%0 Journal Article
%T A unified Explicit form for difference formulas for fractional and classical derivatives and applications
%J Computational Methods for Differential Equations
%I University of Tabriz
%Z 2345-3982
%A Gunarathna, W. Anura
%A Nasir, Haniffa Mohamed
%A Daundasekera, Wasantha B.
%D 2024
%\ 04/22/2024
%V
%N
%P -
%! A unified Explicit form for difference formulas for fractional and classical derivatives and applications
%K Fractional derivative
%K Shifted Gr"unwald approximation
%K Lubich Generators
%K Compact finite difference formula
%K Boundary value problem
%R 10.22034/cmde.2023.58229.2459
%X 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 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.
%U https://cmde.tabrizu.ac.ir/article_17827_625e23ec220556744eb5a69d35ad943b.pdf