*2.2. Preliminaries on the Caputo Fractional Derivatives*

In this section, the definition of the Caputo fractional derivative and its main properties are introduced.

**Definition 1.** *Let t* >, *a* > 0, *a*, α, *t . The Caputo fractional derivative of the order* α *of function f Cn is expressed as:*

$$\, \_a^C D\_t^a f(t) = \frac{1}{\Gamma(n-\alpha)} \int\_a^t \frac{f^n(\xi)}{(t-\xi)^{n+1-n}} d\xi, \; n-1 < \alpha < n \in \mathbb{N} \tag{10}$$

**Property 1.** *Let <sup>f</sup>*(*t*), *<sup>g</sup>*(*t*) : [*a*, *<sup>b</sup>*] <sup>→</sup>*be such that <sup>C</sup> <sup>a</sup> D*<sup>α</sup> *<sup>t</sup> <sup>f</sup>*(*t*) and *<sup>C</sup> <sup>a</sup> D*<sup>α</sup> *<sup>t</sup> g*(*t*) *exist almost everywhere, and let <sup>c</sup>*1, *<sup>c</sup>*<sup>2</sup> *. Then <sup>C</sup> <sup>a</sup> D*<sup>α</sup> *t c*<sup>1</sup> *f*(*t*) + *c*<sup>2</sup> *g*(*t*) . *exists almost everywhere and*

$$\mathbf{c}\_a^\mathbb{C} D\_t^a \{ \mathbf{c}\_1 f(t) + \mathbf{c}\_2 \mathbf{g}(t) \} = \mathbf{c}\_a^\mathbb{C} D\_t^a f(t) + \mathbf{c}\_2^\mathbb{C} D\_t^a \mathbf{g}(t) \tag{11}$$

**Property 2.** *If f(t)* = *c is a constant function then the fractional derivative of the function is equal to 0, and mathematically it can be expressed as:*

$$\, \, \_a^C D\_t^a c = 0 \tag{12}$$

*We considered the general fractional di*ff*erential equation involving the Caputo derivative below*

$$\,\_{a}^{\mathbb{C}}D\_{t}^{a}\mathbf{x}(t) = f(t, \mathbf{x}(t)), \; a\epsilon(0, 1) \tag{13}$$

*with initial conditions x0* = *x(t0).*

**Definition 2.** *The constant x\* is an equilibrium point of the Caputo fractional dynamic system (13) if, and only if, f (t, x\* )* = *0.*

Here, we introduce the new fractional Atangana–Baleanu derivatives along the non-local and non-singular kernel [37,38].

**Definition 3.** *Let f H*1(*a*, *b*), *b* > *a*, α[0, 1]*, then the new fractional derivatives of the Caputo behavior can be expressed as:*

$$D\_t^{\alpha}(f(t)) = \frac{B(\alpha)}{1-\alpha} \int\_{\mathfrak{a}}^t f(\mathfrak{x}) \exp\left(-\alpha \frac{t-\mathfrak{x}}{1-\alpha}\right) d\mathfrak{x}$$

*where B(*α*) denotes a normalization function obeying B(0)* = *B(1)* = *1.*

In the case when the function does not belong to *H*<sup>1</sup> (*a*,*b*), the derivative is given by

$$D\_t^\alpha(f(t)) = \frac{\alpha B(\alpha)}{1 - \alpha} \int\_a^t (f(t) - f(\mathbf{x})) \exp\left(-\alpha \frac{t - \mathbf{x}}{1 - \alpha}\right) d\mathbf{x}d\mathbf{x}$$

Furthermore, if σ = <sup>1</sup>−<sup>α</sup> <sup>α</sup> [0, <sup>∞</sup>), and <sup>α</sup> = <sup>1</sup> <sup>1</sup>−<sup>σ</sup> [0, 1], then the above Equation becomes

$$D\_t^{\sigma}(f(t)) = \frac{N(\sigma)}{\sigma} \int\_a^t f(\mathbf{x}) \exp(-\frac{t-\mathbf{x}}{\sigma}) d\mathbf{x}, \quad N(0) = N(\infty) = 1$$
