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Solving FDE-IVPs by using Fractional HBVMs: some experiments with the fhbvm code

L. Brugnano, G. Gurioli, F. Iavernaro

TL;DR

This work addresses numerical solution of Caputo fractional differential equation initial value problems (FDE-IVPs) by employing Fractional HBVMs (FHBVMs), a generalization of Hamiltonian Boundary Value Methods. The authors describe a piecewise, memory-aware solver built on Jacobi polynomial expansions and Gauss–Jacobi quadrature, culminating in the FHBVM$(k,s)$ framework. Through four numerical experiments, they demonstrate that the fhbvm code achieves substantially higher accuracy in shorter times compared to a competitive solver, including stiff and high-order cases, thereby validating the method's efficiency and spectral-like performance for FDEs. The findings highlight the practical potential of FHBVMs for high-order, Caputo-type FDE-IVPs and provide a freely available tool for researchers and practitioners.

Abstract

In this paper we report a few numerical tests by using a slight extension of the Matlab code fhbvm in [8], implementing Fractional HBVMs, a recently introduced class of numerical methods for solving Initial Value Problems of Fractional Differential Equations (FDE-IVPs). The reported experiments are aimed to give evidence of its effectiveness.

Solving FDE-IVPs by using Fractional HBVMs: some experiments with the fhbvm code

TL;DR

This work addresses numerical solution of Caputo fractional differential equation initial value problems (FDE-IVPs) by employing Fractional HBVMs (FHBVMs), a generalization of Hamiltonian Boundary Value Methods. The authors describe a piecewise, memory-aware solver built on Jacobi polynomial expansions and Gauss–Jacobi quadrature, culminating in the FHBVM framework. Through four numerical experiments, they demonstrate that the fhbvm code achieves substantially higher accuracy in shorter times compared to a competitive solver, including stiff and high-order cases, thereby validating the method's efficiency and spectral-like performance for FDEs. The findings highlight the practical potential of FHBVMs for high-order, Caputo-type FDE-IVPs and provide a freely available tool for researchers and practitioners.

Abstract

In this paper we report a few numerical tests by using a slight extension of the Matlab code fhbvm in [8], implementing Fractional HBVMs, a recently introduced class of numerical methods for solving Initial Value Problems of Fractional Differential Equations (FDE-IVPs). The reported experiments are aimed to give evidence of its effectiveness.
Paper Structure (8 sections, 27 equations, 4 figures)

This paper contains 8 sections, 27 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Fractional HBVMs
  3. Numerical Tests
  4. Example 1
  5. Example 2
  6. Example 3
  7. Example 4
  8. Conclusions

Figures (4)

  • Figure 1: Work-precision diagram for problem (\ref{['ex1']}), $\alpha=1.3$.
  • Figure 2: Work-precision diagram for problem (\ref{['ex2']}).
  • Figure 3: Work-precision diagram for problem (\ref{['ex3']}), $\alpha=1.25$.
  • Figure 4: Work-precision diagram for problem (\ref{['ex4']}), $\alpha=0.25$.