functions (33-XX) 3 Harmonic analysis on Euclidean spaces (42-XX) 3 Calculus of variations and optimal control; optimization (49-XX) 3 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 3 Systems theory; control (93-XX) 2 Mechanics of deformable solids (74-XX) 2 Optics, electromagnetic theory (78-XX) 2 Classical thermodynamics, heat transfer (80-XX) 2 Geophysics (86-XX) 1 General and overarching topics; collections (00-XX) 1 Difference and functional equations (39-XX) 1 Integral transforms, operational calculus (44-XX) 1 Global analysis, analysis on manifolds (58-XX) 1 Statistics (62-XX) 1 Computer science (68-XX)