Abstract
This paper aims to provide the high order numerical schemes for the time-space tempered fractional Fokker-Planck equation in a finite domain. The high order difference operators, called the tempered and weighted and shifted Lubich difference operators, are used to approximate the time tempered fractional derivative. The spatial operators are discretized by the central difference methods. We apply the central difference methods to the spatial operators and obtain that the numerical schemes are convergent with orders O(tau^{q} + h^{2}) (q = 1,2,3,4,5). The stability and convergence of the first order numerical scheme are rigorously analyzed. And the effectiveness of the presented schemes is testified with several numerical experiments. Additionally, some physical properties of this diffusion system are simulated.
Highlights
In recent decades, fractional partial differential equations have become a powerful tool to model the particle transport in anomalous diffusion in various fields
Some important achievements have been made for the fractional differential equations [ – ]
Tempered fractional derivatives and the corresponding tempered fractional differential equations have played a key role in physics [ ], ground water hydrology [ ], finance [ ], poroelasticity [ ], and so on
Summary
Fractional partial differential equations have become a powerful tool to model the particle transport in anomalous diffusion in various fields. Some important achievements have been made for the fractional differential equations [ – ]. The tempered anomalous diffusion equations [ – ] have drawn the wide interests of the researchers. It is closer to reality in the sense of the finite life span or bounded physical space of the diffusion particles. The detailed introductions about the definitions and properties of the tempered fractional calculus can be seen in [ – ] and the references therein. As the generalization of fractional calculus, tempered fractional calculus does not have the properties of the fractional calculus, but can describe some of the other complex dynamics [ ]. Tempered fractional derivatives and the corresponding tempered fractional differential equations have played a key role in physics [ ], ground water hydrology [ ], finance [ ], poroelasticity [ ], and so on
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