2. Fractional derivatives

To understand the need and the concept of a fractional derivative and its importance in the context of modeling physical processes, let us start with a simple example (Figure 5).

Imagine a portion of material that is principally formed of two different regions. In these regions, two similar physical processes φ<sup>1</sup> and φ<sup>2</sup> occur (for the time being it does not matter what is the process under study), but, at different rates, dφ1=dt ¼ 0:1 for Region I and dφ2=dt ¼ 1 for Region II. If we look at the portion of material as a whole, one would naturally choose the rate of 1 as representative of the material's behavior, because this region is bigger

Figure 5. Material formed by two regions where the same physical process occurs at different rates.

(when compared to Region I). However, this clearly neglects entirely the local variation in the deformation associated with the neighboring Region I. With the help of fractional calculus, we may define derivatives/rates of non-integer order, and we may have (for example) a rate given by d<sup>β</sup> <sup>φ</sup>=dt<sup>β</sup> with <sup>β</sup> <sup>¼</sup> <sup>0</sup>:<sup>9</sup> (possibly better representing the material behavior as <sup>a</sup> whole, by providing intermediate rates).

Although we have not defined yet what a fractional derivative is, the fact of having the possibility of non-integer derivatives seems quite attractive, allowing the creation of a continuous path between integer-order derivatives that may lead to a better description of the different rates of a certain physical process occurring in the same material. This means that fractional derivatives can transport more and more precise local information from the microscopic world to the continuum description.
