Finite difference approximations for fractional advection-dispersion flow equations
Journal of Computational and Applied Mathematics
A second-order accurate numerical approximation for the fractional diffusion equation
Journal of Computational Physics
A second-order accurate numerical method for the two-dimensional fractional diffusion equation
Journal of Computational Physics
The Runge phenomenon and spatially variable shape parameters in RBF interpolation
Computers & Mathematics with Applications
Finite difference/spectral approximations for the time-fractional diffusion equation
Journal of Computational Physics
On choosing a radial basis function and a shape parameter when solving a convective PDE on a sphere
Journal of Computational Physics
A Stable Algorithm for Flat Radial Basis Functions on a Sphere
SIAM Journal on Scientific Computing
Matrix approach to discrete fractional calculus II: Partial fractional differential equations
Journal of Computational Physics
Finite difference approximations for two-sided space-fractional partial differential equations
Applied Numerical Mathematics
A Space-Time Spectral Method for the Time Fractional Diffusion Equation
SIAM Journal on Numerical Analysis
Stable computation of multiquadric interpolants for all values of the shape parameter
Computers & Mathematics with Applications
Least squares finite-element solution of a fractional order two-point boundary value problem
Computers & Mathematics with Applications
A direct O(Nlog2N) finite difference method for fractional diffusion equations
Journal of Computational Physics
Numerical approximations for fractional diffusion equations via splines
Computers & Mathematics with Applications
Multigrid method for fractional diffusion equations
Journal of Computational Physics
Stable Computations with Gaussian Radial Basis Functions
SIAM Journal on Scientific Computing
Compact alternating direction implicit method for two-dimensional time fractional diffusion equation
Journal of Computational Physics
Journal of Computational Physics
Numerical treatment for solving the perturbed fractional PDEs using hybrid techniques
Journal of Computational Physics
Hi-index | 31.45 |
One of the ongoing issues with fractional diffusion models is the design of an efficient high-order numerical discretization. This is one of the reasons why fractional diffusion models are not yet more widely used to describe complex systems. In this paper, we derive a radial basis functions (RBF) discretization of the one-dimensional space-fractional diffusion equation. In order to remove the ill-conditioning that often impairs the convergence rate of standard RBF methods, we use the RBF-QR method [1,33]. By using this algorithm, we can analytically remove the ill-conditioning that appears when the number of nodes increases or when basis functions are made increasingly flat. The resulting RBF-QR-based method exhibits an exponential rate of convergence for infinitely smooth solutions that is comparable to the one achieved with pseudo-spectral methods. We illustrate the flexibility of the algorithm by comparing the standard RBF and RBF-QR methods for two numerical examples. Our results suggest that the global character of the RBFs makes them well-suited to fractional diffusion equations. They naturally take the global behavior of the solution into account and thus do not result in an extra computational cost when moving from a second-order to a fractional-order diffusion model. As such, they should be considered as one of the methods of choice to discretize fractional diffusion models of complex systems.