Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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Fractional-Order Derivatives and Integrals: Introductory Overview and Recent Developments

  • Received : 2019.02.01
  • Accepted : 2019.10.29
  • Published : 2020.03.31
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Abstract

The subject of fractional calculus (that is, the calculus of integrals and derivatives of any arbitrary real or complex order) has gained considerable popularity and importance during the past over four decades, due mainly to its demonstrated applications in numerous seemingly diverse and widespread fields of mathematical, physical, engineering and statistical sciences. Various operators of fractional-order derivatives as well as fractional-order integrals do indeed provide several potentially useful tools for solving differential and integral equations, and various other problems involving special functions of mathematical physics as well as their extensions and generalizations in one and more variables. The main object of this survey-cum-expository article is to present a brief elementary and introductory overview of the theory of the integral and derivative operators of fractional calculus and their applications especially in developing solutions of certain interesting families of ordinary and partial fractional "differintegral" equations. This general talk will be presented as simply as possible keeping the likelihood of non-specialist audience in mind.

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References

  1. A. Alsaedi, D. Baleanu, S. Etemad and S. Rezapour, On coupled systems of time-fractional differential problems by using a new fractional derivative, J. Funct. Spaces, (2016), Art. ID 4626940, 8 pp.
  2. M. A. Asiru, Sumudu transform and the solution of integral equations of convolution type, Internat. J. Math. Ed. Sci. Tech., 32(2001), 906-910. https://doi.org/10.1080/002073901317147870
  3. A. Atangana, On the new fractional derivative and application to nonlinear Fisher's reaction diffusion equation, Appl. Math. Comput., 273(2016), 948-956. https://doi.org/10.1016/j.amc.2015.10.021
  4. A. Atangana and B. S. T. Alkahtani, Extension of the RLC electrical circuit to fractional derivative without singular kernel, Adv. Mech. Engrg., 7(6)(2015), 1-6.
  5. A. Atangana and B. S. T. Alkahtani, Analysis of the Keller-Segel model with a fractional derivative without singular kernel, Entropy, 17(2015), 4439-4453. https://doi.org/10.3390/e17064439
  6. A. Atangana and B. S. T. Alkahtani, New model of groundwater flowing within a confine aquifer: application of Caputo-Fabrizio derivative, Arabian J. Geosci., 9(2016), Article ID 8; (DOI: 10.1007/s12517-015-2060-8).
  7. A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and nonsingular kernel: theory and application to heat transfer model, Thermal Sci., 20(2016), 763-769. https://doi.org/10.2298/TSCI160111018A
  8. A. Atangana and J. J. Nieto, Numerical solution for the model of RLC circuit via the fractional derivative without singular kernel, Adv. Mech. Engrg., 7(10)(2015), 1-7.
  9. F. B. M. Belgacem and A. A. Karaballi, Sumudu transform fundamental properties investigations and applications, J. Appl. Math. Stoch. Anal., (2006), Article ID 91083, 23 pp.
  10. F. B. M. Belgacem, A. A. Karaballi and S. L. Kalla, Analytical investigations of the Sumudu transform and applications to integral production equations, Math. Probl. Eng., 3(2003), 103-118.
  11. R. G. Buschman and H. M. Srivastava, The ${\bar{H}}$ function associated with a certain class of Feynman integrals, J. Phys. A Math. Gen., 23(1990), 4707-4710. https://doi.org/10.1088/0305-4470/23/20/030
  12. M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1(2015), 73-85.
  13. M. Caputo and F. Mainardi, A new dissipation model based on memory mechanism, Pure Appl. Geophys., 91(1971), 134-147. https://doi.org/10.1007/BF00879562
  14. C. Cattani, H. M. Srivastava and X.-J. Yang (Editors), Fractional dynamics, Emerging Science Publishers (De Gruyter Open), Berlin and Warsaw, 2015.
  15. V. B. L. Chaurasia and D. Kumar, On the solution of generalized fractional kinetic equations, Adv. Stud. Theoret. Phys., 4(2010), 773-780.
  16. J. Choi and P. Agarwal, A note on fractional integral operator associated with multiindex Mittag-Leffler functions, Filomat, 30(2016), 1931-1939. https://doi.org/10.2298/FIL1607931C
  17. J. Choi and D. Kumar, Solutions of generalized fractional kinetic equations involving Aleph functions, Math. Commun., 20(2015), 113-123.
  18. A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher transcendental functions, Vols. I and II, McGraw-Hill Book Company, New York, Toronto and London, 1953.
  19. A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Tables of integral transforms, Vol. II, McGraw-Hill Book Company, New York, Toronto and London, 1954.
  20. C. Fox, The G and H functions as symmetrical Fourier kernels, Trans. Amer. Math. Soc. 98(1961), 395-429. https://doi.org/10.1090/S0002-9947-1961-0131578-3
  21. M. Fukuhara, Ordinary differential equations, Vol. II, Iwanami Shoten, Tokyo, 1941.
  22. L. Galue, N-fractional calculus operator $N^{\nu}$ method applied to some second order nonhomogeneous equations, J. Fract. Calc., 16(1999), 85-97.
  23. F. Gao, H. M. Srivastava, Y.-N. Gao and X.-J. Yang, A coupling method involving the Sumudu transform and the variational iteration method for a class of local fractional diffusion equations, J. Nonlinear Sci. Appl., 9(2016), 5830-5835. https://doi.org/10.22436/jnsa.009.11.11
  24. R. Gorenflo, F. Mainardi and H. M. Srivastava, Special functions in fractional relaxation-oscillation and fractional diffusion-wave phenomena, Proceedings of the Eighth International Colloquium on Differential Equations, 195-202, VSP Publishers, Utrecht and Tokyo, 1998.
  25. R. Gorenflo and S. Vessela, Abel integral equations: analysis and applications, Lecture Notes in Mathematics 1461, Springer-Verlag, Berlin, Heidelberg, New York and London, 1991.
  26. H. J. Haubold and A. M. Mathai, The fractional kinetic equation and thermonuclear functions, Astrophys. Space Sci., 273(2000), 53-63. https://doi.org/10.1023/A:1002695807970
  27. R. Hilfer (Editor), Applications of fractional calculus in physics, World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, 2000.
  28. R. Hilfer, Fractional time evolution, Applications of Fractional Calculus in Physics, 87-130, World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, 2000.
  29. R. Hilfer and L. Anton, Fractional master equations and fractal time random walks, Phys. Rev. E, 51(1995), R848-R851. https://doi.org/10.1103/PhysRevE.51.R848
  30. E. L. Ince, Ordinary differential equations, Longmans, Green and Company, London, 1927; Reprinted by Dover Publications, New York, 1956.
  31. A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematical Studies 204, Elsevier (North-Holland) Science Publishers, Amsterdam, London and New York, 2006.
  32. V. Kiryakova, Generalized fractional calculus and applications, Pitman Research Notes in Mathematics 301, Longman Scientific and Technical, Harlow (Essex), 1993.
  33. V. Kiryakova, Multiple (multiindex) Mittag-Leffler functions and relations to generalized fractional calculus, J. Comput. Appl. Math., 118(2000), 214-259. https://doi.org/10.1016/S0377-0427(00)00292-2
  34. D. Kumar, On the solution of generalized fractional kinetic equations, J. Global Res. Math. Arch., 1(4)(2013), 31-39.
  35. D. Kumar and J. Choi, Generalized fractional kinetic equations associated with Aleph functions, Proc. Jangjeon Math. Soc., 19(2016), 145-155.
  36. D. Kumar, J. Choi and H. M. Srivastava, Solution of a general family of kinetic equations associated with the Mittag-Leffler function, Nonlinear Funct. Anal. Appl., 23(2018), 455-471. https://doi.org/10.22771/NFAA.2018.23.03.04
  37. D. Kumar, S. D. Purohit, A. Secer and A. Atangana, On generalized fractional kinetic equations involving generalized Bessel function of the first kind, Math. Probl. Engrg., (2015), Article ID 289387, 7 pp.
  38. D. Kumar, J. Singh and D. Baleanu, A new analysis for fractional model of regularized long-wave equation arising in ion acoustic plasma waves, Math. Methods Appl. Sci., 40(2017), 5642-5653. https://doi.org/10.1002/mma.4414
  39. S.-D. Lin, W.-C. Ling, K. Nishimoto and H. M. Srivastava, A simple fractionalcalculus approach to the solutions of the Bessel differential equation of general order and some of its applications, Comput. Math. Appl., 49(2005), 1487-1498. https://doi.org/10.1016/j.camwa.2004.09.009
  40. S.-D. Lin and K. Nishimoto, N-Method to a generalized associated Legendre equation, J. Fract. Calc., 14(1998), 95-111.
  41. S.-D. Lin and K. Nishimoto, New finding of particular solutions for a generalized associated Legendre equation, J. Fract. Calc., 18(2000), 9-37.
  42. S.-D. Lin, K. Nishimoto, T. Miyakoda and H. M. Srivastava, Some differintegral formulas for power, composite and rational functions, J. Fract. Calc., 32(2000), 87-98.
  43. S.-D. Lin, H. M. Srivastava, S.-T. Tu and P.-Y.Wang, Some families of linear ordinary and partial differential equations solvable by means of fractional calculus, Int. J. Differ. Equ. Appl., 4(2002), 405-421.
  44. S.-D. Lin, Y.-S. Tsai and P.-Y.Wang, Explicit solutions of a certain class of associated Legendre equations by means of fractional calculus, Appl. Math. Comput., 187(2007), 280-289. https://doi.org/10.1016/j.amc.2006.08.152
  45. S.-D. Lin, S.-T. Tu, I.-C. Chen and H. M. Srivastava, Explicit solutions of a certain family of fractional differintegral equations, Hyperion Sci. J. Ser. A Math. Phys. Electric. Engrg., 2(2001), 85-90.
  46. S.-D. Lin, S.-T. Tu and H. M. Srivastava, Explicit solutions of certain ordinary differential equations by means of fractional calculus, J. Fract. Calc., 20(2001), 35-43.
  47. S.-D. Lin, S.-T. Tu and H. M. Srivastava, Certain classes of ordinary and partial differential equations solvable by means of fractional calculus, Appl. Math. Comput., 131(2002), 223-233. https://doi.org/10.1016/S0096-3003(01)00139-4
  48. S.-D. Lin, S.-T. Tu and H. M. Srivastava, Explicit solutions of some classes of non-Fuchsian differential equations by means of fractional calculus, J. Fract. Calc., 21(2002), 49-60.
  49. J. Liouville, Memoire sur quelques de geometrie et de mecanique, et sur un nouveau genre de calcul pour resourdre ces wuetions, J. Ecole Polytech., 13(21)(1832), 1-69.
  50. A. C. McBride, Fractional calculus and integral transforms of generalized functions, Pitman Research Notes in Mathematics, Vol. 31, Pitman Publishing Limited, London, 1979.
  51. N. W. McLachlan, Modern operational calculus with applications in technical mathematics, Macmillan, London, 1948.
  52. K. S. Miller and B. Ross, An introduction to fractional calculus and fractional differential equations, A Wiley-Interscience Publication, John Wiley and Sons, New York, Chichester, Brisbane, Toronto and Singapore, 1993.
  53. G. M. Mittag-Leffler, Sur la nouvelle fonction $E_{\alpha}(x)$, C. R. Acad. Sci. Paris, 137(1903), 554-558.
  54. K. Nishimoto, Fractional Calculus, Vols. I, II, III, IV, V, Descartes Press, Koriyama, 1984, 1987, 1989, 1991, and 1996.
  55. K. Nishimoto, An essence of Nishimoto's fractional calculus (Calculus of the 21st century): integrations and differentiations of arbitrary order, Descartes Press, Koriyama, 1991.
  56. K. Nishimoto, Operator $N^{\nu}$ method to nonhomogeneous Gauss and Bessel equations, J. Fract. Calc., 9(1996), 1-15.
  57. K. Nishimoto, J. Aular de Duran and L. Galue, N-Fractional calculus operator $N^{\nu}$ method to nonhomogeneous Fukuhara equations, I, J. Fract. Calc., 9(1996), 23-31.
  58. K. Nishimoto and S. Salinas de Romero, N-Fractional calculus operator $N^{\nu}$ method to nonhomogeneous and homogeneous Whittaker equations, I, J. Fract. Calc., 9(1996), 17-22.
  59. K. Nishimoto, S. Salinas de Romero, J. Matera and A. I. Prieto, N-Method to the homogeneous Whittaker equations, J. Fract. Calc., 15(1999), 13-23.
  60. K. Nishimoto, S. Salinas de Romero, J. Matera and A. I. Prieto, N-Method to the homogeneous Whittaker equations (revise and supplement), J. Fract. Calc., 16(1999), 123-128.
  61. K. Nishimoto, H. M. Srivastava and S.-T. Tu, Application of fractional calculus in solving certain classes of Fuchsian differential equations, J. College Engrg. Nihon Univ. Ser. B, 32(1991), 119-126.
  62. K. Nishimoto, H. M. Srivastava and S.-T. Tu, Solutions of some second-order linear differential equations by means of fractional calculus, J. College Engrg. Nihon Univ. Ser. B, 33(1992), 15-25.
  63. K. B. Oldham and J. Spanier, The fractional calculus: theory and applications of differentiation and integration to arbitrary order, Academic Press, New York and London, 1974.
  64. I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering 198, Academic Press, New York, London, Sydney, Tokyo and Toronto, 1999.
  65. A. I. Prieto, S. Salinas de Romero and H. M. Srivastava, Some fractional calculus results involving the generalized Lommel-Wright and related functions, Appl. Math. Lett., 20(2007), 17-22. https://doi.org/10.1016/j.aml.2006.02.018
  66. B. Rubin, Fractional integrals and potentials, Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman Scientific and Technical, Harlow (Essex), 1996.
  67. K. M. Saad and A. A. Al-Shomrani, An application of homotopy analysis transform method for Riccati differential equation of fractional order, J. Fract. Calc. Appl., 7(2016), 61-72.
  68. A. Saichev and M. Zaslavsky, Fractional kinetic equations: solutions and applications, Chaos, 7(1997), 753-764. https://doi.org/10.1063/1.166272
  69. S. Salinas de Romero and K. Nishimoto, N-Fractional calculus operator $N^{\nu}$ method to nonhomogeneous and homogeneous Whittaker equations II, some illustrative examples, J. Fract. Calc., 12(1997), 29-35.
  70. S. Salinas de Romero and H. M. Srivastava, An application of the N-fractional calculus operator method to a modified Whittaker equation, Appl. Math. Comput., 115(2000), 11-21. https://doi.org/10.1016/S0096-3003(99)00130-7
  71. V. P. Saxena, A trivial extension of Saxena's I-function, Nat. Acad. Sci. Lett., 38(2015), 243-245. https://doi.org/10.1007/s40009-014-0330-8
  72. R. K. Saxena, J. P. Chauhan, R. K. Jana and A. K. Shukla, Further results on the generalized Mittag-Leffler function operator, J. Inequal. Appl., (2015), 2015:75, 12 pp. https://doi.org/10.1186/s13660-015-0589-4
  73. R. K. Saxena and S. L. Kalla, On the solutions of certain fractional kinetic equations, Appl. Math. Comput., 199(2008), 504-511. https://doi.org/10.1016/j.amc.2007.10.005
  74. R. K. Saxena, A. M. Mathai and H. J. Haubold, On fractional kinetic equations, Astrophys. Space Sci., 282(2002), 281-287. https://doi.org/10.1023/A:1021175108964
  75. R. K. Saxena, A. M. Mathai and H. J. Haubold, On generalized fractional kinetic equations, Phys. A, 344(2004), 653-664.
  76. R. K. Saxena, A. M. Mathai and H. J. Haubold, Unified fractional kinetic equation and a fractional diffusion equation, Astrophys. Space Sci., 290(2004), 299-310. https://doi.org/10.1023/B:ASTR.0000032531.46639.a7
  77. R. K. Saxena and K. Nishimoto, N-Fractional calculus of generalized Mittag-Leffler functions, J. Fract. Calc., 37(2010), 43-52.
  78. R. K. Saxena, J. Ram and D. Kumar, Alternative derivation of generalized kinetic equations, J. Fract. Calc. Appl., 4(2013), 322-334.
  79. R. K. Saxena, J. Ram and M. Vishnoi, Fractional differentiation and fractional integration of the generalized Mittag-Leffler function, J. Indian Acad. Math., 32(2010), 153-162.
  80. J. L. Schiff, The Laplace transform: theory and applications, Springer-Verlag, Berlin, Heidelberg and New York, 1999.
  81. A. K. Shukla and J. C. Prajapati, On a generalization of Mittag-Leffler function and its properties, J. Math. Anal. Appl., 336(2007), 797-811. https://doi.org/10.1016/j.jmaa.2007.03.018
  82. H. M. Srivastava, On an extension of the Mittag-Leffler function, Yokohama Math. J., 16(1968), 77-88.
  83. H. M. Srivastava and R. G. Buschman, Convolution integral equations with special function kernels, Halsted Press, John Wiley and Sons, New York, 1977.
  84. H. M. Srivastava and R. G. Buschman, Theory and applications of convolution integral equations, Kluwer Series on Mathematics and Its Applications 79, Kluwer Academic Publishers, Dordrecht, Boston and London, 1992.
  85. H. M. Srivastava and J. Choi, Zeta and q-zeta functions and associated series and integrals, Elsevier Science Publishers, Amsterdam, London and New York, 2012.
  86. H. M. Srivastava, A. K. Golmankhaneh, D. Baleanu and X.-J. Yang, Local fractional Sumudu transform with application to IVPs on Cantor sets, Abstr. Appl. Anal., (2014), Art. ID 620529, 7 pp.
  87. H. M. Srivastava, K. C. Gupta and S. P. Goyal, The H-functions of one and two variables with applications, South Asian Publishers, New Delhi and Madras, 1982.
  88. H. M. Srivastava and P. W. Karlsson, Multiple Gaussian hypergeometric series, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1985.
  89. H. M. Srivastava and B. R. K. Kashyap, Special functions in queuing theory and related stochastic processes, Academic Press, New York, London and Toronto, 1982.
  90. H. M. Srivastava, S.-D. Lin, Y.-T. Chao and P.-Y. Wang, Explicit solutions of a certain class differential equations by means of fractional calculus, Russian J. Math. Phys., 14(2007), 357-365. https://doi.org/10.1134/S1061920807030090
  91. H. M. Srivastava and H. L. Manocha, A treatise on generating functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1984.
  92. H. M. Srivastava and S. Owa (Editors), Univalent functions, fractional calculus, and their applications, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1989.
  93. H. M. Srivastava and S. Owa (Editors), Current topics in analytic function theory, World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, 1992.
  94. H. M. Srivastava, S. Owa and K. Nishimoto, Some fractional differintegral equations, J. Math. Anal. Appl., 106(1985), 360-366. https://doi.org/10.1016/0022-247X(85)90117-9
  95. H. M. Srivastava and K. M. Saad, Some new models of the time-fractional gas dynamics equation, Adv. Math. Models Appl., 3(1)(2018), 5-17.
  96. H. M. Srivastava and Z. Tomovski, Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel, Appl. Math. Comput., 211(2009), 198-210. https://doi.org/10.1016/j.amc.2009.01.055
  97. Z. Tomovski, R. Hilfer and H. M. Srivastava, Fractional and operational calculus with generalized fractional derivative operators and Mittag-Leffler type functions, Integral Transforms Spec. Funct., 21 (2010), 797-814. https://doi.org/10.1080/10652461003675737
  98. F. G. Tricomi, Funzioni ipergeometriche confluenti, Edizioni Cremonese, Rome, 1954.
  99. S.-T. Tu, D.-K. Chyan and H. M. Srivastava, Some families of ordinary and partial fractional differintegral equations, Integral Transforms Spec. Funct., 11(2001), 291-302. https://doi.org/10.1080/10652460108819319
  100. S.-T. Tu, Y.-T. Huang, I.-C. Chen and H. M. Srivastava, A certain family of fractional differintegral equations, Taiwanese J. Math., 4(2000), 417-426. https://doi.org/10.11650/twjm/1500407258
  101. S.-T. Tu, S.-D. Lin, Y.-T. Huang and H. M. Srivastava, Solutions of a certain class of fractional differintegral equations, Appl. Math. Lett., 14(2)(2001), 223-229. https://doi.org/10.1016/S0893-9659(00)00140-3
  102. S.-T. Tu, S.-D. Lin and H. M. Srivastava, Solutions of a class of ordinary and partial differential equations via fractional calculus, J. Fract. Calc., 18(2000), 103-110.
  103. J.-R. Wang, A. G. Ibrahim and M. Feckan, Nonlocal Cauchy problems for semilinear differential inclusions with fractional order in Banach spaces, Commun. Nonlinear Sci. Numer. Simul., 27(2015), 281-293. https://doi.org/10.1016/j.cnsns.2015.03.009
  104. J.-R. Wang and Y. Zhang, On the concept and existence of solutions for fractional impulsive systems with Hadamard derivatives, Appl. Math. Lett., 39(2015), 85-90. https://doi.org/10.1016/j.aml.2014.08.015
  105. J.-R. Wang, Y. Zhou and Z. Lin, On a new class of impulsive fractional differential equations, Appl. Math. Comput., 242(2014), 649-657. https://doi.org/10.1016/j.amc.2014.06.002
  106. P.-Y. Wang, S.-D. Lin and H. M. Srivastava, Explicit solutions of Jacobi and Gauss differential equations by means of operators of fractional calculus, Appl. Math. Comput., 199(2008), 760-769. https://doi.org/10.1016/j.amc.2007.10.037
  107. P.-Y. Wang, S.-D. Lin and S.-T. Tu, A survey of fractional-calculus approaches the solutions of the Bessel differential equation of general order, Appl. Math. Comput., 187(2007), 544-555. https://doi.org/10.1016/j.amc.2006.09.005
  108. G. N. Watson, A treatise on the theory of Bessel functions, Second edition, Cambridge University Press, Cambridge, London and New York, 1944.
  109. G. K. Watugala, Sumudu transform: A new integral transform to solve differential equations and control engineering problems, Math. Engrg. Industr., 6(1998), 319-329.
  110. E. T. Whittaker and G. N. Watson, A course of modern analysis: an introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions, Fourth edition, Cambridge University Press, Cambridge, London and New York, 1927.
  111. A. Wiman, Uber den fundamentalsatz in der theorie der funcktionen $E_{\alpha}(x)$, Acta Math., 29(1905), 191-201. https://doi.org/10.1007/BF02403202
  112. X.-J. Yang, D. Baleanu and H. M. Srivastava, Local fractional integral transforms and their applications, Academic Press (Elsevier Science Publishers), Amsterdam, Heidelberg, London and New York, 2016.