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Multi-domain spectral approach to rational-order fractional derivatives

C. Klein, N. Stoilov

TL;DR

The paper develops a multi-domain, spectrally accurate framework for computing the Riesz fractional derivative on the real line by exploiting $Z_{q}$-curve structure and Puiseux parameters to render integrands smooth. It combines domain-wise transformations with Clenshaw-Curtis quadrature and Chebyshev polynomials, enabling high-precision numerical evaluation and parallelizable computation. The method is validated against known results and applied to solitary waves of the fractional KdV equation, achieving machine-precision accuracy and extending the reachable range of $\alpha$ toward the energy-critical value $\alpha_c=\tfrac{1}{3}$. The work demonstrates improved accuracy over DFT-based approaches in this setting and provides a robust tool for studying nonlocal dispersive PDEs on the whole real line, with potential extensions to other fractional models and time-dependent problems.

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.

Multi-domain spectral approach to rational-order fractional derivatives

TL;DR

The paper develops a multi-domain, spectrally accurate framework for computing the Riesz fractional derivative on the real line by exploiting -curve structure and Puiseux parameters to render integrands smooth. It combines domain-wise transformations with Clenshaw-Curtis quadrature and Chebyshev polynomials, enabling high-precision numerical evaluation and parallelizable computation. The method is validated against known results and applied to solitary waves of the fractional KdV equation, achieving machine-precision accuracy and extending the reachable range of toward the energy-critical value . The work demonstrates improved accuracy over DFT-based approaches in this setting and provides a robust tool for studying nonlocal dispersive PDEs on the whole real line, with potential extensions to other fractional models and time-dependent problems.

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 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.
Paper Structure (21 sections, 58 equations, 13 figures, 1 table)

This paper contains 21 sections, 58 equations, 13 figures, 1 table.

Table of Contents

  1. Introduction
  2. Overview and Preliminaries
  3. Numerical approaches
  4. Fractional integrals
  5. $Z_{q}$ curves
  6. Multi-domain approach
  7. Domain I:
  8. Domain II:
  9. Domain III:
  10. The integral $I^{+}$
  11. Numerical approach
  12. Chebyshev polynomials
  13. Clenshaw-Curtis integration in the multi-domain approach
  14. Example
  15. Comparison with DFT
  16. ...and 6 more sections

Figures (13)

  • Figure 1: Difference of the numerical solution for the fractional derivative (\ref{['exact']}) and its exact expression, on the left for domain II, on the right for domain III.
  • Figure 2: Difference of the numerical solution for the fractional derivative (\ref{['exact']}) with an FFT approach and its exact expression on the left, and the DFT coefficients for the fractional derivative on the right.
  • Figure 3: Difference of the numerical solution for the fractional derivative of order $1/2$ of a Gaussian with an FFT approach and the multi-domain approach on the left, and the DFT coefficients for the Gaussian on the right.
  • Figure 4: Difference of the numerical solution for the fractional derivative (\ref{['example']}) for $\alpha=2/5$ with an FFT approach and the multi-domain approach on the left, and the DFT coefficients for the fractional derivative on the right.
  • Figure 5: Solitary wave of the fKdV equation for $c=1$ and several values of $\alpha$.
  • ...and 8 more figures

Theorems & Definitions (2)

  • Remark 2.1
  • Remark 3.1