In short, fractional derivatives has memory. There are mainly two ways to approximate fractional derivatives: in time domain, and in frequency domain. Please refer to the following links:

Time domain numerical method by Diethelm el al.:

http://link.springer.com/article/10.1023/A%3A1016592219341

Comparison of frequency domain methods:

http://www.sciencedirect.com/science/article/pii/S0362546X07004348

Hello Jessica,

Please refer these links for Fractional derivatives:

http://www.gauge-institute.org/calculus/FractionalCalculus.pdf.

and for its applicability check this:

http://www.hindawi.com/journals/aaa/2013/279681/

Hope it helps

Hi Jessica,

Maybe try to find some meaning of the Caputo derivative at the initial point: aDa[f(a)] from the Mean-Value Theorem ( like Proposition 2.2 in Fractional Taylor Series for Caputo Fractional Derivatives. Construction of Numerical Schemes by David Usero).

If we let x->a in the mean value theorem we obtain a similar formula to the definition of classical derivatives.

Another place to try is Geometric and Physical Interpretation of Fractional Integration and Fractional Differentiation by Igor Podlubny.

Attached picture you may find an example how fraction derivatives are more preferable than ordinary derivatives. The title of the paper is shown on top of the picture.

In short, fractional derivatives has memory. There are mainly two ways to approximate fractional derivatives: in time domain, and in frequency domain. Please refer to the following links:

Time domain numerical method by Diethelm el al.:

http://link.springer.com/article/10.1023/A%3A1016592219341

Comparison of frequency domain methods:

http://www.sciencedirect.com/science/article/pii/S0362546X07004348

Let's talk about fractional calculus applied in control theory. I will not be talking about what fractional calculus is.

Control systems are usually represented using integer-order differentiation equations or transfer functions. You could have a simple first-order model for a DC motor that is contorolled by a simple PID controller. The combination of your controller and DC motor could be situation in a simple unity-feedback system and we have learned in various textbooks that we could derive its frequency and time responses as we may.

Fractional calculus gives a twist in this area. Since fractional calculus means "non-integer-order", the you could have 4 possibilities:

1.) Integer-order controller + integer-order plant (the typical one)

2.) Integer-order controller + fractoinal-order plant

3.) Fractional-order controller + integer-order plant

4.) Fractional-order controller + fractional-order plant

So what is the sense of having, for example, a fractional-order controller? If you have a fractional-order controller, then the entire control system will have dynamics following fractional-order differential equations. It has been found from various researchers that there are many advantages of incorporating fractional-order dynamics in a system:

1.) You can produce iso-damping property in your system. Iso-damping means the overshoot of the system doesn't change even if there are gain variations in your plant. Thus, your system becomes robusts in gain changes which could be achieved only with fractional dynamics.

2.) You can satisfy more than 3 robustness criteria using fractional-order PID controller (FOPID). An FOPID has 5 degrees of freedom which you can tune as necessary. Therefore, you can optimize your feedback control system and satisfy 5 parameters of interest (usually 5 robustness criteria) by just tuning your 5 parameters. Classical PID controller on has 3, i.e. Kp, Ki, and Kd, while FOPID has five, i.e. Kp, Ki, Kd, diff_order, integ_order.

Above are just examples. So, does it make any sense of having fractional calculus in control theory? The answer is a big yes :D

Hope this helps.

Emm