Introduction

1  Introduction

The concept of the differentiation operator $\mathscr{D}=\dif/\dif x$ is a well-known fundamental tool of modern calculus. For a suitable function $f$ the $n$-th derivative is well defined as $\mathscr{D}^n f(x)=\dif f(x)/\dif x^n$, where $n$ is a positive integer. However, what would happen if we extended this concept to a situation, when $n$ is arbitrary, e.g. fractional? This was the very same question L’Hôpital addressed to Leibniz in a letter in 1695. Since then the concept of fractional calculus has drawn the attention of many famous mathematicians, including Euler, Laplace, Fourier, Liouville, Riemann, Abel and Laurent. But it was not until 1884 that the theory of generalized operators reached a satisfactory level of development for the point of departure for the modern mathematician [1].

However, fractional-order calculus was not particularly popular until recent years when benefits stemming from using its concepts became evident in various scientific fields, including system modeling and automatic control. The rise of interest to the topic of fractional differentiation is also related to accessibility of more efficient and powerful computational tools. The introduction of computer algebra systems, such as MATLAB and Mathematica, led to new possibilities for evaluating the theoretical aspects of fractional calculus in specific applications.

Recent findings support the notion that fractional-order calculus should be employed where more accurate modeling and robust control are concerned. Specifically, fractional-order calculus found its way into complex mathematical and physical problems [2, 3]. In general, fractional-order calculus may be useful when modeling any system which has memory and/or hereditary properties [4]. In the field of automatic control fractional calculus is used to obtain more accurate models, develop new control strategies and enhance the characteristics of control systems.

2  Definitions of the fractional operator

Fractional calculus is a generalization of integration and differentiation to non-integer order operator $_a\mathscr{D}_t^\alpha$, where $a$ and $t$ denote the limits of the operation and $\alpha$ denotes the fractional order such that

where generally it is assumed that $\alpha\in\mathbb{R}$, but it may also be a complex number [5].

One of the reasons why fractional calculus is not yet found in elementary texts is a certain degree of controversy found in the theory [1]. This is why there exist several definitions for the fractional-order differintegral operator. Next, several popular definitions are given.

Definition 1. (Riemann-Liouville definition)

where $m-1\lt\alpha\lt m$, $m\in\mathbb{N}$, $\alpha\in\mathbb{R}^{+}$ and $\Gamma\left(\cdot\right)$ is Euler’s gamma function.

Definition 2. (Caputo definition)

where $m-1\lt\alpha\lt m$, $m\in\mathbb{N}$.

Definition 3. (Grünwald-Letnikov definition)

where $\left[\cdot\right]$ means the integer part.

3  Fractional operator properties

Fractional-order differentiation has the following properties [6, 7, 4]:

1. If $f(t)$ is an analytic function, then the fractional-order differentiation $_{0}\mathscr{D}_{t}^{\alpha}f(t)$ is also analytic with respect to $t$.
2. If $\alpha=n$ and $n\in\mathbb{Z}^{+}$, then the operator $_{0}\mathscr{D}_{t}^{\alpha}$ can be understood as the usual operator $\dif\,^{n}/\dif\, t^{n}$.
3. Operator of order $\alpha=0$ is the identity operator: $_{0}\mathscr{D}_{t}^{0}f(t)=f(t)$.
4. Fractional-order differentiation is linear; if $a,\, b$ are constants, then
   \begin{equation} _{0}\mathscr{D}_{t}^{\alpha}\left[af(t)+bg(t)\right]=a\,_{0}\mathscr{D}_{t}^{\alpha}f(t)+b\,{}_{0}\mathscr{D}_{t}^{\alpha}g(t). \tag{5}\end{equation}
5. For the fractional-order operators with $\Re(\alpha)>0,\,\Re(\beta)>0$, and under reasonable constraints on the function $f(t)$ it holds the additive law of exponents:
   \begin{equation} _{0}\mathscr{D}_{t}^{\alpha}\left[_{0}\mathscr{D}_{t}^{\beta}f(t)\right]={}_{0}\mathscr{D}_{t}^{\beta}\left[_{0}\mathscr{D}_{t}^{\alpha}f(t)\right]={}_{0}\mathscr{D}_{t}^{\alpha+\beta}f(t) \tag{6}\end{equation}
6. The fractional-order derivative commutes with integer-order derivative
   \begin{equation} \frac{\dif\,^{n}}{\dif\, t^{n}}\,\left(_{a}\mathscr{D}_{t}^{\alpha}f(t)\right)={}_{a}\mathscr{D}_{t}^{\alpha}\left(\frac{\dif\,^{n}f(t)}{\dif\, t^{n}}\right)={}_{a}\mathscr{D}_{t}^{\alpha+n}f(t), \tag{7}\end{equation}
   under the condition $t=a$ we have $f^{(k)}(a)=0,\,(k=0,\,1,\,2,\,…,\, n-1)$.

4  Computation examples

Example 1. Let us compute¹ the Riemann-Liouville fractional derivative of order $\alpha=\frac{1}{2}$ for an elementary function $f(t)=t^2$ taking $a=0$:

Let us show that using Caputo’s definition yields the same result in this case:

Example 2. Compute the fractional derivative of order $\alpha=\frac{1}{3}$ for function $f_{1}(t)=\mathrm{e}^{5t}$ and the fractional derivative of order $\alpha=\frac{1}{2}$ for function $f_{2}(t)=\sin(3t)$. In this case, in order to obtain the derivative, we use the Riemann-Liouville definition taking $a=-\infty$:

We can compute the fractional derivative for the function $f_2(t)$ in the same way:

5  Thoughts on the meaning of fractional operator

The reader might be wondering, why the title of this section begins with “thoughts”. Could there be no proper explanation for the physical and geometrical meaning of fractional differentiation? Unfortunately, there is no clear, intuitive interpretation so far. Now, there exist several papers that shed some light on this matter, e.g. [8, 9].

Additionally, one may look for interpretation of fractional operators in other directions as well:

• Fractal theory;
• Сorrespondence to integer-order derivatives, which may be considered as a particular case of fractional derivatives.

It is also important to note the apparent reason for the difficulties with understanding the fractional derivative. The integer-order derivatives were developed with a clear application in mind. That is, they were the primary object of development, while the applications of using fractional derivatives, while considered by e.g. Leibniz and L’Hôpital, were not clear at the time.

Since the field of applications of fractional calculus is rapidly growing, it is perhaps safe to say that a clear geometric and physical interpretation of the fractional-order derivative will eventually arise.

References