TY - JOUR

T1 - Trapezoidal rule and its error analysis for the Grünwald-Letnikov operator

AU - Harker, Matthew

AU - O'Leary, Paul

PY - 2017/3/1

Y1 - 2017/3/1

N2 - In this paper, the trapezoidal rule for the Grünwald-Letnikov operator is derived. It is a trapezoidal rule in the sense that the formula yields the exact Grünwald-Letnikov derivative/integral of a piecewise linear function. Firstly, the formula for evenly spaced points is derived, and is used as a basis to derive the equivalent formula for arbitrary abscissae. Further, an analytic bound on the residual error is derived, which depends on a bound on the second derivative of the function. The derived trapezoidal rule can therefore be used to compute fractional integrals and derivatives to within a given error tolerance. Through numerical testing it is shown that the new formula yields results that are orders-of-magnitude more accurate than the classical formula, even for arbitrary functions. A simple adaptive algorithm is proposed for computing the result of applying the Grünwald-Letnikov operator to a function to within a desired accuracy.

AB - In this paper, the trapezoidal rule for the Grünwald-Letnikov operator is derived. It is a trapezoidal rule in the sense that the formula yields the exact Grünwald-Letnikov derivative/integral of a piecewise linear function. Firstly, the formula for evenly spaced points is derived, and is used as a basis to derive the equivalent formula for arbitrary abscissae. Further, an analytic bound on the residual error is derived, which depends on a bound on the second derivative of the function. The derived trapezoidal rule can therefore be used to compute fractional integrals and derivatives to within a given error tolerance. Through numerical testing it is shown that the new formula yields results that are orders-of-magnitude more accurate than the classical formula, even for arbitrary functions. A simple adaptive algorithm is proposed for computing the result of applying the Grünwald-Letnikov operator to a function to within a desired accuracy.

KW - Error analysis

KW - Grünwald-Letnikov operator

KW - Trapezoidal rule

UR - http://www.scopus.com/inward/record.url?scp=85013766110&partnerID=8YFLogxK

U2 - 10.1007/s40435-016-0236-z

DO - 10.1007/s40435-016-0236-z

M3 - Article

AN - SCOPUS:85013766110

VL - 5.2017

SP - 18

EP - 29

JO - International Journal of Dynamics and Control

JF - International Journal of Dynamics and Control

SN - 2195-268X

IS - March

ER -