*4.1. Shooting Method*

The shooting technique along with the Runge Kutta method of the fourth order is employed in order to obtain the numerical solutions of Equations (11)–(13) subject to the boundary conditions. Shooting method helps to reduce the third order ODEs (11)–(13) into the first-order ODEs, such that

$$p = f', q = p', \; r = h'; \; q' = \frac{1}{1+K} \{2p^2 - fq - \mathcal{K}r\} \tag{30}$$

$$r = h; \; r' = \frac{2}{2 + K} \{ 3ph - fr + K(2h + q) \} \tag{31}$$

$$\mathbf{s} = \theta' \mathbf{;} \; s' = \Pr\{p\theta - f\mathbf{s}\} \tag{32}$$

with conditions

$$f(0) = \text{S; } p(0) = -1; \; q(0) = a\_1; \; h(0) = -na\_1; \\ r(0) = a\_2; \; \theta(0) = 1; \\ s(0) = a\_3$$

where α1, α2, and α<sup>3</sup> are called as unknown initial conditions. These three missing values α1, α2, and α<sup>3</sup> have to be obtained by using different shoots; this process of shoots will be continue until the profiles of the *f* 0 (η) → 0; *h*(η) → 0; and θ(η) → 0 are satisfied the boundary condition η → ∞. Maple (18) software has been used to convert the system of the third order ODEs into the system of the first order ODEs, for this process shootlib function is built-in Maple. Using RK method solves the system of the first order ODEs. Further, a detailed discussion about the shooting method with Maple software can be seen in the paper of Meade et al. [34].
