A Finite Element Method for the Fractional Sturm-Liouville Problem

In this work, we propose an efficient finite element method for solving fractional Sturm-Liouville problems involving either the Caputo or Riemann-Liouville derivative of order $\alpha\in(1,2)$ on the unit interval $(0,1)$. It is based on novel variational formulations of the eigenvalue problem. Error estimates are provided for the finite element approximations of the eigenvalues. Numerical results are presented to illustrate the efficiency and accuracy of the method. The results indicate that the method can achieve a second-order convergence for both fractional derivatives, and can provide accurate approximations to multiple eigenvalues simultaneously.Comment: 30 pages, 7 figure

Charge and current oscillations in Fractional quantum Hall systems with edges

Stationary solutions of the Chern-Simons effective field theory for the fractional quantum Hall systems with edges are presented for Hall bar, disk and annulus. In the infinitely long Hall bar geometry (non compact case), the charge density is shown to be monotonic inside the sample. In sharp contrast, spatial oscillatory modes of charge density are found for the two circular geometries, which indicate that in systems with compact geometry, charge and current exist also far from the edges.Comment: 16 pages, 6 figures Revte

An inverse Sturm-Liouville problem with a fractional derivative

In this paper, we numerically investigate an inverse problem of recovering the potential term in a fractional Sturm-Liouville problem from one spectrum. The qualitative behaviors of the eigenvalues and eigenfunctions are discussed, and numerical reconstructions of the potential with a Newton method from finite spectral data are presented. Surprisingly, it allows very satisfactory reconstructions for both smooth and discontinuous potentials, provided that the order $\alpha\in(1,2)$ of fractional derivative is sufficiently away from 2.Comment: 16 pages, 6 figures, accepted for publication in Journal of Computational Physic

An efficient implementation of an implicit FEM scheme for fractional-in-space reaction-diffusion equations

Fractional differential equations are becoming increasingly used as a modelling tool for processes with anomalous diffusion or spatial heterogeneity. However, the presence of a fractional differential operator causes memory (time fractional) or nonlocality (space fractional) issues, which impose a number of computational constraints. In this paper we develop efficient, scalable techniques for solving fractional-in-space reaction diffusion equations using the finite element method on both structured and unstructured grids, and robust techniques for computing the fractional power of a matrix times a vector. Our approach is show-cased by solving the fractional Fisher and fractional Allen-Cahn reaction-diffusion equations in two and three spatial dimensions, and analysing the speed of the travelling wave and size of the interface in terms of the fractional power of the underlying Laplacian operator

Computationally efficient methods for solving time-variable-order time-space fractional reaction-diffusion equation

Fractional differential equations are becoming more widely accepted as a powerful tool in modelling anomalous diffusion, which is exhibited by various materials and processes. Recently, researchers have suggested that rather than using constant order fractional operators, some processes are more accurately modelled using fractional orders that vary with time and/or space. In this paper we develop computationally efficient techniques for solving time-variable-order time-space fractional reaction-diffusion equations (tsfrde) using the finite difference scheme. We adopt the Coimbra variable order time fractional operator and variable order fractional Laplacian operator in space where both orders are functions of time. Because the fractional operator is nonlocal, it is challenging to efficiently deal with its long range dependence when using classical numerical techniques to solve such equations. The novelty of our method is that the numerical solution of the time-variable-order tsfrde is written in terms of a matrix function vector product at each time step. This product is approximated efficiently by the Lanczos method, which is a powerful iterative technique for approximating the action of a matrix function by projecting onto a Krylov subspace. Furthermore an adaptive preconditioner is constructed that dramatically reduces the size of the required Krylov subspaces and hence the overall computational cost. Numerical examples, including the variable-order fractional Fisher equation, are presented to demonstrate the accuracy and efficiency of the approach

A Class of Second Order Difference Approximation for Solving Space Fractional Diffusion Equations

A class of second order approximations, called the weighted and shifted Gr\"{u}nwald difference operators, are proposed for Riemann-Liouville fractional derivatives, with their effective applications to numerically solving space fractional diffusion equations in one and two dimensions. The stability and convergence of our difference schemes for space fractional diffusion equations with constant coefficients in one and two dimensions are theoretically established. Several numerical examples are implemented to testify the efficiency of the numerical schemes and confirm the convergence order, and the numerical results for variable coefficients problem are also presented.Comment: 24 Page