**2. Definitions and preliminaries**

transfer, viscoelasticity and damping, electronics, robotics, electromagnetism, signal processing, telecommunications, control systems, traffic systems, system identification, chaos and fractals, biology, genetic algorithms, filtration, modelling and identification, chemistry,

Modelling of different physical phenomena gave rise to a special differential equation known as Riccati differential equation that was named after an Italian mathematician Count Jacopo Francesco Riccati. Due to many applications of fractional Riccati differential equations such as in stochastic controls and pattern formation, many researchers studied it to get the exact or approximate solutions. Such as fractional variational iteration method was applied in [6] to give an approximate analytical solution of non-linear fractional Riccati differential equation. His modified homotopy perturbation method (MHPM) was used on quadratic Riccati differential equation of fractional order [7]. The results of fractional Riccati differential equation were also obtained on the basis of Taylor collocation method [8]. Fractional Riccati differential equations were solved by means of variational iteration method and homotopy perturbation Pade technique [9, 10], and the numerical results were attained by using Chebyshev finite difference method [11]. Adomian decomposition method was presented for fractional Riccati differential equation [12], the problem was described by means of Bernstein collocation method [13], and enhanced homotopy perturbation method (EHPM) was used to study this problem [14]. Recently, artificial neural network and sequential quadratic programming have been utilized to obtain the solution of Riccati differential equation [15]. The problem was also explained by Legendre wavelet operational matrix method [16] and the results of fractional Riccati differential equation by new

In recent years, artificial neural network (ANN) is one of the methods that are attaining massive attention of researchers in the area of mathematics as well as in different physical sciences. The concept of ANN started to develop in 1943 when a neurophysiologist and a young mathematician [18] gave the idea on working of a neuron with the help of an electric circuit. Later, a book [19] was written to clarify the working of neurons then in 1949. Bernard Widrow and Marcian Hoff developed a model MEDALIN that was used to study the first real-world problem of neural network. Researchers continued to study the singlelayered neural network, but in 1975, the concept of multilayer perceptron (MLP) was introduced, which was computationally exhaustive due to multilayer architecture. The excessive training time and high computational complexity of MLP gave rise to functional neural network by which the complexity of multilayers was overcome by introducing variable functions [20]. Functional link neural network has been implemented to several problems such as modified functional link neural network for denoising of image [21], active control of non-linear noise processes through functional link neural network [22], and the problem of channel equalization in a digital communication system was solved by functional link neural network [23]. Due to less computational effort with easy to implement procedure, functional link neural network was also implemented to solve differential

irreversibility, as well as economy and finance [3–5].

98 Numerical Simulation - From Brain Imaging to Turbulent Flows

homotopy perturbation method (NHPM) [17] were obtained.

equations [24, 25].

The Riemann-Liouville, Grünwald-Letnikov and Caputo definitions of fractional derivatives of order *α* >0 are used more frequently among several definitions of fractional derivatives and integrals, but in this chapter, the Caputo definition will be used for working out the fractional derivative in a subsequent procedure. The definitions of commonly used fractional differential operators are discussed in the study of Sontakke and Shaikh [26].

**Definition 1:** The Riemann-Liouville fractional derivative operator can be defined as follows:

$$D^{\alpha} \mathbf{g}(\mathbf{x}) = \frac{1}{\Gamma\left(\xi - \alpha\right)} \frac{d^{\frac{\omega}{\xi}}}{d\alpha^{\frac{\omega}{\xi}}} \int\_{a}^{\mathbf{x}} \frac{\mathbf{g}\left(\beta\right)}{\left(\mathbf{x} - \beta\right)^{\alpha + 1 - \frac{\omega}{\xi}}} d\beta, \quad \xi \text{-} \mathbf{l} < a < \xi$$

where here *α* >0, *x* >*a*, *α*, *a* and*x* ∈*R*.

**Definition 2:** The definition of fractional differential operator was introduced by Caputo in late 1960s that can be defined as follows [27]:

$$D\_\*^\alpha \mathbf{g}(\mathbf{x}) = \frac{\mathbf{l}}{\Gamma(\xi - \alpha)} \prod\_{\alpha}^\times \frac{\mathbf{g}^{(\xi)}(\beta)}{\left(\mathbf{x} - \beta\right)^{\alpha + 1 - \xi}} d\beta, \quad \xi \text{-} \mathbf{l} < \alpha < \xi$$

where here *α* >0, *x* >*a*, *α*, *a* and*x* ∈*R*.

The Caputo fractional derivative satisfies the important attribute of being zero when applied to a constant. In addition, it is attained by computing an ordinary derivative followed by the fractional integral, while the Riemann-Liouville is acquired in the contrary order.
