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Spectral Approximation to Fractional Integral Operators

Xiaolin Liu, Kuan Xu

TL;DR

This work develops a fast, stable spectral framework for approximating the fractional integral operator $\mathcal{I}^{\mu}$ on $[-1,1]$ using Chebyshev-based Jacobi polynomials $Q_n^{\alpha,\beta}(x)$ and a recurrence for mapped Chebyshev polynomials. The authors derive an infinite-dimensional factorization $\mathcal{I}^{\mu}u = \mathbf{Q}^{\alpha,\beta} \mathcal{S} \hat{u}$ with $\mathcal{S} = \frac{2^{-\alpha}}{\Gamma(\mu)} \mathcal{M} \mathcal{R}$, where $\mathcal{M}$ is a multiplication operator and $\mathcal{R}$ contains recurrence coefficients; an adaptive procedure yields $\mathcal{S}$ with overall $O(N^2)$ complexity and no need for extended precision. The method is demonstrated across a suite of problems including fractional integral equations, fractional differential equations (with Abel, BBO, and Airy examples), initial-value problems via spectral deferred correction, and fractional eigenvalue problems, outperforming existing approaches in accuracy, conditioning, and efficiency. The results show exponential-like convergence, robust conditioning, and the ability to handle general fractional orders, highlighting the practical impact for high-precision simulations and operator analysis in fractional calculus. The work also provides a Julia implementation, suggesting broad applicability to linear operator approximation beyond the presented cases.

Abstract

We propose a fast and stable method for constructing matrix approximations to fractional integral operators applied to series in the Chebyshev fractional polynomials. This method utilizes a recurrence relation satisfied by the fractional integrals of mapped Chebyshev polynomials and significantly outperforms existing methods. Through numerical examples, we highlight the broad applicability of these matrix approximations, including the solution of boundary value problems for fractional integral and differential equations. Additional applications include fractional differential equation initial value problems and fractional eigenvalue problems.

Spectral Approximation to Fractional Integral Operators

TL;DR

This work develops a fast, stable spectral framework for approximating the fractional integral operator on using Chebyshev-based Jacobi polynomials and a recurrence for mapped Chebyshev polynomials. The authors derive an infinite-dimensional factorization with , where is a multiplication operator and contains recurrence coefficients; an adaptive procedure yields with overall complexity and no need for extended precision. The method is demonstrated across a suite of problems including fractional integral equations, fractional differential equations (with Abel, BBO, and Airy examples), initial-value problems via spectral deferred correction, and fractional eigenvalue problems, outperforming existing approaches in accuracy, conditioning, and efficiency. The results show exponential-like convergence, robust conditioning, and the ability to handle general fractional orders, highlighting the practical impact for high-precision simulations and operator analysis in fractional calculus. The work also provides a Julia implementation, suggesting broad applicability to linear operator approximation beyond the presented cases.

Abstract

We propose a fast and stable method for constructing matrix approximations to fractional integral operators applied to series in the Chebyshev fractional polynomials. This method utilizes a recurrence relation satisfied by the fractional integrals of mapped Chebyshev polynomials and significantly outperforms existing methods. Through numerical examples, we highlight the broad applicability of these matrix approximations, including the solution of boundary value problems for fractional integral and differential equations. Additional applications include fractional differential equation initial value problems and fractional eigenvalue problems.

Paper Structure

This paper contains 15 sections, 8 theorems, 63 equations, 6 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Constructing the matrix
  3. Recurrence relation satisfied by $\varphi_n(x)$
  4. Recursing for $\mathcal{R}$
  5. Adaptive construction of $\mathcal{S}$
  6. Numerical examples
  7. FIE boundary value problems
  8. Fractional Abel integral equation
  9. Integral equation of multiple fractional orders with variable coefficients
  10. FDE boundary value problems
  11. Basset--Boussinesq--Oseen equation
  12. Fractional Airy equation
  13. FDE initial value problem
  14. Fractional eigenvalue problem
  15. Conclusion and outlook

Key Result

Lemma 2.1

For Chebyshev polynomials of the first kind $T_n(x)$ and the second kind $U_n(x)$,

Figures (6)

  • Figure 1: (A) Errors in the solutions to \ref{['abel']} for $\lambda = 2$ obtained by the SS, JFP, and GLOFPG methods. For the GLOFPG method, we use the GLOF basis with parameters $\alpha = 0$, $\beta = 5$, and $\lambda = 0$, where $\alpha$, $\beta$, and $\lambda$ follow the notations in che2 and are not those used elsewhere in this paper. (B) Execution times for the GLOFPG method and for the JFP method constructed via different algorithms.
  • Figure 2: Error in the numerical solution to \ref{['var']} obtained by the JFP method.
  • Figure 3: (A) Numerical solution to \ref{['bbo']}. (B) The JFP coefficients and the error of the computed solution.
  • Figure 4: The (A) real and (B) imaginary parts of the numerical solution to \ref{['airy']}. (C) Cauchy error.
  • Figure 5: Convergence of fractional SDC method based on polynomial and JFP bases.
  • ...and 1 more figures

Theorems & Definitions (14)

  • Lemma 2.1
  • Theorem 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 4 more