José A. Tenreiro MACHADO

The Evolution of Fractional Calculus

Fractional Calculus started in 1695 with Leibniz discussing the meaning of $D^ny$ for $n = 1/2$. Many mathematicians developed the theoretical concepts, but the area remained somewhat unknown from applied sciences. During the eighties FC emerged associated with phenomena such as fractal and chaos and, consequently, in nonlinear dynamical. In the last years, Fractional Calculus became a popular tool for the modeling of complex dynamical systems with nonlocality and long memory effects.

Keywords:

fractional calculus, non-locality long range memory,

PDF

___

B˘aleanu, D. and A. M. Lopes, editors, 2019a Handbook of Fractional Calculus with Applications: Applications in Engineering, Life and Social Sciences, Part A, volume 7 of De Gruyter Reference. De Gruyter, Berlin.
B˘aleanu, D. and A. M. Lopes, editors, 2019b Handbook of Fractional Calculus with Applications: Applications in Engineering, Life and Social Sciences, Part B, volume 8 of De Gruyter Reference. De Gruyter, Berlin.
Karniadakis, G. E., editor, 2019 Handbook of Fractional Calculus with Applications: Numerical Methods, volume 3 of De Gruyter Reference. De Gruyter, Berlin.
Kochubei, A. and Y. Luchko, editors, 2019a Handbook of Fractional Calculus with Applications: Basic Theory, volume 1 of De Gruyter Reference. De Gruyter, Berlin.
Kochubei, A. and Y. Luchko, editors, 2019b Handbook of Fractional Calculus with Applications: Fractional Differential Equations, volume 2 of De Gruyter Reference. De Gruyter, Berlin.
Machado, J. and V. Kiryakova, 2017 The chronicles of fractional calculus. Fractional Calculus and Applied Analysis 20: 307–336.
Machado, J. A. T., 2003 A probabilistic interpretation of the fractional-order differentiation. Fractional Calculus & Applied Analysis 6: 73–80.
Machado, J. A. T., 2021 The bouncing ball and the Grünwald- Letnikov definition of fractional derivative. Fractional Calculus and Applied Analysis 24: 1003–1014.
Machado, J. A. T. and V. Kiryakova, 2019 Recent history of the fractional calculus: data and statistics. In Handbook of Fractional Calculus with Applications: Basic Theory, edited by A. Kochubei and Y. Luchko, pp. 1–22, De Gruyter.
Machado, J. A. T., V. Kiryakova, and F. Mainardi, 2010 A poster about the recent history of fractional calculus. Fractional Calculus and Applied Analysis 13: 329–334.
Machado, J. T., 1997 Analysis and design of fractional-order digital control systems. Systems Analysis, Modelling, Simulation 27: 107–122.
Machado, J. T., 2001 Discrete-time fractional-order controllers. Fractional Calculus & Applied Analysis 4: 47–66.
Machado, J. T. and A. M. Lopes, 2020a Multidimensional scaling and visualization of patterns in prime numbers. Communications in Nonlinear Science and Numerical Simulation 83: 105128.
Machado, J. T. and A. M. Lopes, 2020b Multidimensional scaling locus of memristor and fractional order elements. Journal of Advanced Research 25: 147–157.
Petráš, I., editor, 2019 Handbook of Fractional Calculus with Applications: Applications in Control, volume 6 of De Gruyter Reference. De Gruyter, Berlin.
Tarasov, V. E., editor, 2019a Handbook of Fractional Calculus with Applications: Applications in Physics, Part A, volume 4 of De Gruyter Reference. De Gruyter, Berlin.
Tarasov, V. E., editor, 2019b Handbook of Fractional Calculus with Applications: Applications in Physics, Part B, volume 5 of De Gruyter Reference. De Gruyter, Berlin.
Valério, D., J. Machado, and V. Kiryakova, 2014 Some pioneers of the applications of fractional calculus. Fractional Calculus and Applied Analysis 17: 552–578.
van Eck, N. J. and L. Waltman, 2009 Software survey: VOSviewer, a computer program for bibliometric mapping. Scientometrics 84: 523–538.
van Eck, N. J. and L.Waltman, 2017 Citation-based clustering of publications using CitNetExplorer and VOSviewer. Scientometrics 111: 1053–1070.