162

Bioinformatics of the Brain

6.4

Fractional Operators

The commonly used expression “fractional calculus,” which relates to the cal-

culus of non-integer order, is as ancient as the calculus of integer order, that

was established individually by Newton and Leibniz [11]. Only in 1974 was

fractional calculus designated as a distinct field of mathematics, in contrast to

calculus of integer order. Fractional calculus began to get interest from schol-

ars in the 1980s, and explicit applications started to show up in a number

of domains. One reason fractional calculus has gained popularity is that its

applications are more practical. In mathematics, fractional calculus may be

regarded as a frontier field as its applications have been studied just as thor-

oughly as those of calculus of integer order. First, the notion of the fractional

integral in the Liouville sense—a specific instance of the Riemann-Liouville

sense—will be discussed. This may be thought of as an extension of the in-

tegral of integer order. We will then talk about the concepts of derivatives

proposed by Riemann-Liouville and Caputo [11].

6.4.1

Caputo Derivative

The Cauchy-Riemann integral may be produced as an extension of the frac-

tional integral of Riemann-Liouville, which is an integral that generalizes the

idea of an integral in the classical sense. First, the definition of the Riemann-

Liouville fractional integral was established. This was followed by a description

of the fractional derivative of Riemann-Liouville and the fractional integral

in the sense of Riemann-Liouville. Mathematicians, in particular, utilize this

term the most when solving problems where beginning conditions are not

involved [11]. The differential operator of non-integer order in the Riemann-

Liouville sense and the differential operator of non-integer order in the Caputo

sense are comparable. The key distinction is that, in the Riemann-Liouville

sense, the derivatives operate on the integral, meaning that we compute the

derivative after evaluating the integral, but in the Caputo view, the derivative

acts first on the function. When compared to the Riemann-Liouville deriva-

tive, the Caputo derivative is more restrictive. We also see that the Riemann-

Liouvile fractional integral is used to define both derivatives. The significance

of this derivative lies in the fact that it may be applied in the Caputo sense,

for instance, when dealing with fractional differential equations that have

well-established beginning conditions, like integral order calculus. There are

multiple studies relevant to this [12–21].

6.4.2

Fractional Equations

Applications where a particle cloud expands faster than expected by a classical

equation could profit by the use of fractional diffusion equations [22].