Journal ArticleDOI
Finite difference/finite element method for two-dimensional space and time fractional Bloch-Torrey equations
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TLDR
In this paper, a class of two-dimensional space and time fractional Bloch-Torrey equations (2D-STFBTEs) are considered and a semi-discrete variational formulation for 2D- STFB TEs is obtained by finite difference method and Galerkin finite element method.About:
This article is published in Journal of Computational Physics.The article was published on 2015-07-15. It has received 123 citations till now. The article focuses on the topics: Spectral element method & Mixed finite element method.read more
Citations
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Journal ArticleDOI
A fast linearized conservative finite element method for the strongly coupled nonlinear fractional Schrödinger equations
TL;DR: An iterative algorithm is proposed, by which the coefficient matrix is independent of the time level, and thus it leads to Toeplitz-like linear systems that can be efficiently solved by Krylov subspace solvers with circulant preconditioners.
Journal ArticleDOI
Galerkin finite element method for nonlinear fractional Schrödinger equations
TL;DR: By Brouwer fixed point theorem and fractional Gagliardo-Nirenberg inequality, it is proved the fully discrete system is uniquely solvable.
Journal ArticleDOI
Numerical methods for fractional partial differential equations
Changpin Li,An Chen +1 more
TL;DR: This review paper is mainly concerned with the finite difference methods, the Galerkin finite element methods, and the spectral methods for fractional partial differential equations (FPDEs), which are divided into the time-fractionsal, space-fractional, and space-time- fractional partial partial differential equation (PDEs).
Journal ArticleDOI
Finite element multigrid method for multi-term time fractional advection diffusion equations
TL;DR: Two fully discrete schemes of MTFADEs with different definitions on multi-term time fractional derivative are obtained and a V-cycle multigrid method is proposed to solve the resulting linear systems.
Journal ArticleDOI
Unstructured-mesh Galerkin finite element method for the two-dimensional multi-term time–space fractional Bloch–Torrey equations on irregular convex domains
TL;DR: An unstructured-mesh Galerkin finite element method for the two-dimensional multi-term time–space fractional diffusion equation with Riesz fractional operators on irregular convex domains is proposed and the stability and convergence of the numerical scheme are rigorously established.
References
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Book
An Introduction to the Fractional Calculus and Fractional Differential Equations
Kenneth S. Miller,Bertram Ross +1 more
TL;DR: The Riemann-Liouville Fractional Integral Integral Calculus as discussed by the authors is a fractional integral integral calculus with integral integral components, and the Weyl fractional calculus has integral components.
Journal ArticleDOI
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.
Book
The Mathematical Theory of Finite Element Methods
TL;DR: In this article, the construction of a finite element of space in Sobolev spaces has been studied in the context of operator-interpolation theory in n-dimensional variational problems.
Journal ArticleDOI
The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics
Ralf Metzler,Joseph Klafter +1 more
TL;DR: Fractional dynamics has experienced a firm upswing during the past few years, having been forged into a mature framework in the theory of stochastic processes as mentioned in this paper, and a large number of research papers developing fractional dynamics further, or applying it to various systems have appeared since our first review article on the fractional Fokker-Planck equation.
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