Finite difference approximation for two-dimensional time fractional diffusion equation
TL;DR: In this paper, an implicit difference approximation for the 2D-TFDE is presented, and stability and convergence of the method are discussed using mathematical induction, and a numerical example is given.
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Abstract: Fractional diffusion equations have recently been used to model problems in physics, hydrology, biology and other areas of application. In this paper, we consider a two-dimensional time fractional diffusion equation (2D-TFDE) on a finite domain. An implicit difference approximation for the 2D-TFDE is presented. Stability and convergence of the method are discussed using mathematical induction. Finally, a numerical example is given. The numerical result is in excellent agreement with our theoretical analysis.
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Citations
The Use of Finite Difference/Element Approaches for Solving the Time-Fractional Subdiffusion Equation
TL;DR: In this paper, two finite difference/element approaches for the time-fractional subdiffusion equation with Dirichlet boundary conditions are developed, in which the time direction is approximated by the fractional linear multistep method and the space direction by the finite element method.
330
Finite difference methods for fractional differential equations
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Alternating direction implicit schemes for the two-dimensional fractional sub-diffusion equation
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TL;DR: Two new alternating direction implicit schemes based on the L"1 approximation and backward Euler method are considered for the solution of a two-dimensional anomalous sub-diffusion equation with time fractional derivative, and the solvability, unconditional stability and H^1 norm convergence are proved.
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A novel compact numerical method for solving the two-dimensional non-linear fractional reaction-subdiffusion equation
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TL;DR: A novel compact numerical method which is second-order temporal accuracy and fourth-order spatial accuracy is derived from the two-dimensional non-linear fractional reaction-subdiffusion equation.
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Alternating direction implicit Galerkin finite element method for the two-dimensional fractional diffusion-wave equation
TL;DR: In these methods, Galerkin finite element is used for the spatial discretization, and, for the time stepping, new alternating direction implicit (ADI) method based on the Crank-Nicolson method are considered.
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The random walk's guide to anomalous diffusion: a fractional dynamics approach
Ralf Metzler,Joseph Klafter +1 more
TL;DR: Fractional kinetic equations of the diffusion, diffusion-advection, and Fokker-Planck type are presented as a useful approach for the description of transport dynamics in complex systems which are governed by anomalous diffusion and non-exponential relaxation patterns.
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Finite difference approximations for fractional advection-dispersion flow equations
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Fractional diffusion and wave equations
W. R. Schneider,W. Wyss +1 more
TL;DR: In this article, the Green's function of fractional diffusion is shown to be a probability density and the corresponding Green's functions are obtained in closed form for arbitrary space dimensions in terms of Fox functions and their properties are exhibited.
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Finite difference approximations for two-sided space-fractional partial differential equations
TL;DR: In this paper, the authors examined some practical numerical methods to solve a class of initial-boundary value fractional partial differential equations with variable coefficients on a finite domain, and the stability, consistency, and (therefore) convergence of the methods are discussed.
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