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Review of Fractional Differentiation

  • Kolade M. Owolabi
  • Abdon Atangana
Chapter
Part of the Springer Series in Computational Mathematics book series (SSCM, volume 54)

Abstract

Fractional calculus can be classified as applicable mathematics. The property and theory of this fractional operator is proper object of study in its own right. Scientists and applied mathematicians, in the last decades, found the fractional calculus useful in various fields such as diffusion, elasticity, electrochemistry, rheology, quantitative biology, probability, scattering theory, transport theory and potential theory.

References

  1. 1.
    N.H. Abel, Auflösung einer mechanischen Aufgabe. Journal für reine und angewandte Mathematik 1, 153–157 (1826)MathSciNetGoogle Scholar
  2. 2.
    N.H. Abel, Solution de quelques problèmes à l’aide d’intégrales définies. Oeuvres Completètes 1, 16–18 (1881)Google Scholar
  3. 3.
    R.P. Agarwal, A propos d’une note de M. Pierre Humbert, C. R. Sćances Acad. Sci., 236, 2031–2032 (1953)Google Scholar
  4. 4.
    O.P. Agrawal, Generalized variational problems and Euler-Lagrange equations. Comput. Math. Appl. 59, 1852–1864 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    A. Almeida, S. Samko, Fractional and hypersingular operators in variable exponent spaces on metric measure spaces. Mediterranean J. Math. 6, 215–232 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    T.M. Atanacković, S. Pilipović, B. Stanković, D. Zorica, Fractional Calculus with Applications in Maechanics (John Willey & Sons Inc, Hoboken, USA, 2014)zbMATHCrossRefGoogle Scholar
  7. 7.
    A. Atangana, A. Secer, A note on fractional order derivatives and table of fractional derivatives of some special functions. Abstract Appl. Anal. 2013, 8 p. (2013).  https://doi.org/10.1155/2013/279681MathSciNetzbMATHGoogle Scholar
  8. 8.
    A. Atangana, On the new fractional derivative and application to nonlinear Fisher’s reaction-diffusion equation. Appl. Math. Comput. 273, 948–956 (2016)MathSciNetzbMATHGoogle Scholar
  9. 9.
    A. Atangana, D. Baleanu, New fractional derivatives with non-local and non-singular kernel: theory and application to heat transfer model. Thermal Sci. 20, 763–769 (2016)CrossRefGoogle Scholar
  10. 10.
    A. Atangana, J.F. Gómez-Aguilar, A new derivative with normal distribution kernel: theory, methods and applications. Physica A 476, 1–14 (2017)MathSciNetCrossRefGoogle Scholar
  11. 11.
    A. Atangana, I. Koca, Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order. Chaos Solitons Fractals 89, 447–454 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    B. Baeumer, M.M. Meerschaert, Tempered stable Lévy motion and transient super-diffusion. J. Comput. Appl. Math. 233, 243–248 (2010)zbMATHCrossRefGoogle Scholar
  13. 13.
    T. Bakkyaraj, R. Sahadevan, Invariant analysis of nonlinear fractional ordinary differential equations with Riemann-Liouville fractional derivative. Nonlinear Dyn. 80, 447–455 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    L. Bourdin, T. Odzijewicz, D.F.M. Torres, Existence of minimizers for generalized Lagrangian functionals and a necessary optimality condition-application to fractional variational problems. Differ. Integral Equ. 27, 743–766 (2014)MathSciNetzbMATHGoogle Scholar
  15. 15.
    D. Brockmann, L. Hufnagel, T. Geisel, The scaling laws of human travel. Nature 439(2006), 462–465 (2006)CrossRefGoogle Scholar
  16. 16.
    A. Bueno-Orovio, D. Kay, K. Burrage, Fourier spectral methods for fractional-in-space reaction-diffusion equations. BIT Numer. Math. 54, 937–954 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    P.I. Butzer, U. Westphal, An introduction to fractional calculus, in Applications of Fractional Calculus in Physics, ed. by R. Hilfer (World Scientific, Singapore, 2000), pp. 1–85zbMATHGoogle Scholar
  18. 18.
    M. Caputo, Linear models of dissipation whose \(Q\) is almost frequency independent II. Geophys. J. Royal Astron. Soc. 13, 529–539 (1967)CrossRefGoogle Scholar
  19. 19.
    M. Caputo, Elasticita e Dissipazione (Zanichelli, Bologna, 1969)Google Scholar
  20. 20.
    M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel. Progress Fract. Diff. Appl. 1, 73–85 (2015)Google Scholar
  21. 21.
    M. Caputo, M. Fabrizio, Applications of new time and spatial fractional derivatives with exponential kernels. Progress Fract. Differ. Appl. 2, 1–11 (2016)CrossRefGoogle Scholar
  22. 22.
    A. Cartea, D. del-Castillo-Negrete, Fractional diffusion models of option prices in markets with jumps. Phys. A, 374, 749–763 (2007)CrossRefGoogle Scholar
  23. 23.
    J. Chen, Z. Zeng, P. Jiang, Global Mittag-Leffler stability and synchronization of memristor-based fractional-order neural networks. Neural Netw. 51, 1–8 (2014)zbMATHCrossRefGoogle Scholar
  24. 24.
    C.F.M. Coimbra, Mechanics with variable-order differential operators. Ann. Phys. 12, 692–703 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    C. Conrick, S. Hanson, Normal distribution. Probab. Modern Financ. Theory 1, 93–109 (2013)Google Scholar
  26. 26.
    J. Cresson, Fractional embedding of differential operators and Lagrangian systems. J. Math. Phys. 48 033504, 34 p. (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    E.C. de Oliveira and J.A.T. Machado, A Review of Definitions for fractional derivatives and integral. Math. Problems Eng. 2014 (2014) Article ID 238459, 6 p.,  https://doi.org/10.1155/2014/238459MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    J. Deng, L. Zhao, Y. Wu, Fast predictor-corrector approach for the tempered fractional ordinary differential equations. Numer. Algorithms 74, 717–754 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    G. Diaz, C.F.M. Coimbra, Nonlinear dynamics and control of a variable order oscillator with application to the van der Pol equation. Nonlinear Dyn. 56, 145–157 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    K. Diethelm, The Analysis of Fractional Differential Equations: An Application-oriented Exposition using Differential Operators of Caputo type (Springer Lecture Notes in Mathematics, Springer, Berlin Heidelberg, 2010)zbMATHCrossRefGoogle Scholar
  31. 31.
    A. Erdélyi, On fractional integration and its application on the theory of Hankel transforms. Quart. J. Math. 11, 293–303 (1940)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    A. Erdélyi, H. Kober, Some remarks on Hankel transforms. Quart. J. Math. 11, 212–221 (1940)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    B.A. Faycal, About Non-differentiable functions. J. Math. Anal. Appl. 263, 721–737 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    R. Friedrich, F. Jenko, A. Baule, S. Eule, Anomalous diffusion of inertial, weakly damped particles. Phys. Rev. Lett. 96, 230601 (2006)CrossRefGoogle Scholar
  35. 35.
    E.F.D. Goufo, A. Atangana, Analytical and numerical schemes for a derivative with filtering property and no singular kernel with applications to diffusion. Eur. Phys. J. Plus 131, 269 (2016).  https://doi.org/10.1140/epjp/i2016-16269-1CrossRefGoogle Scholar
  36. 36.
    I. Goychuk, V.O. Kharchenko, R. Metzler, Molecular motors pulling cargos in the viscoelastic cytosol: how power strokes beat subdiffusion. Phys. Chem. Chem. Phys. 16, 16524–16535 (2014)CrossRefGoogle Scholar
  37. 37.
    I. Gradshteyn, I. Ryzhik, Table of Interals, Series, and Producs (Academic Press, New York, 1980)Google Scholar
  38. 38.
    D.S. Grebenkov, M. Vahabi, E. Bertseva, L. Forró, S. Jeney, Hydrodynamic and subdiffusive motion of tracers in a viscoelastic medium. Phys. Rev. E 88, 071 (2013)Google Scholar
  39. 39.
    D.S. Grebenkov, M. Vahabi, E. Bertseva, L. Forró, S. Jeney, Hydrodynamic and subdiffusive motion of tracers in a viscoelastic medium. Phys. Rev. E 88, 071 (2013)Google Scholar
  40. 40.
    A.K. Grünwald, Ueber ’begrenzte’ Derivationen und deren Anwendung. Z. angew. Math. und Phys. 12, 441–480 (1867)Google Scholar
  41. 41.
    B. Guo, X. Pu, F. Huang, Fractional Partial Differential Equations and their Numerical Solutions (World Scientific, Singapore, 2011)zbMATHGoogle Scholar
  42. 42.
    A. Hanyga, Wave propagation in media with singular memory. Math. Comput. Modell. 34, 1399–1421 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    H.J. Haubold, A.M. Mathai, R.K. Saxena, Mittag-Leffler functions and their applications. J. Appl. Math. 2011, 298628 (2011).  https://doi.org/10.1155/2011/298628MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    R. Hilfer, Applications of Fractional Calculus in Physics (World Scientific Publishing, River Edge, 2000)zbMATHCrossRefGoogle Scholar
  45. 45.
    R. Hilfer, H.J. Seybold, Computation of the generalized Mittag-Leffler function and its inverse in the complex plane. Integral Transf. Spec. Funct. 17, 637–652 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    J. Hristov, Diffusion models with weakly singular kernels in the fading memories: how the integral-balance method can be applied? Thermal Sci. 19, 947–957 (2015)CrossRefGoogle Scholar
  47. 47.
    C. Ingo, T.R. Barrick, A.G. Webb, I. Ronen, Accurate Padé global approximation for the Mittag–Leffler functions, its inverse, and its partial derivatives to efficiently compute convergent power series. Int. J. Appl. Comput. Math. 16 p. (2016).  https://doi.org/10.1007/s40819-016-0158-7zbMATHCrossRefGoogle Scholar
  48. 48.
    C. Ionescu, J.T. Machado, R. De Keyser, J. Decruyenaere, M.M.R.F. Struys, Nonlinear dynamics of the patients response to drug effect during general anesthesia. Commun. Nonlinear Sci. Numerical Simul. 20, 914–926 (2015)CrossRefGoogle Scholar
  49. 49.
    H. Jafari, H. Tajadodi, S.A. Matikolai, Homotopy perturbation pade technique for solving fractional Riccati differential equations. Int. J. Nonlinear Sci. Numer. Simul. 11, 271–276 (2010)MathSciNetzbMATHGoogle Scholar
  50. 50.
    U.N. Katugampola, New approach to a generalized fractional integral. Appl. Math. Comput. 218, 860–865 (2011)MathSciNetzbMATHGoogle Scholar
  51. 51.
    A.A. Kilbas, M. Saigo, Generalized Mittag-Leffler function and generalized fractional calculus operators. Integral Transf. Spec. Funct. 15, 31–49 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations (Elsevier, Netherlands, 2006)zbMATHGoogle Scholar
  53. 53.
    V.S. Kiryakova, Multiple (multiindex) Mittag-Leffler functions and relations to generalized fractional calculus. J. Comput. Appl. Math. 118, 241–259 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    M. Klimek, Lagrangian fractional mechanics–a noncommutative approach. Czechoslovak J. Phys. 55, 1447–1453 (2005)MathSciNetCrossRefGoogle Scholar
  55. 55.
    M. Klimek, On Solutions of Linear Fractional Differential Equations of a Variational type (The Publishing Office of Czestochowa University of Technology, Czestochowa, 2009)Google Scholar
  56. 56.
    K. Kumar, R.K. Pandey, S. Sharma, Comparative study of three numerical schemes for fractional integro-differential equations. J. Comput. Appl. Math. 315, 287–302 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    K. Kumar, R.K. Pandey, S. Sharma, Approximations of fractional integrals and Caputo derivatives with application in solving Abel’s integral equations. J. King Saud Univ. Sc. (2018).  https://doi.org/10.1016/j.jksus.2017.12.017CrossRefGoogle Scholar
  58. 58.
    K.R. Lang, Astrophysical Formulae, in Space, Time, Matter and Cosmology, 2 (Springer, New York, USA, 1999)zbMATHGoogle Scholar
  59. 59.
    H. Laurent, Sur le calcul des dérivées à indicies quelconques. Nouvelles Annales de Mathématiques 3, 240–252 (1884)Google Scholar
  60. 60.
    H. Laurent, Sur le calcul des dérivées à indicies quelconques. Nouvelles Annales de Mathématiques 3, 240–252 (1884)Google Scholar
  61. 61.
    J.L. Lavoie, T.J. Osler, R. Tremblay, Frational derivatives and special functions. SIAM Rev. 18, 240–268 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  62. 62.
    A.V. Letnikov, Theory of differentiation with an arbitrary index (Russian). Moscow Matem. Sbornik 3, 1–66 (1868)Google Scholar
  63. 63.
    A.V. Letnikov, Theory of differentiation with an arbitrary index (Russian). Moscow Matem. Sbornik 3, 1–66 (1868)Google Scholar
  64. 64.
    A.V. Letnikov, An explanation of the concepts of the theory of differentiation of arbitrary index (Russian). Moscow Matem. Sbornik 6, 413–445 (1872)Google Scholar
  65. 65.
    C. Li, W.H. Deng, L.J. Zhao, Well-posedness and numerical algorithm for the tempered fractional ordinary differential equations. arXiv:1501.00376v1 (2015)
  66. 66.
    C. Li, D. Qian, Y. Q. Chen, On Riemann–Liouville and Caputo derivatives. Discrete Dyn. Nature Soc. 2011 Article ID 562494, 15 p. (2011).  https://doi.org/10.1155/2011/562494zbMATHGoogle Scholar
  67. 67.
    J. Liouville, Mémoire sur le calcul des différentielles à indices quelconques. J. l’Ecole Roy. Polytéchn. 13, 71–162 (1832)Google Scholar
  68. 68.
    C.P. Li, W.H. Deng, Remarks on fractional derivatives. Appl. Math. Comput. 187, 777–784 (2007)MathSciNetzbMATHGoogle Scholar
  69. 69.
    C. Li, W. Deng, High order schemes for the tempered fractional diffusion equations. Adv. Comput. Math. 42, 543–572 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  70. 70.
    J. Liouville, Mémoire sur le calcul des différentielles à indices quelconques. J. l’Ecole Roy. Polytéchn. 13, 71–162 (1832)Google Scholar
  71. 71.
    J. Liouville, Mémoire sur quelques questions de géométrie et de mécanique, et su run nouveau genre de calcul pour résoudre ces questions. J. l’Ecole Roy. Polytéchn. 13, 1–69 (1832)Google Scholar
  72. 72.
    J. Liouville, Mémoire sur l’intégration des équations différentielles à indices fractionnaires. J. l’Ecole Roy. Polytéchn. 15, 58–84 (1837)Google Scholar
  73. 73.
    Y. Liu, Z. Fang, H. Li, S. He, A mixed finite element method for a time-fractional fourth-order partial differential equation. Appl. Math. Comput. 243, 703–717 (2014)MathSciNetzbMATHGoogle Scholar
  74. 74.
    C.F. Lorenzo, T.T. Hartley, Variable order and distributed order fractional operators. Nonlinear Dyn. 29, 57–98 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  75. 75.
    J. Losada, J.J. Nieto, Properties of the new fractional derivative without singular kernel. Progress Fract. Differ. Appl. 1, 87–92 (2015)Google Scholar
  76. 76.
    S. Ma, Y. Xu, W. Yue, Numerical solutions of a variable-order fractional financial system. J. Appl. Math. 2012, Article ID 417942, 14 pp (2012).  https://doi.org/10.1155/2012/417942MathSciNetzbMATHGoogle Scholar
  77. 77.
    J.T. Machado, V. Kiryakova, F. Mainardi, Recent history of fractional calculus. Commun. Nonlinear Sci. Numer. Simul. 16, 1140–1153 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  78. 78.
    R.L. Magin, Fractional Calculus in Bioengineering (Begell House Publishers, Danbury, CT, 2006)Google Scholar
  79. 79.
    F. Mainardi, R. Gorenflo, On Mittag-Leffler-type functions in fractional evolution processes. J. Comput. Appl. Math. 118, 283–299 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  80. 80.
    A.B. Malinowska, T. Odzijewicz, D.F.M. Torres, Advanced Methods in the Fractional Calculus of Variations (SpringerBriefs in Applied Sciences and Technology, New York, 2015)zbMATHCrossRefGoogle Scholar
  81. 81.
    R. Maranon, J.F. Reckelhoff, Sex and gender differences in control of blood pressure. Clin. Sci. 125, 311–318 (2013)CrossRefGoogle Scholar
  82. 82.
    A. Marchaud, Sur les dérivées et sur les différences des fonctions des variables réelles. Journal de Mathématiques Pures et Appliquées 6, 371–382 (1927)zbMATHGoogle Scholar
  83. 83.
    O. Marom, E. Momoniat, A comparison of numerical solutions of fractional diffusion models in & Finance. Nonlinear Anal.: Real World Appl. 10, 3435–3442 (2009)Google Scholar
  84. 84.
    M.M. Meerschaert, A. Sikorskii, Stochastic Models for Fractional Calculus, vol. 43 (De Gruyter, Berlin, 2012)zbMATHGoogle Scholar
  85. 85.
    M.M. Meerschaert, Y. Zhang, B. Baeumer, Tempered anomalous diffusion in heterogeneous systems. Geophys. Res. Lett. 35, L17403 (2008)CrossRefGoogle Scholar
  86. 86.
    M.M. Meerschaert, F. Sabzikar, M.S. Phanikumar, A. Zeleke, Tempered fractional time series model for turbulence in geophysical flows. J. Stat. Mech. Theory Exp. 14, 1742–5468 (2014)Google Scholar
  87. 87.
    R. Metzler, J.H. Jeon, A.G. Cherstvy, E. Barkai, Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking. Phys. Chem. Chem. Phys. 16, 24128–24164 (2014)CrossRefGoogle Scholar
  88. 88.
    K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations (Wiley, New York, NY, USA, 1993)zbMATHGoogle Scholar
  89. 89.
    G. Mittag-Leffler, Sur la representation analytique d’une branche uniforme d’une fonction monogene. Acta Math. 29, 101–181 (1905)MathSciNetzbMATHCrossRefGoogle Scholar
  90. 90.
    T. Odzijewicz, A. B. Malinowska, D.F.M. Torres, Variable order fractional variational calculus for double integrals, \(51^{st}\) IEEE Conference on Decision and Control, December 10-13, 2012, Maui, Hawaii, art. no. 6426489, 6873–6878.  https://doi.org/10.1109/CDC.2012.6426489
  91. 91.
    T. Odzijewicz, A.B. Malinowska, D.F.M. Torres, Noether’s theorem for fractional variational problems of variable order. Central Eur. J. Phys. 11, 691–701 (2013).  https://doi.org/10.2478/s11534-013-0208-2CrossRefGoogle Scholar
  92. 92.
    T. Odzijewicz, A.B. Malinowska, D.F.M. Torres, Fractional variational calculus of variable order. Adv. Harmon. Anal. Oper. Theory 229, 291–301 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  93. 93.
    K.B. Oldham, J. Spanier, The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order (Academic Press, New York, NY, USA, 1974)zbMATHGoogle Scholar
  94. 94.
    M.D. Ortigueira, Fractional Calculus for Scientists and Engineers (Springer, New York, 2011)zbMATHCrossRefGoogle Scholar
  95. 95.
    M.D. Ortigueira, J.A.T. Machado, Fractional signal processing and applications. Signal Process. 83, 2285–2286 (2003)CrossRefGoogle Scholar
  96. 96.
    K.M. Owolabi, A. Atangana, Numerical solution of nonlinear system in Subdiffusive, diffusive and superdiffusive scenarios. J. Comput. Nonlinear Dyn. 12 031010, 7 p. (2017).  https://doi.org/10.1115/1.4035195
  97. 97.
    K.M. Owolabi, Mathematical analysis and numerical simulation of patterns in fractional and classical reaction-diffusion systems. Chaos Solitons and Fractals 93, 89–98 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  98. 98.
    K.M. Owolabi, A. Atangana, Numerical solution of fractional-in-space nonlinear Schrödinger equation with the Riesz fractional derivative. Eur. Phys. J. Plus 131, 335 (2016).  https://doi.org/10.1140/epjp/i2016-16335-8CrossRefGoogle Scholar
  99. 99.
    I. Petras, Fractional-Order Nonlinear Systems: Modeling Analysis and Simulation (Springer, Berlin, 2011)zbMATHCrossRefGoogle Scholar
  100. 100.
    partial fractional differential equations, I. Podlubny, A. Chechkin, T. Skovranek, Y. Chen and B. M. Vinagre Jara, Matrix approach to discrete fractional calculus II. J. Comput. Phys. 228, 3137–3153 (2009)MathSciNetCrossRefGoogle Scholar
  101. 101.
    I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, 1999)zbMATHGoogle Scholar
  102. 102.
    H. Qi, X. Guo, Transient fractional heat conduction with generalized Cattaneo model. Int. J. Heat Mass Transf. 76, 535–539 (2014)CrossRefGoogle Scholar
  103. 103.
    L.E.S. Ramirez, C.F.M. Coimbra, On the selection and meaning of variable order operators for dynamic modeling. Int. J. Differ. Equ. (846107), 16 (2010)Google Scholar
  104. 104.
    L.E.S. Ramirez, C.F.M. Coimbra, On the variable order dynamics of the nonlinear wake caused by a sedimenting particle. Physica D 240, 1111–1118 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  105. 105.
    M. Riesz, L’intégrales de Riemann-Liouville et le probléme de Cauchy. Acta Math. 81, 1–223 (1949)MathSciNetzbMATHCrossRefGoogle Scholar
  106. 106.
    M. Riesz, L’intégrales de Riemann-Liouville et solution invariante du probléme de Cauchy pour l’equation des ondes. C. R. Congrés Intern. Math. 2, 44–45 (1936)Google Scholar
  107. 107.
    M. Riesz, L’intégrales de Riemann-Liouville et le probléme de Cauchy. Acta Math. 81, 1–223 (1949)MathSciNetzbMATHCrossRefGoogle Scholar
  108. 108.
    F. Sabzikar, M.M. Meerschaert, J. Chen, Tempered fractional calculus. J. Comput. Phys. 292, 14–28 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  109. 109.
    S.G. Samko, B. Ross, Integration and differentiation to a variable fractional order. Integral Transf. Spec. Funct. 1, 277–300 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  110. 110.
    S. Samko, A. Kilbas, O. Marichev, Fractional Integrals and derivatives: Theory and Applications (Gordon and Breach, Amsterdam, 1993)zbMATHGoogle Scholar
  111. 111.
    W.E. Schiesser, Numerical Method of Lines Integration of Partial Differential Equations (Academic Press, San Diego, 1991)zbMATHGoogle Scholar
  112. 112.
    N.Y. Sonin, On differentiation with arbitrary index. Moscow Matem. Sbornik 6, 1–38 (1869)Google Scholar
  113. 113.
    H.G. Sun, X. Hao, Y. Zhang, D. Baleanu, Fractional derivative defined by non-singular kernels to capture anomalous relaxation and diffusion. arXiv:1606.04844v2 [cond-mat.stat-mech] (2016)
  114. 114.
    H.G. Sun, W. Chen, C.P. Li, Y.Q. Chen, Fractional differential models for anomalous diffusion. Physica A 389, 2719–2724 (2010)CrossRefGoogle Scholar
  115. 115.
    N. Tatar, The decay rate for a fractional differential equation. J. Math. Anal. Appl. 295, 303–314 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  116. 116.
    D. Tavares, R. Almeida, D.F.M. Torres, Caputo derivatives of fractional variable order: numerical approximations. Commun. Nonlinear Sci. Numer. Simul. 35, 69–87 (2016)MathSciNetCrossRefGoogle Scholar
  117. 117.
    L. Turgeman, S. Carmi, E. Barkai, Fractional Feynman-Kac equation for non-Brownian functionals. Phys. Rev. Lett. 103, 190201 (2009)MathSciNetCrossRefGoogle Scholar
  118. 118.
    V.V. Uchaikin, Fractional Derivatives for Physicists and Engineers (Higher Education Press and Springer Verlag, Beijing/Berlin, Background and Theory, 2013)zbMATHCrossRefGoogle Scholar
  119. 119.
    H. Vandebroek, C. Vanderzande, Transient behaviour of a polymer dragged through a viscoelastic medium. J. Chem. Phys. 141, 114910 (2014)CrossRefGoogle Scholar
  120. 120.
    Y. Watanabe, Notes on the generalized derivative of Riemann–Liouville and its application to Leibnitz’s formula. I and II, Tohoku Math. J. 34, 8–41 (1931)Google Scholar
  121. 121.
    H. Weyl, Bemerkungen zum Begriff des differential quotienten gebrochener Ordnung. Vierteljshr. Naturforsch. Gesellsch. Zürich 62, 296–302 (1917)zbMATHGoogle Scholar
  122. 122.
    H. Weyl, Bemerkungen zum Begriff des Differential quotienten gebrochener Ordnung. Vierteljshr. Naturforsch. Gesellsch. Zürich 62, 296–302 (1917)zbMATHGoogle Scholar
  123. 123.
    S. Yadav, R.K. Pandey, A.K. Shukla, Numerical approximations of Atangana-Baleanu Caputo derivative and its application. Chaos, Solitons and Fractals 118, 58–64 (2019)MathSciNetCrossRefGoogle Scholar
  124. 124.
    G.M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics (Oxford University Press, New York, 2005)zbMATHGoogle Scholar
  125. 125.
    S. Zhang, Y. Yu, H. Wang, Mittag–Leffler stability of fractional-order Hopfield neural networks. Nonlinear Anal.: Hybrid Syst. 16, 104–121 (2014)MathSciNetzbMATHGoogle Scholar
  126. 126.
    Y.Y. Zheng, C.P. Li, Z. G. Zhao, A fully discrete discontinuous Galerkin method for nonlinear fractional fokker-planck equation. Math. Problems Eng. 2010, Article ID 279038, 26 p. (2010)Google Scholar
  127. 127.
    B. Zheng, \((G^{\prime }/G)\)-expansion method for solving fractional partial differential equations in the theory of mathematical physics. Commun. Theor. Phys. 58, 623–630 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  128. 128.
    P. Zhuang, F. Liu, V. Anh, I. Turner, New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation. SIAM J. Numer. Anal. 46, 1079–1095 (2008)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  • Kolade M. Owolabi
    • 1
  • Abdon Atangana
    • 2
  1. 1.Institute for Groundwater StudiesUniversity of the Free StateBloemfonteinSouth Africa
  2. 2.Institute for Groundwater StudiesUniversity of the Free StateBloemfonteinSouth Africa

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