Multi-domain spectral approach to rational-order fractional derivatives

2024/01/09 by Christian Klein, Klein, C., N. Stoilov +1
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #Fractional Differential Equations Solutions #Mathematical Analysis and Transform Methods #Nonlinear Waves and Solitons #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.2401.04461

openalex publication_date 2024/01/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

We propose a method to numerically compute fractional derivatives (or the fractional Laplacian) on the whole real line via Riesz fractional integrals. The compactified real line is divided into a number of intervals, thus amounting to a multi-domain approach; after transformations in accordance with the underlying Z_q curve ensuring analyticity of the respective integrands, the integrals over the different domains are computed with a Clenshaw-Curtis algorithm. As an example, we consider solitary waves for fractional Korteweg-de Vries equations and compare these to results obtained with a discrete Fourier transform.

Related

• A fast and spectrally convergent algorithm for rational-order fractional\n integral and differential equations
• A fast convolution method for the fractional Laplacian in ℝ
• Rational spectral methods for PDEs involving fractional Laplacian in\n unbounded domains
• Spectral Reconstruction in Fractional Derivative Order
• Spectral approximation of ψ-fractional differential equation based on mapped Jacobi functions