A spectrally accurate step-by-step method for the numerical solution of fractional differential equations

TL;DR

Addresses the numerical solution of fractional differential equations with Caputo derivatives by introducing a step-by-step graded-mesh method that expands the vector field on an orthonormal Jacobi basis $P_j(c)$, producing a piecewise quasi-polynomial approximation $\sigma^{(\alpha)}$ and solving small nonlinear systems per step via Gauss-Jacobi quadrature. The approach yields spectral-like accuracy under mild assumptions, supported by a detailed error analysis and practical algorithms to compute the required integrals $J_j^\alpha(x)$ and coefficients $\gamma_j$, with a Runge-Kutta–type interpretation and fixed-point convergence guarantees. Numerical tests across several benchmark FDEs demonstrate high accuracy, efficiency, and resilience to initial-time singularities, confirming the method’s potential for adaptive and parallel extensions. The framework naturally extends to multiple fractional orders by using different Jacobi bases, enabling scalable solutions for complex FDE systems and guiding future work on solver efficiency, adaptive meshing, and parallel implementations.

Abstract

In this paper we consider the numerical solution of fractional differential equations. In particular, we study a step-by-step graded mesh procedure based on an expansion of the vector field using orthonormal Jacobi polynomials. Under mild hypotheses, the proposed procedure is capable of getting spectral accuracy. A few numerical examples are reported to confirm the theoretical findings.

Figures (2)

• Figure 1: Gauss-Jacobi abscissae for $\alpha=0.5$.
• Figure 2: solution (continuous line) and vector field (dashed line) for problem (\ref{['prob2']}).

Theorems & Definitions (16)

• Remark 1
• Remark 2
• Lemma 1
• Remark 3
• Corollary 1
• Lemma 2
• Corollary 2
• Lemma 3
• Corollary 3
• Theorem 1