A Multi-Order Extension of Fractional HBVMs (FHBVMs)

TL;DR

This paper extends Fractional HBVMs (FHBVMs), a class of Runge-Kutta methods, to efficiently solve multi-order fractional differential equations, with a new Matlab implementation for two different fractional orders.

Contribution

The paper introduces a multi-order extension of FHBVMs, enabling the numerical solution of systems with different fractional derivative orders, which was not previously addressed.

Findings

The extended FHBVMs effectively solve multi-order fractional differential equations.

A Matlab code for two fractional orders demonstrates high efficiency.

The approach broadens the applicability of FHBVMs in practical problems.

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.