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Generalized Differential and Integral Operators

  • Trifce Sandev
  • Živorad Tomovski
Chapter
Part of the Developments in Mathematics book series (DEVM, volume 61)

Abstract

From the time of discovery of calculus by Leibniz, he studied the problem of fractional differentiations. 30 September 1695, the day when Leibniz sent a letter to L’Hôpital with a reply of the L’Hôpital’s question related to the differentiation of a function of order n = 1∕2, became a birthday of the fractional calculus. By using the Leibniz product rule and the binomial theorem he obtained some paradoxical results. Euler partially resolved the Leibniz paradox by introducing the gamma function as 1 ⋅ 2 ⋅… ⋅ n = n! = Γ(n + 1). Therefore, the fractional calculus has attracted attention to a range of celebrated mathematicians and physicists, such as Leibniz, Euler, Laplace, Lacroix, Fourier, Abel, Liouville, Riemann, Grünwald, Letnikov, to name but a few.

References

  1. 1.
    Abatangelo, N., Valdinoci, E.: Getting acquainted with the fractional Laplacian. ArXiv: 1710.11567Google Scholar
  2. 2.
    Allen, M., Caffarelli, L., Vasseur, A.: Arch. Ration. Mech. Anal. 221, 603 (2016)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bucur, C., Valdinoci, E.: Nonlocal Diffusion and Applications. Springer, Basel (2016)zbMATHCrossRefGoogle Scholar
  4. 4.
    Bukur, C.: ESAIM Control Optim. Calc. Var. 23, 1361 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bulavatsky, V.M.: Cybern. Syst. Anal. 53, 204 (2017)CrossRefGoogle Scholar
  6. 6.
    Caffarelli, L., Silvestre, L.: Commun. Part. Diff. Equ. 32, 1245 (2007)CrossRefGoogle Scholar
  7. 7.
    Caffarelli, L.A., Stinga, P.R.: Ann. I. H. Poincaré 33, 767 (2016)CrossRefGoogle Scholar
  8. 8.
    Camargo, R.F., Chiacchio, A.O., Charnet, R., Capelas de Oliveira, E.: J. Math. Phys. 50, 063507 (2009)Google Scholar
  9. 9.
    Caputo, M.: Elasticita e Dissipazione. Zanichelli Printer, Bologna (1969)Google Scholar
  10. 10.
    Colombaro, I., Giusti, A., Vitali, S.: Mathematics 6, 15 (2018)CrossRefGoogle Scholar
  11. 11.
    Cottone, G., Di Paola, M.: Prob. Eng. Mech. 24, 321 (2009)CrossRefGoogle Scholar
  12. 12.
    Cottone, G., Di Paola, M., Zingales, M.: Physica E 42, 95 (2009)CrossRefGoogle Scholar
  13. 13.
    Cottone, G., Di Paola, M., Zingales, M.: Fractional mechanical model for the dynamics of non-local continuum. In: Mastorakis, N., Sakellaris, J. (eds.) Advances in Numerical Methods. Lecture Notes in Electrical Engineering, vol. 11. Springer, Boston (2009)Google Scholar
  14. 14.
    dos Santos, M.A.F.: Physics 1, 40 (2019)CrossRefGoogle Scholar
  15. 15.
    D’Ovidio, M., Polito, F.: Theory Probab. Appl. 62, 552 (2018). arXiv:1307.1696 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions, vol. 3. McGraw-Hill, New York (1955)zbMATHGoogle Scholar
  17. 17.
    Feller, W.: An Introduction to Probability Theory and Its Applications, vol. II. Wiley, New York (1968)zbMATHGoogle Scholar
  18. 18.
    Figueiredo Camargo, R., Capelas de Oliveira, E., Vaz, J. Jr.: J. Math. Phys. 50, 123518 (2009)Google Scholar
  19. 19.
    Garra, R., Gorenflo, R., Polito, F., Tomovski, Z.: Appl. Math. Comput. 242, 576 (2014)MathSciNetGoogle Scholar
  20. 20.
    Garrappa, R.: Commun. Nonlinear Sci. Numer. Simul. 38, 178 (2016)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Garrappa, R., Maione, G.: Fractional Prabhakar derivative and applications in anomalous dielectrics: a numerical approach. In: Babiarz, A., Czornik, A., Klamka, J., Niezabitowski, M. (eds.) Theory and Applications of Non-integer Order Systems. Lecture Notes in Electrical Engineering, vol. 407. Springer, Cham (2017)zbMATHGoogle Scholar
  22. 22.
    Garrappa, R., Mainardi, F., Maione, G.: Fract. Calc. Appl. Anal. 19, 1105 (2016)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Giusti, A., Colombaro, I.: Commun. Nonlinear Sci. Numer. Simul. 56, 138 (2018)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Gorenflo, R., Kilbas, A.A., Mainardi, F., Rogosin, S.V.: Mittag-Leffler Functions, Related Topics and Applications. Springer, Heidelberg (2014)Google Scholar
  25. 25.
    Hilfer, R.: Phys. Rev. E 48, 2466 (1993)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Hilfer, R.: Physica A 221, 89 (1995)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Hilfer, R.: Chaos Solitons and Fractals 5, 1475 (1995)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Hilfer, R.: Fractals 3, 211 (1995)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Hilfer, R.: Application of Fractional Calculus in Physics. World Scientific, Singapore (2000)zbMATHCrossRefGoogle Scholar
  30. 30.
    Hilfer, R.: Chem. Phys. 284, 399 (2002)CrossRefGoogle Scholar
  31. 31.
    Hilfer, R.: Fractals 11, 251 (2003)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Hilfer, R., Luchko, Y., Tomovski, Z.: Fract. Calc. Appl. Anal. 12, 299 (2009)MathSciNetGoogle Scholar
  33. 33.
    Iqbal, S., Krulić, K., Pečarić, J.: J. Inequal. Appl. 2010, 264347 (2010)CrossRefGoogle Scholar
  34. 34.
    Kilbas, A.A., Saigo, M., Saxena, R.K.: Integral Transforms Spec. Funct. 15, 31 (2004)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematical Studies, vol. 204. Elsevier/North-Holland, Amsterdam (2006)zbMATHCrossRefGoogle Scholar
  36. 36.
    Kochubei, A.: Integral Equ. Oper. Theory 71, 583 (2011)CrossRefGoogle Scholar
  37. 37.
    Luchko, Y., Yamamoto, M.: Fract. Calc. Appl. Anal. 19, 676 (2016)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Mainardi, F., Pagnini, G., Saxena, R.K.: J. Comput. Appl. Math. 178, 321 (2005)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Mathai, A.M., Saxena, R.K.: The H-Function with Applications in Statistics and Other Disciplines. Wiley Halsted, New York (1978)zbMATHGoogle Scholar
  40. 40.
    Podlubny, I.: Fractional Differential Equations. Academic, San Diego (1999)zbMATHGoogle Scholar
  41. 41.
    Polito, F., Scalas, E.: Electron. Commun. Probab. 21, 1 (2016)CrossRefGoogle Scholar
  42. 42.
    Polito, F., Tomovski, Z.: Fract. Diff. Calc. 6, 73 (2016)CrossRefGoogle Scholar
  43. 43.
    Prabhakar, T.R.: Yokohama Math. J. 19, 7 (1971)MathSciNetGoogle Scholar
  44. 44.
    Prudnikov, A.P., Brychkov, Y.A., Marichev, O.I.: Integrals and Series. More Special Functions, vol. 3. Gordon and Breach, New York (1989)Google Scholar
  45. 45.
    Saichev, A., Zaslavsky, G.: Chaos 7, 753 (1997)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Sajid, I., Pečarić, J., Samraiz, M., Tomovski, Z.: Tbilisi Math. J. 10, 75 (2017)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Sandev, T.: Mathematics 5, 66 (2017)CrossRefGoogle Scholar
  48. 48.
    Sandev, T., Iomin, A.: Europhys. Lett. 124, 20005 (2018)CrossRefGoogle Scholar
  49. 49.
    Sandev, T., Tomovski, Z.: Phys. Scr. 82, 065001 (2010)CrossRefGoogle Scholar
  50. 50.
    Sandev, T., Metzler, R., Tomovski, Z.: J. Phys. A: Math. Theor. 44, 255203 (2011)CrossRefGoogle Scholar
  51. 51.
    Sandev, T., Tomovski, Z., Dubbeldam, J.L.A.: Physica A 390, 3627 (2011)CrossRefGoogle Scholar
  52. 52.
    Sandev, T., Petreska, I., Lenzi, E.K.: J. Math. Phys. 55, 092105 (2014)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Sandev, T., Chechkin, A., Kantz, H., Metzler, R.: Fract. Calc. Appl. Anal. 18, 1006 (2015)MathSciNetCrossRefGoogle Scholar
  54. 54.
    Sandev, T., Sokolov, I.M., Metzler, R., Chechkin, A.: Chaos Solitons Fractals 102, 210 (2017)MathSciNetCrossRefGoogle Scholar
  55. 55.
    Sandev, T., Deng, W., Xu, P.: J. Phys. A: Math. Theor. 51, 405002 (2018)CrossRefGoogle Scholar
  56. 56.
    Srivastava, H.M., Saxena, R.K.: Appl. Math Comput. 118, 1 (2001)MathSciNetGoogle Scholar
  57. 57.
    Srivastava, H.M., Tomovski, Z.: Appl. Math. Comput. 211, 198 (2009)MathSciNetGoogle Scholar
  58. 58.
    Tomovski, Z.: Nonlinear Anal. 75, 3364 (2012)MathSciNetCrossRefGoogle Scholar
  59. 59.
    Tomovski, Z., Pogány, T.K., Srivastava, H.M.: J. Franklin Inst. 351, 5437 (2014)MathSciNetCrossRefGoogle Scholar
  60. 60.
    Viñales, A.D., Despósito, M.A.: Phys. Rev. E 75, 042102 (2007)CrossRefGoogle Scholar
  61. 61.
    Viñales, A.D., Wang, K.G., Despósito, M.A.: Phys. Rev. E 80, 011101 (2009)CrossRefGoogle Scholar
  62. 62.
    Xu, J.: J. Phys. A: Math. Theor. 50, 195002 (2017)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Trifce Sandev
    • 1
  • Živorad Tomovski
    • 2
  1. 1.Research Center for Computer Science and Information TechnologiesMacedonian Academy of Sciences and ArtsSkopjeNorth Macedonia
  2. 2.Faculty of Natural Sciences and Mathematics, Institute of MathematicsSs. Cyril and Methodius University in SkopjeSkopjeNorth Macedonia

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