2021/05/27 by W. A. Gunarathna, Gunarathna, W. A., Haniffa Mohamed Nasir +3
Engineering · Mathematics · #26A33 #34B0 #65L12 #Advanced Control Systems Design #FOS: Mathematics #Fractional Differential Equations Solutions #Nonlinear Differential Equations Analysis #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2105.12997
openalex publication_date 2021/05/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Grünwald 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.