Efficient high order algorithms for fractional integrals and fractional differential equations
This work addresses the need for efficient and high-order numerical methods for fractional integrals and differential equations, which are important in various scientific and engineering applications.
The paper proposes an efficient high-order algorithm for fractional integrals using Runge-Kutta based convolution quadrature, achieving among the best memory and computational cost to date. It also provides error analysis for fractional diffusion equations when coupled with FEM.
We propose an efficient algorithm for the approximation of fractional integrals by using Runge--Kutta based convolution quadrature. The algorithm is based on a novel integral representation of the convolution weights and a special quadrature for it. The resulting method is easy to implement, allows for high order, relies on rigorous error estimates and its performance in terms of memory and computational cost is among the best to date. Several numerical results illustrate the method and we describe how to apply the new algorithm to solve fractional diffusion equations. For a class of fractional diffusion equations we give the error analysis of the full space-time discretization obtained by coupling the FEM method in space with Runge--Kutta based convolution quadrature in time.