New numerical computation formula and error analysis of some existing formulae in fractional derivatives and integrals

摘要

Fractional calculus is non-integer calculus, and mainly used in engineering field. For instance, it is used for simulation of viscosity model or control system. This paper analyzes numerical computation formulae for calculating fractional derivatives and integrals. Those formulae are derived from Grunwald-Letkoniv de.nition. Two numerical computation formulae derived from Grunwald-Letkoniv de.nition have been suggested. One is the"seemingly" first order accuracy formula, and the other is the "seemingly" second order accuracy formula, but their orders were not proved. This paper veri.es the seemingly first order accuracy formula has actually first order accuracy, and disproves second order accuracy of the seemingly second order formula. In addition, this paper proposes a new formula with second order accuracy and verifies its order. The proposed numerical computation formulae derived from Grunwald-Letkoniv de.nition are easier to calculate than formulae derived from Riemann-Liouville definition, and have a property that they are extended version of Newton-Cotes rules and finite difference methods. Numerical computation formulae this paper verifies help deeper understandings of fractional calculus.