Fractional Derivatives, Fractional Integrals, and Fractional Differential Equations in Matlab

The term fractional calculus is more than 300 years old. It is a generalization of the ordinary differentiation and integration to non-integer (arbitrary) order. The subject is as old as the calculus of differentiation and goes back to times when Leibniz, Gauss, and Newton invented this kind of calculation. In a letter to L’Hospital in 1695 Leibniz raised the following question (Miller and Ross, 1993): “Can the meaning of derivatives with integer order be generalized to derivatives with non-integer orders?" The story goes that L’Hospital was somewhat curious about that question and replied by another question to Leibniz. “What if the order will be 1/2?" Leibniz in a letter dated September 30, 1695 replied: “It will lead to a paradox, from which one day useful consequences will be drawn." The question raised by Leibniz for a fractional derivative was an ongoing topic in the last 300 years. Several mathematicians contributed to this subject over the years. People like Liouville, Riemann, and Weyl made major contributions to the theory of fractional calculus. The story of the fractional calculus continued with contributions from Fourier, Abel, Leibniz, Grünwald, and Letnikov. Nowadays, the fractional calculus attracts many scientists and engineers. There are several applications of this mathematical phenomenon in mechanics, physics, chemistry, control theory and so on (Caponetto et al., 2010; Magin, 2006; Monje et al., 2010; Oldham and Spanier, 1974; Oustaloup, 1995; Podlubny, 1999). It is natural that many authors tried to solve the fractional derivatives, fractional integrals and fractional differential equations in Matlab. A few very good and interesting Matlab functions were already submitted to the MathWorks, Inc. Matlab Central File Exchange, where they are freely downloadable for sharing among the users. In this chapter we will use some of them. It is worth mentioning some addition to Matlab toolboxes, which are appropriate for the solution of fractional calculus problems. One of them is a toolbox created by CRONE team (CRONE, 2010) and another one is the Fractional State–Space Toolkit developed by Dominik Sierociuk (Sierociuk, 2005). Last but not least we should also mention a Matlab toolbox created by Dingyü Xue (Xue, 2010), which is based on Matlab object for fractional-order transfer function and some manipulation with this class of the transfer function. Despite that the mentioned toolboxes are mainly for control systems, they can be “abused" for solutions of general problems related to fractional calculus as well. 10