arXiv subjects

W. A. Gunarathna

Publications and source records attributed to W. A. Gunarathna.

2 recordsLinked to original sources

arXiv2021

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

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.

math.NA↗

arXiv2018

A general explicit form for higher order approximations for fractional derivatives and its consequences

A general explicit form for generating functions for approximating fractional derivatives is derived. To achieve this, an equivalent characterisation for consistency and order of approximations established on a general generating function is used to form a linear system of equations with Vandermonde matrix for the coefficients of the generating function which is in the form of power of a polynomial. This linear system is solved for the coefficients of the polynomial in the generating function. These generating functions completely characterise Grünwald type approximations with shifts and order of accuracy. Incidentally, the constructed generating functions happen to be generalization of the previously known Lubich forms of generating functions without shift. As a consquence, a general explicit form for new finite difference formulas for integer-order derivatives with any order of accuracy are derived.

math.NA↗