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Numerical solution of FDE-IVPs by using Fractional HBVMs: the fhbvm code

Luigi Brugnano, Gianmarco Gurioli, Felice Iavernaro

TL;DR

The paper addresses numerical integration of initial-value problems for fractional differential equations using Caputo derivatives. It develops Fractional HBVMs (FHBVMs), an extension of HBVMs to fractional dynamics, combining a quasi-polynomial local approximation with a fully discrete Runge-Kutta formulation to achieve spectral accuracy in time. The Matlab implementation fhbvm includes a graded/uniform mesh strategy, precomputation of fractional integrals via efficient Gauss-type quadratures, an error estimator based on mesh doubling, and a robust blended Newton-type nonlinear solver; it demonstrates the method's efficiency and accuracy on a suite of test problems, including a fractional Brusselator. Overall, the work provides a practical, high-accuracy tool for solving $y^{(\alpha)}(t)=f(y(t))$ with $0<\alpha<1$, with potential extensions to other $(k,s)$ configurations and $\alpha$ ranges.

Abstract

In this paper we describe the efficient numerical implementation of Fractional HBVMs, a class of methods recently introduced for solving systems of fractional differential equations. The reported arguments are implemented in the Matlab code fhbvm, which is made available on the web. An extensive experimentation of the code is reported, to give evidence of its effectiveness.

Numerical solution of FDE-IVPs by using Fractional HBVMs: the fhbvm code

TL;DR

The paper addresses numerical integration of initial-value problems for fractional differential equations using Caputo derivatives. It develops Fractional HBVMs (FHBVMs), an extension of HBVMs to fractional dynamics, combining a quasi-polynomial local approximation with a fully discrete Runge-Kutta formulation to achieve spectral accuracy in time. The Matlab implementation fhbvm includes a graded/uniform mesh strategy, precomputation of fractional integrals via efficient Gauss-type quadratures, an error estimator based on mesh doubling, and a robust blended Newton-type nonlinear solver; it demonstrates the method's efficiency and accuracy on a suite of test problems, including a fractional Brusselator. Overall, the work provides a practical, high-accuracy tool for solving with , with potential extensions to other configurations and ranges.

Abstract

In this paper we describe the efficient numerical implementation of Fractional HBVMs, a class of methods recently introduced for solving systems of fractional differential equations. The reported arguments are implemented in the Matlab code fhbvm, which is made available on the web. An extensive experimentation of the code is reported, to give evidence of its effectiveness.
Paper Structure (15 sections, 4 theorems, 95 equations, 5 figures)

This paper contains 15 sections, 4 theorems, 95 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Fractional HBVMs
  3. Quasi-polynomial approximation
  4. The fully discrete method
  5. Implementation issues
  6. Graded or uniform mesh?
  7. Approximating the fractional integrals
  8. Error estimation
  9. The nonlinear iteration
  10. Numerical Tests
  11. Example 1
  12. Example 2
  13. Example 3
  14. Example 4
  15. Conclusions

Key Result

theorem 1

There exists a unique $\bar{r}>1$ satisfying $\bar{r}=\psi(\bar{r})$, and the iteration (iter) globally converges to this value over the interval $(1, +\infty)$.

Figures (5)

  • Figure 1: Maximum amplification factor (\ref{['ros1']})--(\ref{['xi']}), $k=22$ and $s=20$.
  • Figure 2: Work-precision diagram for problem (\ref{['ex1']}), $\alpha=0.3$.
  • Figure 3: Work-precision diagram for problem (\ref{['ex2']}).
  • Figure 5: True and estimated absolute errors for fhbvm solving problem (\ref{['ex3']}), $M=2$.
  • Figure 6: Phase-plot of the computed solution by using fhbvm solving problem (\ref{['ex4']}), $M=5$.

Theorems & Definitions (14)

  • remark thmcounterremark
  • remark thmcounterremark
  • remark thmcounterremark
  • definition thmcounterdefinition
  • remark thmcounterremark
  • theorem 1
  • proof
  • remark thmcounterremark
  • theorem 2
  • proof
  • ...and 4 more