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On a spectral solver for highly oscillatory and non-smooth solutions of a class of linear fractional differential systems

Amin Faghih

TL;DR

The paper addresses solving linear fractional differential systems with non-constant coefficients that exhibit highly oscillatory and non-smooth behavior. It introduces a Müntz-Jacobi spectral Galerkin method, where solutions are expanded in a Müntz-Jacobi basis and the coefficients are computed via recurrence relations, avoiding large ill-conditioned algebraic systems. The authors provide a regularity analysis guiding the basis choice, prove unique solvability of the recurrence-based scheme, and establish spectral $L^2$-convergence under suitable assumptions. Numerical experiments validate the method's high accuracy and stability for challenging oscillatory problems while comparing favorably to existing collocation and FHBVM approaches.

Abstract

This study discusses a class of linear systems of fractional differential equations with non-constant coefficients, with a particular focus on problems exhibiting highly oscillatory and non-smooth behavior. We first establish the regularity properties of the solutions under specific conditions on the input data. A spectral Galerkin method based on Müntz-Jacobi functions is developed that efficiently handle the non-smooth and highly oscillatory solutions. A key advantage of the proposed approach is the ability to compute the approximate solution via recurrence relations, avoiding the need to solve complex algebraic systems. Moreover, the method remains stable even at higher approximation degrees, effectively capturing highly oscillatory solutions with high accuracy. The well-known exponential accuracy is established in the $L^2$-norm, and some numerical examples are provided to demonstrate both the validity of the theoretical analysis and the efficiency of the proposed algorithm.

On a spectral solver for highly oscillatory and non-smooth solutions of a class of linear fractional differential systems

TL;DR

The paper addresses solving linear fractional differential systems with non-constant coefficients that exhibit highly oscillatory and non-smooth behavior. It introduces a Müntz-Jacobi spectral Galerkin method, where solutions are expanded in a Müntz-Jacobi basis and the coefficients are computed via recurrence relations, avoiding large ill-conditioned algebraic systems. The authors provide a regularity analysis guiding the basis choice, prove unique solvability of the recurrence-based scheme, and establish spectral -convergence under suitable assumptions. Numerical experiments validate the method's high accuracy and stability for challenging oscillatory problems while comparing favorably to existing collocation and FHBVM approaches.

Abstract

This study discusses a class of linear systems of fractional differential equations with non-constant coefficients, with a particular focus on problems exhibiting highly oscillatory and non-smooth behavior. We first establish the regularity properties of the solutions under specific conditions on the input data. A spectral Galerkin method based on Müntz-Jacobi functions is developed that efficiently handle the non-smooth and highly oscillatory solutions. A key advantage of the proposed approach is the ability to compute the approximate solution via recurrence relations, avoiding the need to solve complex algebraic systems. Moreover, the method remains stable even at higher approximation degrees, effectively capturing highly oscillatory solutions with high accuracy. The well-known exponential accuracy is established in the -norm, and some numerical examples are provided to demonstrate both the validity of the theoretical analysis and the efficiency of the proposed algorithm.

Paper Structure

This paper contains 9 sections, 5 theorems, 85 equations, 8 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Regularity analysis
  3. Numerical method
  4. Müntz-Jacobi functions
  5. Müntz-Jacobi Galerkin algorithm
  6. Numerical solvability and complexity analysis
  7. Error analysis
  8. Numerical examples
  9. Conclusion

Key Result

Theorem 2.1

Let $p_{j,r} : \mathbb{R} \to \mathbb{C}$ be periodic, highly oscillatory continuous functions of the form which can be written as where $\bar{p}_{j,r}$ are analytic in a neighborhood of the origin. Then the series representation of the solution $v_j(t)$ of the equation eq1, in a neighborhood of the origin, is given by in which $\psi_{j}(t)=\sum\limits_{k=0}^{\lceil \theta_j \rceil -1}{\frac{v_

Figures (8)

  • Figure 1: Semi-logarithmic plot of the numerical errors versus $N$ for Example \ref{['exm1']}.
  • Figure 2: Plots comparing the approximate solutions at $N=1000$ (red squares) and the exact solutions (solid lines) for Example \ref{['exm1']}.
  • Figure 3: Semi-logarithmic plot of the numerical errors versus $N$ for Example \ref{['exm3']}.
  • Figure 4: Plots comparing the approximate solutions at $N=1200$ (red squares) and the exact solutions (solid lines) for Example \ref{['exm3']}.
  • Figure 5: Semi-logarithmic plot of the numerical errors versus $N$ for Example \ref{['exm2']}.
  • ...and 3 more figures

Theorems & Definitions (17)

  • Theorem 2.1: Regularity
  • proof
  • Corollary 2.2
  • proof
  • Example 2.1
  • Theorem 3.1
  • Theorem 3.2
  • proof
  • Theorem 4.1: Error estimate
  • proof
  • ...and 7 more