Finite difference scheme for multi-term variable-order fractional diffusion equation
TL;DR: In this paper, a multi-term variable-order fractional diffusion equation on a finite domain is considered and a finite difference scheme is proposed to approximate the temporal direction derivative by L1-algorithm and the spatial direction derivative using the standard and shifted Grunwald method, respectively.
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Abstract: In this paper, we consider a multi-term variable-order fractional diffusion equation on a finite domain, which involves the Caputo variable-order time fractional derivative of order $\alpha(x,t) \in(0,1) $
and the Riesz variable-order space fractional derivatives of order $\beta(x,t) \in (0,1)$
, $\gamma(x,t)\in(1,2)$
. Approximating the temporal direction derivative by L1-algorithm and the spatial direction derivative by the standard and shifted Grunwald method, respectively, a characteristic finite difference scheme is proposed. The stability and convergence of the difference schemes are analyzed via mathematical induction. Some numerical experiments are provided to show the efficiency of the proposed difference schemes.
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A review on variable-order fractional differential equations: mathematical foundations, physical models, numerical methods and applications
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The Temporal Second Order Difference Schemes Based on the Interpolation Approximation for Solving the Time Multi-term and Distributed-order Fractional Sub-diffusion Equations
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- 05 Aug 2016
TL;DR: In this article, a special point is found for the interpolation approximation of the linear combination of multi-term fractional derivatives and the derived numerical differentiation formula can achieve at least second order accuracy.
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Approximation methods for solving fractional equations
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References
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Anatoly A. Kilbas,Hari M. Srivastava,Juan J. Trujillo +2 more
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TL;DR: In this article, the authors present a method for solving Fractional Differential Equations (DFE) using Integral Transform Methods for Explicit Solutions to FractionAL Differentially Equations.
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Kenneth S. Miller,Bertram Ross +1 more
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Stefan Samko,Anatoly A. Kilbas,O. I. Marichev +2 more
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TL;DR: Fractional integrals and derivatives on an interval fractional integral integrals on the real axis and half-axis further properties of fractional integral and derivatives, and derivatives of functions of many variables applications to integral equations of the first kind with power and power-logarithmic kernels integral equations with special function kernels applications to differential equations as discussed by the authors.
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Applications Of Fractional Calculus In Physics
Rudolf Hilfer
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TL;DR: An introduction to fractional calculus can be found in this paper, where Butzer et al. present a discussion of fractional fractional derivatives, derivatives and fractal time series.
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