A general explicit form for higher order approximations for fractional derivatives and its consequences

TL;DR

This paper derives a comprehensive explicit form for generating functions that approximate fractional derivatives, generalizing existing methods and enabling new high-accuracy finite difference formulas for derivatives of any order.

Contribution

It introduces a general explicit form for generating functions for fractional derivatives, extending Lubich's forms and enabling high-order approximations.

Findings

Derived a general explicit form for generating functions.

Unified fractional and integer-order derivative approximations.

Enabled construction of high-accuracy finite difference formulas.

Abstract

A general explicit form for generating functions for approximating fractional derivatives is derived. To achieve this, an equivalent characterisation for consistency and order of approximations established on a general generating function is used to form a linear system of equations with Vandermonde matrix for the coefficients of the generating function which is in the form of power of a polynomial. This linear system is solved for the coefficients of the polynomial in the generating function. These generating functions completely characterise Gr\"unwald type approximations with shifts and order of accuracy. Incidentally, the constructed generating functions happen to be generalization of the previously known Lubich forms of generating functions without shift. As a consquence, a general explicit form for new finite difference formulas for integer-order derivatives with any order of accuracy are derived.