A unified Explicit form for difference formulas for fractional and classical derivatives and applications

Document Type : Research Paper

Authors

1 Department of Physical Sciences, College of Applied Sciences, Rajarata University, Sri Lanka.

2 FracDiff Research Group, Department of Mathematics, P. O. Box: 36, Sultan Qaboos University, Al-Khoud 123, Muscat, Sultanate of Oman.

3 Department of Mathematics, University of Peradeniya, Peradeniya, Sri Lanka.

Abstract

A unified explicit form for difference formulas to approximate 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 computational domain. It also gives Gr¨unwald type approximations for fractional derivatives with arbitrary order of approximation at any nodal point. Thus, this explicit form unifies approximations of both types of derivatives. Moreover, for classical derivatives, it also 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 leading 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.

Keywords

Main Subjects

References

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Computational Methods for Differential Equations
Volume 13, Issue 1
January 2025
Pages 307-326

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History

  • Receive Date: 01 September 2023
  • Revise Date: 18 December 2023
  • Accept Date: 06 January 2024

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