I have a problem of understanding the Definitions of the Fractional-Order Calculus which make the foundation of the fractional order controllers.
How I can study and understand this theory in a simpler manner?
Are there any references simplify the understanding of this theory with numerical examples and figures?
Note: I'm an engineer, and I studied Calculus, differential equations, control theory, and some other math courses.
Fractional calculus is my field of research. There are some nice books by
1) An Introduction to the Fractional Calculus and Fractional Differential Equations by Miler and Ross
2) Fractional Order Systems: Modeling and Control Applications by Caponetto and Petras
3) Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation by Petras.
4) Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models by Mainardi
5) Functional Fractional Calculus by Shantanu Das
6) The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order by Oldham and Spanier.
For basic fractional calculus 1, 5 and 6.
For application based: 2, 3, 4.
I wish I could give you more info on it. But I do not have any medium to impart the knowledge to you. By the way, have a look at this simulation of mine. You will enjoy. See link below:
https://www.youtube.com/watch?v=4ZihQc5BamI
For a very quick but not complete tour.. have a look at this link below. It had helped me too.
http://www.xuru.org/fc/Intro.asp
And you will need this MatLab toolbox for designing of fractional-order systems. See link below
http://fomcon.net/fomcon-toolbox/download/
You can use the book: theory and applications Of fractional diferencial Equation
Thank you very much Vikasha.
your recommendations are very helpful to get started.
No problem. Sorry, the link I had given above is broken. You can find the correct one below:
https://www.youtube.com/channel/UCvLjlCpW1g1bm8vzo-WAyuw/videos
Dear Islam Zead,
I would like to refer the following books for your study:
- Oldham, Keith B.; Spanier, Jerome (1974). The Fractional Calculus; Theory and Applications of Differentiation and Integration to Arbitrary Order. Mathematics in Science and Engineering. V. Academic Press. ISBN 0-12-525550-0.
- Miller, Kenneth S.; Ross, Bertram, eds. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley & Sons. ISBN 0-471-58884-9.
- Samko, S.; Kilbas, A.A.; Marichev, O. (1993). Fractional Integrals and Derivatives: Theory and Applications. Taylor & Francis Books. ISBN 2-88124-864-0.
- Carpinteri, A.; Mainardi, F., eds. (1998). Fractals and Fractional Calculus in Continuum Mechanics. Springer-Verlag Telos. ISBN 3-211-82913-X.
- Podlubny, Igor (1998). Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications. Mathematics in Science and Engineering. 198. Academic Press. ISBN 0-12-558840-2.
- West, Bruce J.; Bologna, Mauro; Grigolini, Paolo (2003). Physics of Fractal Operators. Springer Verlag. ISBN 0-387-95554-2.
- Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J. (2006). Theory and Applications of Fractional Differential Equations. Amsterdam, Netherlands: Elsevier. ISBN 0-444-51832-0.
- Tarasov, V.E. (2010). Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer.
- Daftardar-gejji, Varsha (2013). Fractional Calculus: Theory and Applications. Narosa Publishing House.
- Herrmann, R. (2014). Fractional Calculus - An Introduction for Physicists. Singapore: World Scientific.
- Li, C P; Zeng, F H (2015). Numerical Methods for Fractional Calcuus. USA: CRC Press.
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