Applications and Add-ons

  • Edmundo Capelas de OliveiraEmail author
Part of the Studies in Systems, Decision and Control book series (SSDC, volume 240)


We present here examples in which the classical special functions play a fundamental role, as well as examples envolving fractional derivative which give rise to special function proper to fractional calculus. Differential, integral and integrodifferential equations of fractional order will be discussed in specific exercises. As an add-on we address issues that, in our view, would not fit in the main text, because they require more lengthy calculation, for example, fractional integration.


  1. 1.
    Wang, Z.X., Guo, D.R.: Special Functions. World Scientific, Singapore (1989)CrossRefGoogle Scholar
  2. 2.
    de Oliveira, E.C.: Analytical Methods of Integration. Editora Livraria da Física, São Paulo (2012). (in Portuguese)Google Scholar
  3. 3.
    Costa, F.S., Oliveira, D.S., Rodrigues, F.G., Capelas de Oliveira, E.: The fractional space-time radial diffusion equation in terms of the Fox’s \(H\)-function. Phys. A Stat. Mech. Appl. (2018). Scholar
  4. 4.
    Duan, J.S., Guo, A.P., Yun, W.Z.: Similarity solution for fractional diffusion equation. Abs. Appl. Anal. 2014, 548126 (2014)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Lenzi, E.K., Vieira, D.S., Lenzi, M.K., Gonçalves, G., Leitoles, D.P.: Solutions for a fractional diffusion equation with radial symmetry and integrodifferential boundary conditions. Therm. Sci. 19, S1–S6 (2015)CrossRefGoogle Scholar
  6. 6.
    Muslih, S.I., Agrawal, O.P.: Solutions of wave equation in fractional dimensional space. In: Baleanu, D. et al. (eds.) Fractional Dynamics and Control, pp. 217–228 (2012)Google Scholar
  7. 7.
    Grigoletto, E.C., Figueiredo Camargo, R., de Oliveira, E.C.: Three-parameter Mittag-Leffler function with an integral representation on the positive real axis. Proc. Ser. Braz. Soc. Comput. Appl. Math. 6, 010331-1–010331-7 (2018)Google Scholar
  8. 8.
    Inizan, P.: Homogeneous fractional embeddings. J. Math. Phys. 49, 082901 (2008)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Diethelm, K.: The analysis of fractional differential equations. Lecture Notes in Mathematics. Springer, Heidelberg (2010)Google Scholar
  10. 10.
    Herrmann, R.: Fractional Calculus: An Introduction for Physicis. World Scientific, New Jersey (2011)CrossRefGoogle Scholar
  11. 11.
    Gorenflo, R., Kilbas, A.A., Mainardi, F., Rogosin, S.V.: Mittag-Leffler Functions, Related Topics and Applications. Springer, Heidelberg (2014)Google Scholar
  12. 12.
    Shoubia, A.L., Figueiredo Camargo, R., de Oliveira, E.C., Vaz, Jr, J.: Theorem for series in three-parameter Mittag-Leffler function. Fract. Cal. Appl. Anal. 13, 9–20 (2010)Google Scholar
  13. 13.
    Wittaker, E.T., Watson, G.N.: A Course of Modern Analysis, 4th edn. Cambridge Mathematical Library, Cambridge (1996)CrossRefGoogle Scholar
  14. 14.
    Mainardi, F., Gorenflo, R.: On Mittag-Leffler-type functions in fractional evolution processes. J. Comput. Appl. Math. 118, 283–299 (2000)MathSciNetCrossRefGoogle Scholar
  15. 15.
    de Oliveira, E.C., Mainardi, F., Vaz, Jr, J.: Models based on Mittag-Leffler functions for anomalous relaxation in dielectrics. Eur. Phys. J. 193, 161–171 (2011)Google Scholar
  16. 16.
    de Oliveira, E.C., Mainardi, F., Vaz, Jr, J.: Fractional models of anomalous relaxation based on the Kilbas and Saigo function. Meccanica 49, 2049–2060 (2014)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Podlubny, I.: Fractional Differential Equations. Mathematical in Sciences and Engineering, vol. 198. Academic Press, San Diego (1999)Google Scholar
  18. 18.
    Matlob, M.A., Jamali, Y.: The concepts and applications of fractional order differential calculus in modelling of viscoelastic systems: a primer (2017). arXiv:1706.06446v2
  19. 19.
    Camargo, R.F., de Oliveira, E.C.: Fractional Calculus. Editora Livraria da Física, São Paulo (2015). (in Portuguese)Google Scholar
  20. 20.
    Teodoro, G.S.: Fractional derivatives: types and criteria. Ph.D. thesis (2019), Imecc-Unicamp, Campinas. (in Portuguese)Google Scholar
  21. 21.
    Hilfer, R. (ed.): Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)zbMATHGoogle Scholar
  22. 22.
    Tarasov, V.E.: No violation of the Leibniz rule. No fractional derivative. Commun. Nonlinear Sci. Numer. Simul. 18, 2945–2948 (2013)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Mainardi, F., Pironi, P.: The fractional Langevin equation: Brownian motion revisited. Extr. Math. 11, 140–154 (1996)MathSciNetGoogle Scholar
  24. 24.
    Porrà, J.M., Wand, K.G., Masoliver, J.: Generalized Langevin equation: anomalous diffusion and probability distributions. Phys. Rev. E. 53, 5872–5881 (1996)CrossRefGoogle Scholar
  25. 25.
    Camargo, R.F, Chiacchio, A.O., Charnet, R., de Oliveira, E.C.: Solution of the fractional Langevin equation and the Mittag-Leffler functions. J. Math. Phys. 50, 063507 (2009)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Sau Fa, K.: Generalized Langevin equation with fractional derivative and long-time correlation function. Phys. Rev. E. 73, 061104 (2006)CrossRefGoogle Scholar
  27. 27.
    Sau Fa, K.: Fractional Langevin equation and Riemann-Liouville fractional derivative. Eur. Phys. J. E. 24, 139–143 (2007)CrossRefGoogle Scholar
  28. 28.
    Viñales, A.D., Despósito, M.A.: Anomalous diffusion induced by a Mittag-Leffler correlated noise. Phys. Rev. E. 75, 042102 (2007)CrossRefGoogle Scholar
  29. 29.
    Körner, T.W.: Fourier Analysis. Cambridge University Press, Cambridge (1990)Google Scholar
  30. 30.
    Love, E.R.: Changing the order of integration. J. Austral. Math. Soc. 9, 421–432 (1970)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)zbMATHGoogle Scholar
  32. 32.
    de Oliveira, E.C., Rodrigues Jr., W.A.: Analytical Functions with Applications. Editora Livraria da Física, São Paulo (2005). (in Portuguese)Google Scholar
  33. 33.
    Nigmatullin, R.R.: The realization of the generalized transfer equation in a medium with fractal geometry. Phys. Sta. Sol. B 133, 425–430 (1986)CrossRefGoogle Scholar
  34. 34.
    Mainardi, F.: The fundamental solutions for the fractional diffusion-wave equation. Appl. Math. Lett. 9, 23–28 (1996)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Mathai, A.M., Saxena, R.K., Haubold, H.J.: The \(H\)-Function: Theory and Applications. Springer, New York (2010)CrossRefGoogle Scholar
  36. 36.
    Arfken, G.B., Weber, H.J.: Mathematical Methods for Physicists. Academic Press, New York (1995)zbMATHGoogle Scholar
  37. 37.
    Oliveira, D.S., de Oliveira, E.C., Deif, S.: On a sum with a three-parameter Mittag-Leffler function. Int. Transf. Spec. Funct. 27, 639–652 (2016)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Teodoro, G.S, de Oliveira, E.C.: Laplace transform and the Mittag-Leffler function. Int. J. Math. Educ. Sci. Technol. 45, 595–604 (2014)Google Scholar
  39. 39.
    Mittag-Leffler, G.M.: Sur la nouvelle fonction \(E_{\alpha }(x)\). CR Acad. Sci. Paris 137, 554–558 (1903)zbMATHGoogle Scholar
  40. 40.
    Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series and Products. Academic Press Inc, New York (1980)zbMATHGoogle Scholar
  41. 41.
    Prudnikov, A.P., Brychkov, YuA, Marichev, O.I.: Integral and Series (Inverse Laplace Transforms). Gordon and Breach Science Publishers, New York (1992)zbMATHGoogle Scholar
  42. 42.
    Costa, F.S., Plata, A.R.G., de Oliveira, E.C.: Fractional Spacetime Diffusion with Time Dependent Diffusion Coefficients. Submited for Publication (2019)Google Scholar
  43. 43.
    Prudnikov, A.P., Brychkov, YuA, Marichev, O.I.: Integral and Series (Elementary Functions). Gordon and Breach Science Publishers, London (1986)zbMATHGoogle Scholar
  44. 44.
    de Oliveira, E.C.: Special Functions and Applications, 2nd edn. Livraria Editora da Física, São Paulo (2012). (in Portuguese)Google Scholar

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Authors and Affiliations

  1. 1.IMECCUniversidade Estadual de CampinasCampinasBrazil

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