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Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain


Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

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Accuracy Assessment of Fractional Order Derivatives and Integrals Numerical Computations

Journal of Applied Nonlinear Dynamics 4(1) (2015) 53--65 | DOI:10.5890/JAND.2015.03.005

Dariusz W. Brzeziński; P.Ostalczyk

Institute of Applied Computer Science, Lodz University of Technology, 18/22 Stefanowskiego St., 90 -924 Łodź, Poland

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This paper presents results of a numerical experiment, during which different numerical criteria of exact values setting in the accuracy assessment of fractional order derivatives / integrals numerical calculations are tested. Although traditional accuracy criteria in form of relative error expressed in % are applied, the values assumed as exact, necessary for comparison, are now: value of a function, classical 1st derivative, integral of the 1st order and Mittag-Leffler function’s values. For that purpose, fractional order differentiation and integration operators concatenation rules are applied. The methods allow to assess the accuracy of numerical calculations of fractional derivatives and integrals for each required function and not only for ones, for which mathematical formulas are available. The proposed measures are employed to determine proper operation and assess the accuracy of fractional order derivatives and integrals numerical algorithms.The algorithms utilize Riemann-Liouville and Grunwald-Letnikov frac-¨ tional order derivatives and integrals formulas.


The research is supported by the Polish National Science Center in 2013-2015 as a research project (DEC-2012/05/B/ST 6/03647).


  1. [1]  Demidowicz, B.P., Maron, I.A., and Szuwałowa (1965), Metody Numeryczne: Państwowe Wydawnictwo Naukowe, Warszawa. (in Polish).
  2. [2]  Aho, A.V., Ullman, J.D., and Hopcroft, J.E. (1983), Data Structures and Algorithms, Bell Telephone Laboratories.
  3. [3]  Papadimitriou, Ch.H. (1995), Computational Complexity, Addison-Wesley, Inc.
  4. [4]  Taylor, J.R. (1997), An Introduction to Error Analysis. The Study of Uncertainties in Physical Measurements, Second Edition, University Science Books, Sausalito, California, USA.
  5. [5]  Drozdek, A. (2001), Data Structures and Algorithms in C++, Second Edition, Brooks/Cole.
  6. [6]  Burden, R.L. and Faires, J.D. (2003), Numerical Analysis, 5th ed., : Brooks/Cole Cengage Learning, Boston.
  7. [7]  Null, L. and Lobur, J. (2004), The Essentials of Computer Organization and Architecture, Jones and Barlett Publishers, Inc.
  8. [8]  Kythe, P.K. and Schaferkotter, M.R. (2005), Handbook of Computational Methods for Integration, Chapman & Hall/CRC.
  9. [9]  Celier, F.E. and Kofman, E. (2006), Continous System Simulation, : Springer Science+Business Media, Inc.
  10. [10]  Butcher, J.C. (2008), Numerical Methods for Ordinary Differential Equations. Second Edition, John Wiley & Sons, Ltd.
  11. [11]  Meyers, S. (2008), More Effective C++. 35 New Ways to Improve Your Programs and Designs. Second Edition, Pearson Education, Inc.
  12. [12]  Chapra, S.C. and Canale, R.P. (2010), Numerical Methods for Engineers, Sixth Edition, McGraw-Hill Companies, Inc.
  13. [13]  Bateman, H. (1954), Tables of Integral Transforms. Volume II, McGraw-Hill Book Company, Inc., NY,Toronto, London.
  14. [14]  Miller, K. S. and Ross, B. (1993), An Introduction To The Fractional Calculus and Fractional Differential Equations, John Willey and Sons INC., New York, NY.
  15. [15]  Podlubny, I. (1999), Fractional Differential Equations, Academic Press, San Diego, CA.
  16. [16]  Diethelm, K. (2004), The Analysis of Fractional Differential Equations, Springer-Verlag.
  17. [17]  Baleanu, D., Diethelm, K., Scalas, E., and Trujillo, J.J. (2012), Fractional Calculus Models and Numerical Methods, World Scientific Publishing Co.Pte. Ltd., Singapore.
  18. [18]  Ostalczyk, P. (2008), Zarys rachunku różniczkowego i całkowego ulamkowych rzędów: Wydawnictwo Politechniki Łodzkiej, Łódź, Poland. (in Polish).
  19. [19]  Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006), Theory and Applications of Fractional Differential Equations, Elsevier Science.
  20. [20]  Machado, J.T. (2014), Numerical Calculation of The Left and Right Fractional Derivatives: J.Comput. Phys.
  21. [21]  Gorenflo, R. and Mainardi, F. (1997), Fractional Calculus: Integral and Differential Equations of Fractional Order. Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag, Wien and New York, 223-273.
  22. [22]  Gorenflo, R., Loutchko, J. and Luchko, Y. (2002), Computation of the Mittag-Leffler Function and Its Derivative, Fractional Calculus & Applied Analysis, 4, 491-518.
  23. [23]  Valerio, D., Trujillo, J. J., Rivero, M., Machado, J.A.T., and Baleanu, D. (2013), Fractional Calculus: A Survey of Useful Formulas, The European Physical Journal Special Topics, 222, 1827-1846.
  24. [24]  Ortigueira, M.D., Machado, J.A.T. and Sa da Costa, J. (2005), Which Differintegration? IEE Proceedings - Vision, Image and Signal Processing, 152(6).
  25. [25]  Ortigueira, M.D. and Machado, J.A.T. (2014), What is a Fractional Derivative, J. Comput. Phys.
  26. [26]  Hjorton-Jensen, M. (2009), Computational Physics, University of Oslo.
  27. [27]  Valerio, D. and Machado, J.A.T. (2014), On the Numerical Computation of the Mittag-Leffler Function, Commun Nonlinear Sci Numer Simulat.
  28. [28]  Brzeziński, D.W. and Ostalczyk,P. (2013), Evaluation of Efficient Methods of Fractional Order Derivatives and Integrals Numerical Calculations, Proceedings of the XIV Conference on System Modelling and Control, September 23-24 2013, L∩odz, Poland.
  29. [29]  Brzeziński, D.W. and Ostalczyk,P. (2014), High-accuracy Numerical IntegrationMethods for Fractional Order Derivatives and Integrals Computations, Bulletin of the Polish Academy of Sciences Technical Sciences, 62(4).
  30. [30]  Takahasi, H.(1973), Quadrature Formulas Obtained by Variable Transformation, Numerische Mathematik, 21.
  31. [31]  Mori, M. (1990), Development in the Double Exponential Formulas for Numerical Integration, Proceedings of the International Congress of Mathematicians, Kyoto, Japan, 1990.
  32. [32]  Mori, M. (2005), Discovery of the Double Exponential Transformation and its Developments, RIMS, Kyoto Univ..
  33. [33]  Schwartz, Ch. (1969), Numerical Integration of Analytic Functions, Journal of Computational Physics,4.
  34. [34]  Ostalczyk, P., Duch, P. and Sankowski, D. (2011), Fractional Order Backward Difference Grunwald-Letnikov and Horner Simplified Forms Evaluation Accuracy Analysis, Automatyka,15(3).
  35. [35]  Brzeziński, D.W. and Ostalczyk, P. (2012), The Grünwald-Letnikov Formula and Its Horner's Equivalent Form Accuracy Comparison and Evaluation For Application To Fractional Order PID Controler, IEEE Explore Digital Library: IEEE Conference Publications-17th International Conference On Methods and Models In Automation and Robotics (MMAR).
  36. [36]  Overton, M.(2001), Numerical Computing with IEEE Floating Point Arithmetic, SIAM.
  37. [37]  Wilkinson, J.H. (1994), Rounding Errors in Algebraic Processes, Dover, New York,
  38. [38]  Brisebarre, N. and Muller, J.M. (2007), Correct Rounding of Algebraic Functions, Theoretical Informatics and Applications, Jan-March 2007, 47, 71-83.
  39. [39]  Brisebarre, N. and Muller, J.M. (2008), Correctly Rounded Multiplication by Arbitrary Precision Constants, IEEE Transactions on Computers, 57(2), 165-174.
  40. [40]  Muller, J.M., Brisebarre, N., Dinechin De, F., Jeannerod, C.P., Lefevre, V., Melquiond, G., Revol, N., Stehle, D., and Torres, S. (2010), Handbook of Floating-Point Arithmetic, Birkhauser Boston, New York, NY.
  41. [41]  Ghazi, K.R., Lefevre, V., Theveny, P., and Zimmermann, P. (2001), Why and how to use arbitrary precision, IEEE Computer Society, 12(3).
  42. [42]  The GNU Multiple Precision Floating-Point Reliable Library,
  43. [43]  The GNU Multiple Precision Arithmetic Library,