## 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 derivative with a desired order of accuracy

at any nodal point in the computational domain. It also gives Grunwald type approximations for fractional

derivatives with arbitrary order of approximation at any point. Thus, this formulation unifies approximations

of both types of derivatives. Moreover, classical derivatives, provide 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 which leads 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.

The formula gives finite difference approximations for any classical derivative with a desired order of accuracy

at any nodal point in the computational domain. It also gives Grunwald type approximations for fractional

derivatives with arbitrary order of approximation at any point. Thus, this formulation unifies approximations

of both types of derivatives. Moreover, classical derivatives, provide 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 which leads 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.

Original language | English |
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Number of pages | 21 |

Journal | Computational Methods for Differential Equations |

DOIs | |

Publication status | E-pub ahead of print - Apr 22 2024 |

## Keywords

- Fractional derivative
- Shifted Grunwald approximation
- Lubich Generators
- Compact finite difference formula
- Boundary value problem

## ASJC Scopus subject areas

- Mathematics(all)