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Recent advances in the numerical solution of multi-order fractional differential equations

Luigi Brugnano, Gianmarco Gurioli, Felice Iavernaro, Mikk Vikerpuur

TL;DR

This work extends Fractional HBVMs to multi-order fractional differential equations by employing multiple orthogonal polynomials (MOPs) to construct shared Jacobi-Piñeiro abscissae, enabling memory terms to be evaluated on a single abscissa set. The approach yields spectrally accurate solutions while reducing computational cost compared to naïve multi-order quadrature schemes. A new MATLAB code, fhbvm22, implements FHBVM(22,22) for ν=1,2 with efficient iterative strategies and mixed meshes, and numerical tests show superior performance against existing solvers. The results demonstrate significant gains in accuracy and speed for challenging multi-order FDEs with applications across physics, biology, and engineering.

Abstract

The efficient numerical solution of fractional differential equations has been recently tackled through the definition of Fractional HBVMs (FHBVMs), a class of Runge-Kutta type methods. Corresponding Matlab (c) codes have been also made available on the internet, proving to be very competitive w.r.t. existing ones. However, so far, FHBVMs have been given for solving systems of fractional differential equations with the same order of fractional derivative, whereas the numerical solution of multi-order problems (i.e., problems in which different orders of fractional derivatives occur) has not been handled, yet. Due to their relevance in applications, in this paper we propose an extension of FHBVMs for addressing fractional multi-order problems, providing full details for such an approach. A corresponding Matlab (c) code, handling the case of two different fractional orders, is also made available, proving very effective for numerically solving these problems.

Recent advances in the numerical solution of multi-order fractional differential equations

TL;DR

This work extends Fractional HBVMs to multi-order fractional differential equations by employing multiple orthogonal polynomials (MOPs) to construct shared Jacobi-Piñeiro abscissae, enabling memory terms to be evaluated on a single abscissa set. The approach yields spectrally accurate solutions while reducing computational cost compared to naïve multi-order quadrature schemes. A new MATLAB code, fhbvm22, implements FHBVM(22,22) for ν=1,2 with efficient iterative strategies and mixed meshes, and numerical tests show superior performance against existing solvers. The results demonstrate significant gains in accuracy and speed for challenging multi-order FDEs with applications across physics, biology, and engineering.

Abstract

The efficient numerical solution of fractional differential equations has been recently tackled through the definition of Fractional HBVMs (FHBVMs), a class of Runge-Kutta type methods. Corresponding Matlab (c) codes have been also made available on the internet, proving to be very competitive w.r.t. existing ones. However, so far, FHBVMs have been given for solving systems of fractional differential equations with the same order of fractional derivative, whereas the numerical solution of multi-order problems (i.e., problems in which different orders of fractional derivatives occur) has not been handled, yet. Due to their relevance in applications, in this paper we propose an extension of FHBVMs for addressing fractional multi-order problems, providing full details for such an approach. A corresponding Matlab (c) code, handling the case of two different fractional orders, is also made available, proving very effective for numerically solving these problems.

Paper Structure

This paper contains 11 sections, 4 theorems, 98 equations, 6 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Fractional HBVMs (FHBVMs)
  3. Extending FHBVMs to the multi-order case
  4. Multiple Orthogonal Polynomials (MOPs)
  5. Some computational remarks
  6. Solving the discrete problems
  7. The case $\nu=1$
  8. The case $\nu>1$
  9. The code fhbvm2 2
  10. Numerical tests
  11. Conclusions

Key Result

Theorem 1

Assume (mop2) holds true. Then the interpolatory quadratures with weights satisfy

Figures (6)

  • Figure 1: WPD for Problem (\ref{['prob1']}).
  • Figure 3: Solution (upper-plot) and vector field (lower-plot) for Problem (\ref{['prob3']}).
  • Figure 4: WPD for Problem (\ref{['prob3']}).
  • Figure 6: Solution in the phase space (upper-plot) and versus time (lower-plot) for Problem (\ref{['prob4']}).
  • Figure 7: Solution of Problem (\ref{['prob4']}), ${\alpha}_1={\alpha}_2=0.7$ (circles) and ${\alpha}_1=0.7$, ${\alpha}_2=0.7+10^{-4}$ (crosses).
  • ...and 1 more figures

Theorems & Definitions (9)

  • Remark 1
  • Remark 2
  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Remark 3
  • Theorem 3
  • Remark 4
  • Remark 5