Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Fractional Calculus and Applied Analysis

Editor-in-Chief: Kiryakova, Virginia

6 Issues per year

The journal celebrates now its 20 years!

IMPACT FACTOR 2016: 2.034
5-year IMPACT FACTOR: 2.359

CiteScore 2016: 2.18

SCImago Journal Rank (SJR) 2016: 1.372
Source Normalized Impact per Paper (SNIP) 2016: 1.492

Mathematical Citation Quotient (MCQ) 2016: 0.61

See all formats and pricing
More options …

Fractional calculus and Sinc methods

Gerd Baumann
  • Mathematics Department, German University in Cairo, New Cairo City, Egypt
  • Department of Mathematical Physics, University of Ulm, Albert-Einstein-Allee 11, D - 89069, Ulm, Germany
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Frank Stenger
Published Online: 2011-09-29 | DOI: https://doi.org/10.2478/s13540-011-0035-3


Fractional integrals, fractional derivatives, fractional integral equations, and fractional differential equations are numerically solved by Sinc methods. Sinc methods are able to deal with singularities of the weakly singular integral equations of Riemann-Liouville and Caputo type. The convergence of the numerical method is numerically examined and shows exponential behavior. Different examples are used to demonstrate the effective derivation of numerical solutions for different types of fractional differential and integral equations, linear and non-linear ones. Equations of mixed ordinary and fractional derivatives, integro-differential equations are solved using Sinc methods. We demonstrate that the numerical calculation needed in fractional calculus can be gained with high accuracy using Sinc methods.

MSC: Primary 65-XX, 45-XX, 97-XX; Secondary 65D15, 45E10, 44A35, 97N50

Keywords: Sinc method; fractional calculus; approximation

  • [1] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, 1974. Google Scholar

  • [2] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993. Google Scholar

  • [3] A. A. Kilbas, H. M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006. Google Scholar

  • [4] J. Sabatier, O. P. Agrawal, and J. A. T. Machado, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht, 2007; http://www.springerlink.com/content/978-1-4020-6041-0/ #section=345728&page=1 Google Scholar

  • [5] S. Das, Functional Fractional Calculus for System Identification and Controls, Springer-Verlag Berlin Heidelberg, Berlin, 2008; http://www.springerlink.com/content/978-3-540-72702-6/ #section=234316&page=1 Google Scholar

  • [6] N.H. Abel, Auflösung einer mechanischen Aufgabe, J. Reine Angew. Math. 1 (1826), 153–157. http://dx.doi.org/10.1515/crll.1826.1.153CrossrefGoogle Scholar

  • [7] K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, Springer, Heidelberg, 2010; http://www.springer.com/mathematics/dynamical+systems/book/ 978-3-642-14573-5 Google Scholar

  • [8] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their solution and Some of Their Applications, Academic Press, San Diego, 1999. Google Scholar

  • [9] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, London, Hackensack NJ, 2010. http://dx.doi.org/10.1142/9781848163300CrossrefGoogle Scholar

  • [10] A. M. Mathai, R.K. Saxena, and H.J. Haubold, The H-function: Theory and Applications, Springer, New York, 2010; http://www.springerlink.com/content/978-1-4419-0915-2/ #section=534430&page=1 Google Scholar

  • [11] R. Gorenflo and F. Mainardi, Fractional calculus, arXiv:0805.3823v1 [math-ph], 25 May 2008. Google Scholar

  • [12] N. Südland, G. Baumann, and T.F. Nonnenmacher, Open Problem: Who knows about the ℵ-functions ?, Fract. Calc. Appl. Anal. 1, No 4 (1998), 401–402; http://www.math.bas.bg/~fcaa/volume1/pp401-402.gif Google Scholar

  • [13] N. Südland, Fraktionale Differentialgleichungen und Foxsche HFunktionen mit Beispielen aus der Physik, PhD Thesis, University of Ulm, 2000. Google Scholar

  • [14] A. M. Mathai and H. J. Haubold, Special Functions for Applied Scientists, Springer, New York, 2008; http://www.springerlink.com/content/978-0-387-75893-0/ #section=202128&page=1 http://dx.doi.org/10.1007/978-0-387-75894-7Google Scholar

  • [15] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Application, CRC Press, Boca Raton, 1993. Google Scholar

  • [16] A. D. Freed and K. Diethelm, Fractional calculus in biomechanics: a 3D viscoelastic model using regularized fractional derivative kernels with application to the human calcaneal fat pad, Biomech. Mod. Mech. 5 (2006), 203–215. http://dx.doi.org/10.1007/s10237-005-0011-0CrossrefGoogle Scholar

  • [17] Ch. Lubich, Runge-Kutta theory for Volterra and Abel integral equations of the second kind, Math. Comput. 41 (1983), 87–102. http://dx.doi.org/10.1090/S0025-5718-1983-0701626-6CrossrefGoogle Scholar

  • [18] A. Schmidt and L. Gaul, FE Implementation of Viscoelastic Constitutive Stress-Strain Relations Involving Fractional Time Derivatives, Preprint, 2001, 1–11. Google Scholar

  • [19] O. P. Agrawal and P. Kumar, Comparison of five numerical schemes for fractional differential equations, In: J. Sabatier, O. P. Agrawal, and J. A. T. Machado (Ed-s), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht (2007), 43–60. Google Scholar

  • [20] L. Yuan and O. P. Agrawal, A numerical scheme for dynamic systems containing fractional derivatives, J. Vib. Acoust. 124 (2002), 014502–014506. http://dx.doi.org/10.1115/1.1448322CrossrefGoogle Scholar

  • [21] S. Momani and Z. Odibat, Homotopy perturbation method for nonlinear partial differential equations of fractional order, Phys. Lett. A 365 (2007), 345–350. http://dx.doi.org/10.1016/j.physleta.2007.01.046CrossrefGoogle Scholar

  • [22] K. Diethelm, N. J. Ford, A. D. Freed, and Y. Luchko, Algorithms for the fractional calculus: A selection of numerical methods, Comput. Methods Appl. Mech. Eng. 194 (2005), 743–773. http://dx.doi.org/10.1016/j.cma.2004.06.006CrossrefGoogle Scholar

  • [23] J. McNamee, F. Stenger, and E. L. Whitney, Whittaker’s cardinal function in retrospect, Math. Comp. 23 (1971), 141–154. Google Scholar

  • [24] F. Stenger, Collocating convolutions, Math. Comp. 64 (1995), 211–235. http://dx.doi.org/10.1090/S0025-5718-1995-1270624-7CrossrefGoogle Scholar

  • [25] G.-A. Zakeri and M. Navab, Sinc collocation approximation of nonsmooth solution of a nonlinear weakly singular Volterra integral equation, J. Comp. Phys. 229 (2010), 6548–6557. http://dx.doi.org/10.1016/j.jcp.2010.05.010CrossrefGoogle Scholar

  • [26] T. Okayama, T. Matsuo, and M. Sugihara, Sinc-collocation Methods for Weakly Singular Fredholm Integral Equations of the Second Kind, 2009; http://www.keisu.t.u-tokyo.ac.jp/research/techrep/index.html Google Scholar

  • [27] F. Stenger, Handbook of Sinc Numerical Methods, CRC Press, Boca Raton, 2011; http://www.crcpress.com/product/isbn/9781439821589 Google Scholar

  • [28] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Acad. Publ., Dordrecht, 1994. Google Scholar

  • [29] J. T. Edwards, J. A. Roberts, and N. J. Ford, A comparison of Adomiannal differential equations: An application-oriented exposition using differential operators of Caputo type, Numerical Analysis Report, Manchester Centre for Computational Mathematics 309 (1997), 1–18. Google Scholar

  • [30] A. Répaci, Nonlinear dynamical systems: On the accuracy of adomian’s decomposition method, Appl. Math. Lett. 3 (1990), 35–39. http://dx.doi.org/10.1016/0893-9659(90)90042-ACrossrefGoogle Scholar

  • [31] S. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman & Hall/CRC Press, Boca Raton, 2004. Google Scholar

  • [32] J. H. He, Approximate solution of non linear differential equations with convolution product nonlinearities, Comput. Meth. Appl. Mech. Eng. 167 (1998), 69–73. http://dx.doi.org/10.1016/S0045-7825(98)00109-1CrossrefGoogle Scholar

  • [33] J. H. He, Variational iteration method — some recent results and new interpretations, J. Comput. Appl. Math. 207 (2007), 3–17. http://dx.doi.org/10.1016/j.cam.2006.07.009CrossrefGoogle Scholar

  • [34] Z. Odibata, S. Momanib, and V. S. Erturkc, Generalized differential transform method: Application to differential equations of fractional order, Appl. Math. Comp. 197 (2008), 467–477. http://dx.doi.org/10.1016/j.amc.2007.07.068CrossrefGoogle Scholar

  • [35] M. Tataria and M. Dehghan, On the convergence of He’s variational iteration method, J. Comp. Appl. Math. 207 (2007), 121–128. http://dx.doi.org/10.1016/j.cam.2006.07.017CrossrefGoogle Scholar

  • [36] S. Liang and D. J. Jeffreya, Comparison of homotopy analysis method and homotopy perturbation method through an evolution equation, Comm. Non. Sci. Num. Sim. 14 (2009), 4057–4064. http://dx.doi.org/10.1016/j.cnsns.2009.02.016CrossrefGoogle Scholar

  • [37] J.-P. Berrut, Barycentric formulae for cardinal (SINC-)interpolants, Numer. Math. 54 (1989), 703–718. http://dx.doi.org/10.1007/BF01396489CrossrefGoogle Scholar

  • [38] V. Daftardar-Gejji and H. Jafari, Analysis of a system of nonautonomous fractional differential equations involving Caputo derivatives, J. Math. Anal. Appl. 328 (2007), 1026–1033. http://dx.doi.org/10.1016/j.jmaa.2006.06.007CrossrefGoogle Scholar

  • [39] F. Stenger, Summary of Sinc numerical methods, J. Comp. Appl. Math. 121 (2000), 379–420; http://www.mendeley.com/research/summary-sinc-numerical-methods-13/ http://dx.doi.org/10.1016/S0377-0427(00)00348-4CrossrefGoogle Scholar

  • [40] K. Diethelm and N. J. Ford, Numerical solution of the Bagley-Torvik equation, Numerical Analysis Report, Manchester Centre for Computational Mathematics 378 (2003), 1–13. Google Scholar

  • [41] G. Baumann, N. Südland, and T. F. Nonnenmacher, Anomalous relaxation and diffusion processes in complex systems, Trans. Th. Stat. Phys. 29 (2000), 157–171. http://dx.doi.org/10.1080/00411450008205866CrossrefGoogle Scholar

  • [42] M. Caputo and F. Mainardi, A new dissipation model based on memory mechanism, Pure Appl. Geophys. 91 (1971), 134–147; http://www.springerlink.com/content/wv5233j83145/?sortorder=asc&po=10; Reprinted in: Fract. Calc. Appl. Anal. 10, No 3 (2007), 309–324; at http://www.math.bas.bg/~fcaa http://dx.doi.org/10.1007/BF00879562Google Scholar

  • [43] K. Diethelm, Efficient solution of multi-term fractional differential equations using P(EC)mE methods, Computing 71 (2003), 305–319. http://dx.doi.org/10.1007/s00607-003-0033-3CrossrefGoogle Scholar

  • [44] M. A. Kowalski, K. A. Sikorski, and F. Stenger, Selected Topics in Approximation and Computation, Oxford Univ. Press, New York, 1995. Google Scholar

  • [45] F. Stenger, Numerical Methods Based on Sinc and Analytic Functions, Springer, New York, 1993. http://dx.doi.org/10.1007/978-1-4612-2706-9CrossrefGoogle Scholar

  • [46] G. Baumann and M. Mnuk, Elements — An object-oriented approach to industrial software development, The Mathematica Journal 10 (2006), 161–186; http://www.mathematica-journal.com/issue/v10i1/Elements.html Google Scholar

  • [47] G. Baumann, Mathematica® for Theoretical Physics: Electrodynamics, Quantum Mechanics, General Relativity and Fractals, Springer, New York, 2005; http://www.springerlink.com/content/978-0-387-21933-2/ #section=531623&page=1 Google Scholar

  • [48] C. Lubich, Convolution Quadrature and Discretized Operational Calculus: I, Numer. Math. 52 (1988), 129–145. http://dx.doi.org/10.1007/BF01398686CrossrefGoogle Scholar

  • [49] C. Lubich, Convolution Quadrature and Discretized Operational Calculus: II, Numer. Math. 52 (1988), 413–425. http://dx.doi.org/10.1007/BF01462237CrossrefGoogle Scholar

  • [50] K. Diethelm, An improvement of a nonclassical numerical method for the computation of fractional derivatives, J. Vib. Acoust. 131 (2009), 321–325. http://dx.doi.org/10.1115/1.2981167CrossrefGoogle Scholar

  • [51] P. J. Torvik and R. L. Bagley, On the appearance of the fractional derivative in the behavior of real materials, J. Appl. Mech. 51 (1984), 294–298. http://dx.doi.org/10.1115/1.3167615CrossrefGoogle Scholar

  • [52] P. Linz, Analytical and Numerical Methods for Volterra Equations, Soc. for Industrial and Applied Mathematics, Philadelphia, 1985. Google Scholar

  • [53] H. Brunner, The numerical analysis of functional integral and integrodifferential equations of Volterra type, Acta Numerica 13 (2004), 55–145. http://dx.doi.org/10.1017/CBO9780511569975.002CrossrefGoogle Scholar

  • [54] H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations, Cambridge University Press, Cambridge, 2004. http://dx.doi.org/10.1017/CBO9780511543234CrossrefGoogle Scholar

About the article

Published Online: 2011-09-29

Published in Print: 2011-12-01

Citation Information: Fractional Calculus and Applied Analysis, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: https://doi.org/10.2478/s13540-011-0035-3.

Export Citation

© 2011 Diogenes Co., Sofia. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

Gerd Baumann and Frank Stenger
Mathematics, 2017, Volume 5, Number 1, Page 12
Gerd Baumann and Frank Stenger
Mathematics, 2015, Volume 3, Number 2, Page 444
Gerd Baumann and Frank Stenger
Fractional Calculus and Applied Analysis, 2013, Volume 16, Number 3
Shahrokh Esmaeili, Mostafa Shamsi, and Mehdi Dehghan
Open Physics, 2013, Volume 11, Number 10
Lixing Han and Jianhong Xu
Journal of Computational and Applied Mathematics, 2014, Volume 255, Page 805
Zhi Mao, Aiguo Xiao, Zuguo Yu, and Long Shi
Journal of Applied Mathematics, 2014, Volume 2014, Page 1

Comments (0)

Please log in or register to comment.
Log in