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Finite difference scheme for the time-space fractional diffusion equations

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Cao J., Li C.

Open Physics

2013

vol. 11

issue 10

1440-1456

In this paper, we derive two novel finite difference schemes for two types of time-space fractional diffusion equations by adopting weighted and shifted Grünwald operator, which is used to approximate the Riemann-Liouville fractional derivative to the second order accuracy. The stability and convergence of the schemes are analyzed via mathematical induction. Moreover, the illustrative numerical examples are carried out to verify the accuracy and effectiveness of the schemes.

Numerical approach to the Caputo derivative of the unknown function

100%

Zeng F., Li C.

Open Physics

2013

vol. 11

issue 10

1433-1439

If a function can be explicitly expressed, then one can easily compute its Caputo derivative by the known methods. If a function cannot be explicitly expressed but it satisfies a differential equation, how to seek Caputo derivative of such a function has not yet been investigated. In this paper, we propose a numerical algorithm for computing the Caputo derivative of a function defined by a classical (integer-order) differential equation. By the properties of Caputo derivative derived in this paper, we can change the original typical differential system into an equivalent Caputo-type differential system. Numerical examples are given to support the derived numerical method.

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2 Open Physics

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2 Li C.

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1 Zeng F.

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Finite difference scheme for the time-space fractional diffusion equations

Numerical approach to the Caputo derivative of the unknown function