# BazEkon - The Main Library of the Cracow University of Economics

## Main menu

Author
Brzeziński Dariusz W. (Lodz University of Technology, Poland), Ostalczyk Piotr (Lodz University of Technology, Poland)
Title
Accuracy Evaluation of Classical Integer Order Based and Direct Non-integer Order Numerical Algorithms of Non-integer Order Derivatives and Integrals Computations
Source
Annals of Computer Science and Information Systems, 2014, vol. 2, s. 553 - 560, rys., bibliogr. 23 poz.
Keyword
Algorytmy numeryczne, Rachunek różniczkowy i całkowy, Analiza matematyczna
Numeric algorithms, Calculus, Mathematical analysis
Note
summ.
Abstract
In this paper the authors evaluate in context of numerical calculations accuracy classical integer order and direct non-integer based order numerical algorithms of non-integer orders derivatives and integrals computations. Classical integer order based algorithm involves integer and fractional order differentiation and integration operators concatenation to obtain non-integer order. Riemann-Liouville and Caputo formulas are applied to obtain directly derivatives and integrals of non-integer orders. The following accuracy comparison analysis enables to answer the question, which algorithm of the two is burdened with lower computational error. The accuracy is estimated applying non-integer order derivatives and integrals computational formulas of some elementary functions available in the literature of the subject.(original abstract)
Full text
Show
Bibliography
Show
1. Baleanu D., Diethelm K., Scalas E., and Trujillo J. J., "Fractional Calculus Models and Numerical Methods," World Scientific Publishing Co.Pte. Ltd., Singapore, 2012.
2. Brzeziński D. W. and Ostalczyk P., "Evaluation of Efficient Methods of Fractional Order Derivatives and Integrals Numerical Calculations", in Proceedings of the XIV Conference on System Modelling and Control, September 23-24 2013, Łódz, Poland, 2013.
3. Brzeziński D. W. and Ostalczyk P., "High-accuracy Numerical Integration Methods for Fractional Order Derivatives and Integrals Computations", After reviews, awaits publication in Bulletin of the Polish Academy of Sciences Technical Sciences, vol. 4, 2014.
4. Burden R. L., Faires J. D., "Numerical Analysis", 5 th. Ed., Brooks/Cole Cengage Learning, Boston, 2003.
5. C++ wrapper for the GNU Multiple Precision Floating-Point Reliable Library, http://www.holoborodko.com/pavel/mpfr/.
6. Ghazi K. R., Lefevre V., Theveny P. and Zimmermann P., "Why and how to use arbitrary precision" IEEE Computer Society, vol.12, nr 3, 2010, DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/MCSE.2010.73.
7. Gorenflo R. and Mainardi F., "Fractional Calculus: Integral and Differential Equations of Fractional Order", in Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag, Wien and New York, 1997.
8. Hjort-Jensen M., "Computational Physics", University of Oslo, 2009.
9. Kilbas A. A., Srivastava H. M., Trujillo J. J., "Theory and Applications of Fractional Differential Equations", North Holland Mathematics Studies 204, Elsevier, 2006.
10. Krommer R. A., Ueberhuber Ch. W., "Computational Integration", SIAM, Philadelphia, 1986.
11. Kythe P. K., Schäferkotter M. R., "Handbook of Computational Methods For Integration", Chapman & Hall/CRC, 2005.
12. Miller K. S. and Ross B., "An Introduction To The Fractional Calculus and Fractional Differential Equations", John Willey and Sons INC., New York,NY, 1993.
13. Mori M., "Discovery of The Double Exponential Transformation and Its Developments", publ. RIMS, Kyoto Univ., 41, pp. 897-935, 2005.
14. Muller J. M., Brisebarre N., De Dinechin F., Jeannerod C. P., Lefevre V., Melquiond G., Revol N., Stehle D., and Torres D. S., "Handbook of Floating-Point Arithmetic", Birkhauser Boston, New York,NY, 2010.
15. Oldham K. B., Spanier J., "The Fractional Calculus. Theory and Applications of Differentiation and Integration to Arbitrary Order", Academic Press, 1974.
16. Ostalczyk P., Zarys rachunku ró˙zniczkowego i całkowego ułamkowych rze¸dów, Komitet Automatyki i Robotyki Polskiej Akademii Nauk, Wydawnictwo Politechniki Łódzkiej, Łód´z, Poland, 2008.
17. Podlubny I., Fractional Differential Equations, Academic Press, San Diego, CA, 1999.
18. Schwartz C., "Numerical Integration of Analytic Functions", in Journal of Computational Physics, vol. 4, pp. 19-29, 1969.
19. Stroud A. H., Secrest D., "Gaussian Quadrature Formulas", Prentice-Hall, Englewood Cliffs, NJ., 1966.
20. Takahasi H., Quadrature Formulas Obtained by Variable Transformation, Numerische Mathematik, nr 21, 1973.
21. The GNU Multiple Precision Arithmetic Library, https://gmplib.org/.
22. The GNU Multiple Precision Floating-Point Reliable Library, https://mpfr.org/.
23. Waldvogel J., "Towards A General Error Theory of the Trapezoidal Rule", in Approximation and Computation, pp 267-282, Springer Verlag, W.Gautschi, G.Mastroianni and Th.M.Rassias (Eds.), 2011.
Cited by
Show
ISSN
2300-5963
Language
eng