A Fast Time Stepping Method for Evaluating Fractional Integrals

2010/01/01 by Jing-Rebecca Li, Jing‐Rebecca Li · 9 citations
Mathematics · #Fractional Differential Equations Solutions #Mathematical functions and polynomials #Numerical methods for differential equations

paper · doi:10.1137/080736533

openalex publication_date 2010/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30

Abstract

We evaluate the fractional integral I^α[f](t)=(1)/(Γ(α))∫₀^t(t-τ)^α-1 f(τ) dτ, 0<α<1, at time steps t=Δ t,2Δ t,…,NΔ t by making use of the integral representation of the convolution kernel t^α-1=(1)/(Γ(1-α))∫₀^∞e^{-ξ t} ξ^-α dξ. We construct an efficient Q-point quadrature of this integral representation and use it as a part of a fast time stepping method. The new method has algorithmic complexity O(NQ) and storage requirement O(Q). The number of quadrature nodes Q is independent of N and grows like O((-logε-logΔ t)²), where ε is the quadrature error tolerance and Δ t is the size of the time step. The (possible) singularity of f near τ=0 is taken into account. This new method is particularly well-suited for long time simulations.

Citations

• A fast algorithm for the evaluation of heat potentials

Cited by

• Fast Summation by Interval Clustering for an Evolution Equation with Memory
• A linear unconditionally structure-preserving L1 scheme for the time-fractional Allen-Cahn equation
• An Efficient Exponential Sum Approximation of Power-Law Kernels for Solving Fractional Differential Equation
• An Automated Singularity-Capturing Scheme for Fractional Differential Equations
• A kernel compression method for distributed-order fractional partial differential equations
• An accurate spectral method for Maxwell equations in Cole-Cole dispersive media
• A new class of semi-implicit methods with linear complexity for nonlinear fractional differential equations
• High order fast algorithm for the Caputo fractional derivative
• Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations

Related

• A stable fast time-stepping method for fractional integral and derivative operators
• Efficient high order algorithms for fractional integrals and fractional differential equations
• Efficient multistep methods for tempered fractional calculus: Algorithms and Simulations
• Novel Variants of Diffusive Representation of Fractional Integrals: Construction and Numerical Computation
• A fast convolution method for the fractional Laplacian in ℝ