## 3 The Similarity Transformation and Stream Function Formulation  
The proper dimensionless similarity and stream function to convert PDEs into ODEs are given as:  
\[u=\frac{1}{\tilde{\rho}}\frac{\partial\psi}{\partial y},\ \ v=-\frac{1}{ \tilde{\rho}}\frac{\partial\psi}{\partial x},\ H_{1}=\frac{1}{\tilde{\rho}} \frac{\delta\varphi}{\delta y},\ H_{2}=-\frac{1}{\tilde{\rho}}\frac{\delta \varphi}{\delta x},\ \theta=\frac{T-T_{\infty}}{T_{w}-T_{\infty}}. \tag{7}\]  
The stream function \(\phi\) and \(\psi\) are for magnetic and velocity fields, \(\theta\) is dimensilized temperature and similarity parameter is \(\eta\).  
\[u=u_{\infty}f^{\prime},\ \ \ \ \ \ \ \ \ \ v=\frac{-1}{2\tilde{\rho}}\sqrt{ \frac{\overline{u_{\infty}}}{x}}, \tag{8}\]  
\[H_{1}=g^{\prime}H_{0},\ \ H_{2}=\frac{-1}{2\tilde{\rho}}\frac{H_{0}}{U_{ \infty}}\sqrt{\frac{vL_{\infty}}{x}}(g-\eta g^{\prime}),\]  
\[\eta=\int_{0}^{y}\tilde{\rho}dy\sqrt{\frac{u_{\infty}}{\
u x}},\ \ \ \ \psi=\sqrt{\
u xt_{\infty}}f(\eta),\ \varphi=\frac{H_{0}}{U_{\infty}}\sqrt{\
u xt_{\infty}g}\]  
The converted form of the ODEs in Equations (1)-(6) by applying the Equations (7) and (8), we have non linear ODEs given below;  
\[\frac{1}{2}ff^{\prime}+e^{-n\theta}f^{\prime\prime}-n\theta^{\prime}e^{-n \theta}f^{\prime\prime}+\lambda\left(\frac{1-e^{n\theta}}{1-e^{-n}}\right)+e^ {n\theta}Mgg^{\prime\prime}=0 \tag{9}\]  
\[P_{m}\left(e^{-n\theta}g^{\prime\prime}-n\theta^{\prime}e^{-n\theta}g^{\prime \prime}\right)+\frac{fg^{\prime\prime}}{2}-\frac{gf^{\prime\prime}}{2}=0 \tag{10}\]\[e^{-n\theta}\theta^{\prime}-ne^{-n\theta}\theta^{\prime 2}+P_{r}\bigg{(}\frac{1}{2}f \theta^{\prime}-f^{\prime}\theta-Stf^{\prime}\bigg{)}=0 \tag{11}\]  
where \(P_{r}=\frac{\
u}{\alpha}\) is Prandtl parameter, \(P_{m}=\frac{\
u_{m}}{\
u}\) is magnetic Prandtl parameter, \(M=\frac{H_{0}^{2}L^{2}\
u_{0}}{\
u^{2}4\pi\rho}\) is a magnetic parameter, \(\lambda=\frac{g\theta^{*}T_{0}}{U_{\alpha}^{2}}\) is the mixed convection parameter, \(\eta\) is similarity variable, \(\theta\) is dimensionless temperature, \(\
u=\frac{\mu}{\rho}\) is kinematic fluid viscosity, and \(St=\frac{\epsilon}{b}\) is stratification number. The transformed boundary conditions in (6) then becomes,  
\[f^{\prime}=0,\ f=0,\ g=0,\ \theta=1-St,\ g^{\prime}=1\ \ \ \text{at}\ \ \ \eta=0 \tag{12}\]  
\[f^{\prime}\to 1,\ \theta\to 0,\ g^{\prime}\to 0\,\ \ \ \ \ \text{at}\ \ \ \ \ \eta\to\infty.\]  
The formula for skin friction is \(Cf_{x}=\frac{\tau_{m}}{\rho Id_{x}{}^{2}}\), the formula for the Nusselt number is \(Nu_{x}=\frac{\
u_{M}}{\kappa(T_{x}-T_{00})}\), and the formula for the magnetic intensity coefficient is \(Mg_{x}=-\frac{i\pi}{U_{\alpha}{}^{2}}\). The values of \(\tau_{w}\), \(q_{w}\), and \(j_{w}\) are listed below:  
\[\tau_{w}=\mu\bigg{(}\frac{\partial u}{\partial y}\bigg{)}_{y=0}\,\ q_{w}=-\kappa\bigg{(}\frac{ \partial T}{\partial y}\bigg{)}_{y=0}\,\ j_{w}=\
u_{m}\bigg{(}\frac{\partial H}{\partial y}\bigg{)}_{y=0}\]  
Consequently, the values of magnetic intensity, Nusselt number, and skin friction are provided by,  
\[R_{e_{x}}{}^{1/2}C_{f_{x}}=f^{\prime\prime}(0),\ R_{e_{x}}{}^{-1/2}N_{u_{x}}=- \theta^{\prime}(0),\ R_{e_{x}}{}^{-1/2}Mg_{x}=-g^{\prime\prime}(0).\]