## 2 Mathematical Model  
This study is concerned with the steady Newtonian hydromagnetic two-dimensional boundary layer convective heat intensity transport from a inclined surface affected by an aligned magnetic field along with thermal radiation. The sheet (e.g., electrically conducting polymer) is moving in a coplanar manner with a rate of change of distance and temperature inversely correlated with the distance from the leading edge, respectively. An external velocity \(u=u_{e}(x)=Dx^{p}\) is imposed at the free stream. A slit is used at the origin to let the sheet squeeze through the fluid medium. Both the x-axis and the y-axis are pointed at the sheet. Two equal and opposing forces along the x-  
field to be abandoned in contrast to an applied magnetic field. It is assumed that there is no applied electric field and that the Hall effect does not exist. The basic equations for two-dimensional steady incompressible laminar flow are given by [PERSON] [30], [PERSON] et al. [19], and [PERSON] et al. [16]. _Mass conservation (continuity)_  
\[\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0 \tag{1}\]  
_Magnetic field continuity_  
\[\frac{\partial H_{1}}{\partial x}+\frac{\partial H_{2}}{\partial y}=0 \tag{2}\]  
_Momentum_  
\[u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}-\frac{\mu}{4\pi \rho}\bigg{(}H_{1}\frac{\partial H_{1}}{\partial x}+H_{2}\frac{\partial H_{1} }{\partial y}\bigg{)}=v\frac{\partial^{2}u}{\partial y^{2}}+\bigg{(}u_{e} \frac{du_{e}}{dx}-\frac{\mu H_{e}}{4\pi\rho}\;\frac{dH_{e}}{dx}\;\bigg{)} \tag{3}\]  
_Magnetic induction conservation_  
\[u\frac{\partial H_{1}}{\partial x}+v\frac{\partial H_{1}}{\partial y}-H_{1} \frac{\partial u}{\partial x}-H_{2}\frac{\partial u}{\partial y}=\mu_{e} \frac{\partial^{2}H_{1}}{\partial y^{2}} \tag{4}\]  
_Energy conservation_  
\[u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\alpha\frac{ \partial^{2}T}{\partial y^{2}}-\frac{\partial q_{r}}{\partial y} \tag{5}\]  
where the extended surface has cartesian coordinates \(x\) and \(y\) which are perpendicular to it, respectively. The magnetic induction units along the \(x\) and \(y\) axes are \(H_{1}\) and \(H_{2}\) respectively, \(u_{e}\) and \(H_{e}\) are the horizontal-velocity component and component of the horizontally produced magnetic field at the boundary layer's edge, \(v\) is the magnetic polymer's kinematic viscosity, \(p\) is its density and \(\mu\) is its dynamic viscosity, \(k\) is the fluid's thermal conductivity, \(c_{p}\) is its specific heat at constant pressure, and \(q_{r}\) is the radiative heat flux. Thermal diffusivity is symbolized as \(\alpha=\frac{k}{c_{p}}\). Using [PERSON]'s estimation [32; 33; 34; 35; 36; 37; 38; 39], the following expression of the radiative heat flux is generated  
\[q_{r}=-\frac{4\sigma}{3k}\frac{\partial T^{4}}{\partial y} \tag{6}\]  
Figure 1: Magnetic boundary layer convection-radiation flow from a stretching surface. where the [PERSON] mean absorption coefficient is equal to \(k\), and the Stefan-Boltzmann constant is equal to \(\sigma\). According to [32; 33; 34; 35; 36; 37; 38; 39], we disregard higher-order terms under the supposition that inside the flow, temperature variations are sufficiently modest such that Taylor series expansion can show \(T^{4}\) about the free steam temperature \(T_{\infty}\):  
\[T^{4}\approx 4{T_{\infty}}^{3}\;T-3{T_{\infty}}^{4} \tag{7}\]  
Equations (6) and (7) allow us to derive:  
\[\frac{\partial q_{r}}{\partial y}=-\frac{16\sigma_{1}{T_{\infty}}^{3}}{3k_{1} }\frac{\partial^{2}T}{\partial y^{2}} \tag{8}\]  
Using Equation (8) the _energy_, i.e., _thermal boundary layer_ Equation (5) becomes:  
\[u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\alpha\frac{ \partial^{2}T}{\partial y^{2}}-\frac{16\sigma_{1}{T_{\infty}}^{3}}{3k_{1}} \frac{\partial^{2}T}{\partial y^{2}} \tag{9}\]  
The following are suitable border conditions:  
\[\left\{