## 2 Flow Problem and Mathematical Description  
The problem is devoted tonumerical solutions of magneto-thermo analysis of convective heat transfer along thermally and magnetized geometry immersed in thermally stratified medium. The extended issue will change into arrangement of partial differential conditions and then transform into ordinary ones using the stream-function forms. The molded problem will be resolved by using FDM method conjunct with the Keller Box method. The numerical result for considered material properties will be depicted in graphical and tabular form.  
Consider two dimensional steady and incompressible electrical conducting fluid. In Figure 1, where \(x\) and \(y\) are axes along and perpendicular to the vertical magnetized plate immersed in thermally stratified medium, respectively, \(u\), \(H_{x}\) and \(v\), \(H_{y}\) are the velocity-fields in \(x\) and \(y\) orders, respectively. The \(H_{0}\) is the magnetic field strength, \(H_{x}=H_{0}\) indicates that the magnetic field is exact at the surface, \(T\) is the temperature, \(T_{\infty}\) is the free stream temperature, \(\kappa\) is the thermal conductivity of the fluid, and \(C_{p}\) is the specific heat. The \(\rho\) is the fluid density, \(\mu_{o}\) is the megmatic permeability, \(\
u=\mu/\rho\) is the kinematic fluid viscosity, \(\beta^{*}\) is the volumetric coefficient of thermal expansion, and gravity acceleration is \(g\). The following is how the equations for continuity, momentum, magnetic force, and energy are expressed where high temperature difference arises with temperature dependent density function:  
\[\rho=\rho_{\infty}\exp[\beta_{T}(T-T_{\infty})],\ \ \ \ \ \beta_{T}=-\frac{1}{\rho}\bigg{(}\frac{ \partial\rho}{\partial T}\bigg{)}_{p} \tag{1a}\]  
Figure 1: Coordinate system and flow geometry.  
It should be emphasized that the Boussinesq approximation holds true for fluids with \(\beta_{T}\) of the magnitude of \(10^{-3}\) when the temperature differential between the fluid and the surface is of the order of \(O(10^{-2})\). Consequently, leaving out higher-order terms and expanding Equation (1a) into a Taylor series, the above equation becomes \(\rho=\rho_{\infty}[1-\beta_{T}(T_{w}-T_{\infty})]\).  
\[\frac{\partial(\rho u)}{\partial x}+\frac{\partial(\rho v)}{\partial y}=0 \tag{1b}\]  
\[\frac{\partial\rho H_{1}}{\partial x}+\frac{\partial\rho H_{2}}{\partial y}=0 \tag{2}\]  
\[u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\mu\frac{ \partial}{\partial y}\left(\frac{1}{\rho}\frac{\partial u}{\partial y}\right) +\frac{g}{\rho}(\rho_{\infty}-\rho)+\frac{\mu_{0}}{4\pi\rho}\left(H_{1}\frac{ \partial H_{1}}{\partial x}+H_{2}\frac{\partial H_{1}}{\partial y}\right) \tag{3}\]  
\[u\frac{\partial H_{1}}{\partial x}+v\frac{\partial H_{1}}{\partial y}=v_{m} \frac{\partial^{2}H_{1}}{\partial y^{2}}+\left(H_{1}\frac{\partial u}{ \partial x}+H_{2}\frac{\partial u}{\partial y}\right) \tag{4}\]  
\[u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\frac{\kappa}{C_ {p}}\frac{\partial}{\partial y}\left(\frac{1}{\rho}\frac{\partial T}{\partial y }\right) \tag{5}\]  
The boundary conditions of the present model are,  
\[u=0;\ v=0;\ H_{1}=H_{0},\ H_{2}=0;\ T=T_{w}aty=0\ \text{at}\ y=0 \tag{6}\]  
\[U\to U_{\infty}\,\ \ T\to T_{\infty}\,\ \ H_{1}\to 0\ \ \ \text{as}\ \ \ y\to\infty.\]  
The proper boundary conditions for the temperature with temperature-dependent density and the thermal distraction components are described above. The Reynolds number is \(R_{e}x=\frac{U_{\infty}}{v^{\prime}}\), \(U_{\infty}\) being the free steam velocity, \(T_{w}=T_{o}+b\frac{\chi}{L}\) is the variable temperature, and \(T_{\infty}=T_{o}+c\frac{\chi}{L}\) is ambient variable temperature of the thermally and magnetized heated plate, \(b,\ c\) are positive constants and \(T_{o}\) is the temperature constant along the thermally and magnetized heated plate.