*2.1. Generalized Fractional Calculus*

**Definition 1** ([14])**.** *The Caputo derivative of a function v*(*τ*) *of order γ is defined as*

$$(\mathcal{D}\_{0+}^{\gamma}v)(\tau) = (\mathcal{D}^{\delta-\gamma}\mathcal{D}^{\delta}v)(\tau) = \frac{1}{\Gamma(\delta-\gamma)}\int\_{0}^{\tau}(\tau-s)^{\delta-\gamma-1}v^{(\delta)}(s)ds,\ \tau>0,\tag{3}$$

*where <sup>δ</sup>* <sup>−</sup> <sup>1</sup> <sup>&</sup>lt; *<sup>γ</sup>* <sup>≤</sup> *<sup>δ</sup> and <sup>δ</sup>* <sup>∈</sup> <sup>Z</sup>+*.*

**Definition 2** ([14])**.** *A generalized fractional derivative of order γ* > 0 *of a function v*(*τ*) *with respect to weight function ω*(*τ*) *and scale function z*(*τ*) *is defined as*

$$(\mathcal{D}\_{[z\omega;\mathbb{L}]}^{\gamma}v)(\tau) = [\omega(\tau)]^{-1} \left[ \left(\frac{D\_t}{z'(\tau)}\right)^{\gamma} (\omega(\tau)v(\tau)) \right].\tag{4}$$

**Definition 3.** *A generalized fractional integral of a function v*(*τ*) *of fractional order γ* > 0 *is defined as*

$$(\mathcal{Z}\_{[z;\omega]}^{\gamma}v)(\tau) = \frac{[\omega(\tau)]^{-1}}{\Gamma(\gamma)} \int\_0^{\tau} \frac{\omega(s)z'(s)v(s)}{[z(\tau)-z(s)]^{1-\gamma}}ds.\tag{5}$$

**Definition 4** ([14])**.** *The left/forward generalized Caputo fractional derivative of order γ* > 0 *of function v*(*τ*) *with respect to weight function ω*(*τ*) *and scale function z*(*τ*) *is defined by*

$$(\*\mathcal{D}\_{0+}^{\gamma}v)(\tau) = \mathcal{T}\_{0+;[z:\omega]}^{\delta-\gamma}\mathcal{D}\_{0+;[z:\omega;z]}^{\delta} = \frac{[\omega(\tau)]^{-1}}{\Gamma(\delta-\gamma)}\int\_{0}^{\tau} \frac{(\omega(\mathbf{s})v(\mathbf{s}))^{(\delta)}}{(z(\tau)-z(\mathbf{s}))^{\gamma+1-\delta}}d\mathbf{s},\tag{6}$$

*where <sup>δ</sup>* <sup>−</sup> <sup>1</sup> <sup>&</sup>lt; *<sup>γ</sup>* <sup>≤</sup> *<sup>δ</sup>*, *<sup>δ</sup>* <sup>∈</sup> <sup>Z</sup>+*.*

For a particular choice of the weight and scale functions (*z*(*τ*) = *τ*, *ω*(*τ*) = 1), Equation (6) reduces to the Caputo derivative. We have considered the weight and scale sufficiently good such that the integral exists in GFD.
