Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The numerical solution of fractional integral equations via orthogonal polynomials in fractional powers

Tianyi Pu, Marco Fasondini

TL;DR

The paper develops a Jacobi fractional polynomial (JFP) spectral framework for one-sided fractional integral equations on [-1,1], achieving exponential convergence across a wide range of orders, including irrational ones. It introduces two algorithms to compute fractional integration matrices in the JFP basis and employs a pseudo-stabilization technique that leverages high-precision arithmetic to control instability in matrix construction, while keeping the resulting linear systems well conditioned. Through applications to FIEs, FDEs, and fractional PDEs with Mittag–Leffler solutions, the authors show that the JFP method outperforms sparse sum-space approaches in stability and scalability, especially for large fractional orders, and provide a detailed comparison of computational costs and coefficient behavior. The work advances numerical methods for fractional operators by combining orthogonal spectral bases with carefully structured matrix representations and high-precision stabilization, enabling accurate, efficient solutions to complex fractional models in science and engineering.

Abstract

We present a spectral method for one-sided linear fractional integral equations on a closed interval that achieves exponentially fast convergence for a variety of equations, including ones with irrational order, multiple fractional orders, non-trivial variable coefficients, and initial-boundary conditions. The method uses an orthogonal basis that we refer to as Jacobi fractional polynomials, which are obtained from an appropriate change of variable in weighted classical Jacobi polynomials. New algorithms for building the matrices used to represent fractional integration operators are presented and compared. Even though these algorithms are unstable and require the use of high-precision computations, the spectral method nonetheless yields well-conditioned linear systems and is therefore stable and efficient. For time-fractional heat and wave equations, we show that our method (which is not sparse but uses an orthogonal basis) outperforms a sparse spectral method (which uses a basis that is not orthogonal) due to its superior stability.

The numerical solution of fractional integral equations via orthogonal polynomials in fractional powers

TL;DR

The paper develops a Jacobi fractional polynomial (JFP) spectral framework for one-sided fractional integral equations on [-1,1], achieving exponential convergence across a wide range of orders, including irrational ones. It introduces two algorithms to compute fractional integration matrices in the JFP basis and employs a pseudo-stabilization technique that leverages high-precision arithmetic to control instability in matrix construction, while keeping the resulting linear systems well conditioned. Through applications to FIEs, FDEs, and fractional PDEs with Mittag–Leffler solutions, the authors show that the JFP method outperforms sparse sum-space approaches in stability and scalability, especially for large fractional orders, and provide a detailed comparison of computational costs and coefficient behavior. The work advances numerical methods for fractional operators by combining orthogonal spectral bases with carefully structured matrix representations and high-precision stabilization, enabling accurate, efficient solutions to complex fractional models in science and engineering.

Abstract

We present a spectral method for one-sided linear fractional integral equations on a closed interval that achieves exponentially fast convergence for a variety of equations, including ones with irrational order, multiple fractional orders, non-trivial variable coefficients, and initial-boundary conditions. The method uses an orthogonal basis that we refer to as Jacobi fractional polynomials, which are obtained from an appropriate change of variable in weighted classical Jacobi polynomials. New algorithms for building the matrices used to represent fractional integration operators are presented and compared. Even though these algorithms are unstable and require the use of high-precision computations, the spectral method nonetheless yields well-conditioned linear systems and is therefore stable and efficient. For time-fractional heat and wave equations, we show that our method (which is not sparse but uses an orthogonal basis) outperforms a sparse spectral method (which uses a basis that is not orthogonal) due to its superior stability.
Paper Structure (20 sections, 10 theorems, 101 equations, 16 figures, 2 tables)

This paper contains 20 sections, 10 theorems, 101 equations, 16 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quasimatrix notation and fractional monomial bases
  4. Jacobi polynomials
  5. High-precision computation
  6. Jacobi fractional polynomials
  7. Properties of fractional integrals applied to the JFP basis
  8. Algorithms for computing fractional integration matrices
  9. Algorithm 1
  10. Algorithm 2
  11. Spectral methods using Jacobi fractional polynomials
  12. FIEs, FDEs and a fractional PDE with Mittag--Leffler function solutions
  13. More general FIEs and FDEs
  14. Conclusion
  15. Future work and open problems
  16. ...and 5 more sections

Key Result

Proposition 1

Let $\gamma > 0$ and $\mu = k\gamma$, where $k$ is a positive integer (i.e., $k \in \mathbb{N}_+$), then $\mathcal{I}^{\mu}$ maps $\mathbf{M}^{\gamma}_{\delta}$ to itself via an infinite matrix whose $k$-th subdiagonal is nonzero, i.e., $\mathcal{I}^{\mu} \mathbf{M}^{\gamma}_{\delta} = \mathbf{M}^{\

Figures (16)

  • Figure 1: Algebraic convergence of polynomial approximants to a fractional integral. \ref{['flo51']}: Plots of fractional integrals of the unit constant function, $\mathcal{I}^{\mu}_{-1^+}[1](x)$. \ref{['flo52']}: Errors from Chebyshev polynomial approximations to $\mathcal{I}^{\mu}_{-1^+}[1](x)$, which converge as $\mathcal{O}(n^{-2\mu})$, where $n+1$ is the truncation size.
  • Figure 2: An example of JFP basis functions (\ref{['eq31']} with $\alpha = \beta = b =0$ and $p = 2$), which in this case are Legendre polynomials in the mapped variable $y = \sqrt{2(x+1)} - 1$. \ref{['flo20']}: $\frac{1+x}{2}=\left(\frac{1+y}{2}\right)^2$; \ref{['flo21']}: $P^{(0,0)}_n(y)$ for $n=1,2,3,4$. \ref{['flo22']}: $Q^{(0,0,0,2)}_n(x)$ for $n=1,2,3,4$.
  • Figure 3: Errors in $\log_{10}$ scale from using Algorithm 1 to compute $I_{0,2,1/2}^{(0,0)}$.
  • Figure 4: Errors in $\log_{10}$ scale from computing $I_{0,2,1/2}^{(0,0)}$ by solving the banded Sylvester equations \ref{['eq18']} and \ref{['eq:intcomm']} in 256-bit precision. The difference in accuracy between the two solutions is negligible.
  • Figure 5: Errors produced by the JFP method for the problem \ref{['eq:exFIE1']} for different fractional orders with $\alpha=\beta=b=0$. The vertical axes are the maximum errors (calculated by evaluating the computed solution on the grid $[-1\!\!:\!\!0.01\!\!:\!\!1]$) and the horizontal axes give the truncation size of the system \ref{['eq:MLeqsyst']} that is solved to compute the JFP coefficients.
  • ...and 11 more figures

Theorems & Definitions (35)

  • Proposition 1
  • Proposition 2
  • proof
  • Lemma 3
  • proof
  • Corollary 4
  • proof
  • Theorem 5
  • proof
  • Proposition 6: Fractional integral recurrence
  • ...and 25 more