**3. Homotopic Solutions**

For the considered problem, (*Lf* , *Lg*,*L*θ, *L*φ) are the linear operators and (*f*0, *g*0, θ0,φ0) are the initial guesses expressed in the following form:

$$\begin{cases} f\_0(\eta) = \frac{A}{1 + \gamma\_1 - \gamma\_2} (1 - \exp(-\eta)) & g\_0(\eta) = \frac{\beta}{1 + \gamma\_3 - \gamma\_4} (1 - \exp(-\eta)) \\ \text{where} & 1 + \gamma\_1 - \gamma\_2 \neq 0 \ 1 + \gamma\_3 - \gamma\_4 \neq 0 \\ \Theta\_0(\eta) = (1 - S\_1)(1 - \exp(-\eta)) & \phi\_0(\eta) = (1 - S\_2)(1 - \exp(-\eta)) \end{cases} \tag{25}$$

$$\begin{cases} L\_f(f) = \frac{d^3f}{d\eta^3} - \frac{df}{d\eta} & L\_\xi(g) = \frac{d^3\xi}{d\eta^3} - \frac{d\xi}{d\eta} \\ L\_0(\theta) = \frac{d^2\theta}{d\eta^2} - \theta & L\_\phi(\phi) = \frac{d^2\phi}{d\eta^2} - \phi \end{cases} \tag{26}$$

these operators satisfy the following condition:

$$\begin{array}{l} L\_f[\mathbb{C}\_1 + \mathbb{C}\_2 \exp(\eta) + \mathbb{C}\_3 \exp(-\eta)] = 0\\ L\_\mathcal{X}[\mathbb{C}\_4 + \mathbb{C}\_5 \exp(\eta) + \mathbb{C}\_6 \exp(-\eta)] = 0\\ L\_\theta[\mathbb{C}\_9 \exp(\eta) + \mathbb{C}\_{10} \exp(-\eta)] = 0 \end{array} \tag{27}$$
