On the construction of $m$-step methods for FDEs

TL;DR

This paper develops a new approach for constructing m-step methods for fractional differential equations using rational approximations of FBDF generating functions, leading to efficient and accurate numerical solutions.

Contribution

It introduces a novel technique based on rational approximation for creating m-step methods that preserve FBDF properties with computational benefits.

Findings

The proposed methods accurately simulate FBDF properties.

Numerical experiments demonstrate improved efficiency.

The approach offers a flexible framework for fractional differential equations.

Abstract

In this paper we consider the numerical solution of Fractional Differential Equations by means of $m$-step recursions. The construction of such formulas can be obtained in many ways. Here we study a technique based on the rational approximation of the generating functions of Fractional Backward Differentiation Formulas (FBDFs). Accurate approximations allow to define methods which simulate the theoretical properties of the underlying FBDF with important computational advantages. Numerical experiments are presented.