Numerical evaluation of variable - fractional-order derivatives

• Autorzy: Ostalczyk P. , Brzeziński D.

Warianty tytułu

PL

Ocena numerycznego obliczania zmiennych pochodno-całek ułamkowych rzędów

• Języki publikacji: EN

Abstrakty

EN

In this paper we discuss the methods of calculating fractional-order derivatives (FOD) and integrals (FOI), which also can be used to calculate integer order derivatives and integrals. We focus on one of the methods - Riemann-Liouville formula. To calculate FOD/FOI by this method we need tools. The tools in this case were advanced methods of numerical integration. We have chosen a function which is often used in technical applications. The order of derivative to calculate was variable - designated by a function of time. Our goal was to determine which of the methods of numerical integration - Newton-Cotes or Gauss Quadratures allows to obtain the values with lowest level of absolute errors by the smallest number of sample points. As a point reference served the values of absolute error of the values calculated by using the Grunwald-Letnikov formula.

PL

Tematem niniejszego artykułu są numeryczne metody obliczania pochodno-całek ułamkowych rzędów (FOD/FOI), które mogą być również użyte do obliczania wartości klasycznych pochodnych i całek oznaczonych całkowitych rzędów. Główny temat niniejszej pracy to metoda Reimanna-Liouvilla. By ją móc stosować, konieczne są narzędzia w postaci programów całkujących numerycznie. Do testów wybrana została funkcja często używana w zastosowaniach technicznych. Rzędy obliczanych pochodno-całek były zmienne, wyznaczone przez funkcję zmienną w czasie. Celem autorów było stwierdzenie, które z metod całkowania numerycznego - wybrane kwadratury Newtona-Cotesa czy wybrane kwadratury Gaussa - pozwolą na uzyskanie obliczanych wartości z najmniejszym błędem bezwzględnym stosując jak najmniejszą ilość punktów próbkowania (które mogą być kosztowne). Punktem odniesienia były wartości błędów uzyskane za pomocą popularnej w zastosowaniach technicznych metody Grunwalda-Letnikova.

Słowa kluczowe

EN

variable, fractional-order derivative and integrals; numerical integration

PL

zmienne pochodno-całki ułamkowych rzędów

• Wydawca: Wydawnictwa AGH
• Czasopismo: Automatyka / Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie
• Rocznik: 2011
• Tom: T. 15, z. 3
• Strony: 431--441
• Opis fizyczny: Bibliogr. 34 poz., rys., wykr., tab.

Twórcy

autor

Ostalczyk P.

• Computer Engineering Department, Technical University of Lodz, Poland

autor

Brzeziński D.

• Computer Engineering Department, Technical University of Lodz, Poland

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