Algorithm 1. To deduce the approximate solutions of (3) in detail, one can perform the following manner by one of the known software packages like MATHEMATICA 12.

Step I: Write the system (3) in the form $$\left\{\begin{array}{c}{\mathcal{C}}^{\beta }{\underset{\_}{\omega }}_{\rho }\left(t\right)-{\underset{\_}{\mathcal{F}}}_{\rho }\left(t,{\underset{\_}{\omega }}_{\rho }\left(t\right),{\overline{\omega }}_{\rho }\left(t\right)\right)=0\\ {\mathcal{C}}^{\beta }{\overline{\omega }}_{\rho }\left(t\right)-{\overline{\mathcal{F}}}_{\rho }\left(t,{\underset{\_}{\omega }}_{\rho }\left(t\right),{\overline{\omega }}_{\rho }\left(t\right)\right)=0\end{array}\right.$$

Step II: Suppose that the solutions of the system (3) about the initial point $t=0$ have the fractional PS expansion forms $${\underset{\_}{\omega }}_{\rho }\left(t\right)={\displaystyle {\displaystyle \sum }_{k=0}^{\infty }}\frac{c_k{t}^{\beta k}}{{\beta }^{k}k!},{\overline{\omega }}_{\rho }\left(t\right)={\displaystyle {\displaystyle \sum }_{k=0}^{\infty }}\frac{d_k{t}^{\beta k}}{{\beta }^{k}k!}, t\ge 0, 0<\beta \le 1, \rho \in \left[0,1\right].$$

Step III: Set $c_0={\underset{\_}{\omega }}_{\rho }\left(0\right)={\underset{\_}{\delta }}_{\rho }$ and $d_0={\overline{\omega }}_{\rho }\left(0\right)={\overline{\delta }}_{\rho }$, then the $j$-th fractional PS approximate solutions ${\underset{\_}{\omega }}_{\rho }^j\left(t\right)$ and ${\overline{\omega }}_{\rho }^j\left(t\right)$ of ${\underset{\_}{\omega }}_{\rho }\left(t\right) \mathrm{and} {\overline{\omega }}_{\rho }\left(t\right)$ can be written respectively as $${\underset{\_}{\omega }}_{\rho }^j\left(t\right)=c_0+{\displaystyle {\displaystyle \sum }_{k=1}^j}\frac{c_k{t}^{\beta k}}{{\beta }^{k}k!} \mathrm{and} {\overline{\omega }}_{\rho }^j\left(t\right)=d_0+{\displaystyle {\displaystyle \sum }_{k=1}^j}\frac{d_k{t}^{\beta k}}{{\beta }^{k}k!},t\ge 0, 0<\beta \le 1, \rho \in \left[0,1\right].$$

Step IV: Define the $j$-th fractional residual functions ${\underset{¯}{Res}}_{\rho }^j\left(t\right)$ and ${\overline{Res}}_{\rho }^j\left(t\right)$ such that $$\begin{array}{c}{\underset{¯}{Res}}_{\rho }^j\left(t\right)={\mathcal{C}}^{\beta }{\underset{\_}{\omega }}_{\rho }^j\left(t\right)-{\underset{\_}{\mathcal{F}}}_{\rho }\left(t,{\underset{\_}{\omega }}_{\rho }^j\left(t\right),{\overline{\omega }}_{\rho }^j\left(t\right)\right),\\ {\overline{Res}}_{\rho }^j\left(t\right)={\mathcal{C}}^{\beta }{\overline{\omega }}_{\rho }^j\left(t\right)-{\overline{\mathcal{F}}}_{\rho }\left(t,{\underset{\_}{\omega }}_{\rho }^j\left(t\right),{\overline{\omega }}_{\rho }^j\left(t\right)\right).\end{array}$$

Step V: Substitute ${\underset{\_}{\omega }}_{\rho }^j\left(t\right)$ and ${\overline{\omega }}_{\rho }^j\left(t\right)$ in ${\underset{¯}{Res}}_{\rho }^j\left(t\right)$ and ${\overline{Res}}_{\rho }^j\left(t\right)$ so that $$\begin{array}{c}{\underset{¯}{Res}}_{\rho }^j\left(t\right)={\mathcal{C}}^{\beta }\left(c_0+{\displaystyle {\displaystyle \sum }_{k=1}^j}\frac{c_k{t}^{\beta k}}{{\beta }^{k}k!}\right)-{\underset{\_}{\mathcal{F}}}_{\rho }\left(t,c_0+{\displaystyle {\displaystyle \sum }_{k=1}^j}\frac{c_k{t}^{\beta k}}{{\beta }^{k}k!},d_0+{\displaystyle {\displaystyle \sum }_{k=0}^j}\frac{d_k{t}^{\beta k}}{{\beta }^{k}k!}\right),\\ {\overline{Res}}_{\rho }^j\left(t\right)={\mathcal{C}}^{\beta }\left(d_0+{\displaystyle {\displaystyle \sum }_{k=0}^j}\frac{d_k{t}^{\beta k}}{{\beta }^{k}k!}\right)-{\overline{\mathcal{F}}}_{\rho }\left(t,c_0+{\displaystyle {\displaystyle \sum }_{k=1}^j}\frac{c_k{t}^{\beta k}}{{\beta }^{k}k!},d_0+{\displaystyle {\displaystyle \sum }_{k=0}^j}\frac{d_k{t}^{\beta k}}{{\beta }^{k}k!}\right).\end{array}$$

Step VI: Consider $j=1$, in Step V, then solve ${\underset{¯}{Res}}_{\rho }^1\left(t\right)=0$ and ${\overline{Res}}_{\rho }^1\left(t\right)=0$ at $t=0$ for $c_1$ and $d_1$. Therefore, the first fractional PS approximate solutions ${\underset{\_}{\omega }}_{\rho }^1\left(t\right)$ and ${\overline{\omega }}_{\rho }^1\left(t\right)$ will be obtained.

Step VII: For $j=2,3,\dots ,r$ in Step V, apply the operator $\left(j-1\right)\beta$-th on both sides of the resulting fractional equations such that ${\mathcal{C}}^{\left(j-1\right)\beta }{\underset{¯}{Res}}_{\rho }^j\left(t\right)$ and ${\mathcal{C}}^{\left(j-1\right)\beta }{\overline{Res}}_{\rho }^j\left(t\right)$. Then, by solving ${\mathcal{C}}^{\left(j-1\right)\beta }{\underset{¯}{Res}}_{\rho }^j\left(0\right)=0$ and ${\mathcal{C}}^{\left(j-1\right)\beta }{\overline{Res}}_{\rho }^j\left(0\right)=0$, $c_j$ and $d_j$ can be obtained.

Step VIII: Write the forms of the obtained coefficients $c_j$ and $d_j$ in terms of $j$-th fractional PS expansions ${\underset{\_}{\omega }}_{\rho }^j\left(t\right)$ and ${\overline{\omega }}_{\rho }^j\left(t\right)$ and repeat the above steps to reach a closed-form in terms of infinite series as in Step II. Elsewhere, the solution obtained will be representing the $j$-th fractional PS approximate solutions of the crisp system (3).