A unified explicit form for difference formulas for fractional and classical derivatives

TL;DR

This paper introduces a unified explicit difference formula framework that approximates both fractional and classical derivatives with customizable accuracy, enabling versatile and efficient numerical solutions for differential equations.

Contribution

It presents a novel unified explicit formula that encompasses various classical and fractional derivative approximations, streamlining computational methods.

Findings

Provides finite difference formulas for classical derivatives of any order.

Offers Grünwald type approximations for fractional derivatives.

Facilitates automated and efficient computation of difference coefficients.

Abstract

A unified explicit form for difference formulas to approximate the fractional and classical derivatives is presented. The formula gives finite difference approximations for any classical derivatives with a desired order of accuracy at nodal point in the computational domain. It also gives Gr\"unwald type approximations for fractional derivatives with arbitrary order of approximation at any point. Thus, this explicit unifies approximations of both types of derivatives. Moreover, for classical derivatives, it provides various finite difference formulas such as forward, backward, central, staggered, compact, non-compact etc. Efficient computations of the coefficients of the difference formulas are also presented that lead to automating the solution process of differential equations with a given higher order accuracy. Some basic applications are presented to demonstrate the usefulness of this unified formulation.