Discretized fractional calculus
SIAM Journal on Mathematical Analysis
A numerical method for a partial integro-differential equation
SIAM Journal on Numerical Analysis
The accuracy and stability of an implicit solution method for the fractional diffusion equation
Journal of Computational Physics
Theory and Applications of Fractional Differential Equations, Volume 204 (North-Holland Mathematics Studies)
Journal of Computational and Applied Mathematics
An approximate method for numerical solution of fractional differential equations
Signal Processing - Fractional calculus applications in signals and systems
Finite difference/spectral approximations for the time-fractional diffusion equation
Journal of Computational Physics
A fully discrete difference scheme for a diffusion-wave system
Applied Numerical Mathematics
A Space-Time Spectral Method for the Time Fractional Diffusion Equation
SIAM Journal on Numerical Analysis
Least squares finite-element solution of a fractional order two-point boundary value problem
Computers & Mathematics with Applications
A high order schema for the numerical solution of the fractional ordinary differential equations
Journal of Computational Physics
Fractional Sturm-Liouville eigen-problems: Theory and numerical approximation
Journal of Computational Physics
Journal of Computational Physics
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Current discretizations of fractional differential equations lead to numerical solutions of low order of accuracy. Here, we present different methods for fractional ODEs that lead to exponentially fast decay of the error. First, we develop a Petrov-Galerkin (PG) spectral method for Fractional Initial-Value Problems (FIVPs) of the form Dt@n0u(t)=f(t) and Fractional Final-Value Problems (FFVPs) DT@ntu(t)=g(t), where @n@?(0,1), subject to Dirichlet initial/final conditions. These schemes are developed based on a new spectral theory for fractional Sturm-Liouville problems (FSLPs), which has been recently developed in [1]. Specifically, we obtain solutions to FIVPs and FFVPs in terms of the new fractional (non-polynomial) basis functions, called Jacobi polyfractonomials, which are the eigenfunctions of the FSLP of first kind (FSLP-I). Correspondingly, we employ another space of test functions as the span of polyfractonomial eigenfunctions of the FSLP of second kind (FSLP-II). Subsequently, we develop a Discontinuous Spectral Method (DSM) of Petrov-Galerkin sense for the aforementioned FIVPs and FFVPs, where the basis functions do not satisfy the initial/final conditions. Finally, we extend the DSM scheme to a Discontinuous Spectral Element Method (DSEM) for efficient longer time-integration and adaptive refinement. In these discontinuous schemes, we employ the asymptotic eigensolutions to FSLP-I & -II, which are of Jacobi polynomial forms, as basis and test functions. Our numerical tests confirm the exponential/algebraic convergence, respectively, in p- and h-refinements, for various test cases with integer- and fractional-order solutions.