## 6 Analysis and Discussion of Results  
The present analysis addresses the temperature-dependent density effect on magnetohydrodynamic convective heat transfer phenomena of electrical-conductive fluid flow along the vertical magnetized and heated plate placed in thermally stratified medium. The non-linear coupled PDE equations for the above fluid flow mechanism are formulated and then converted into non-similar formulation by applying an appropriate and well-known similarity transformation for integration. The final non-similar forms are integrated numerically by employing Keller Box scheme method. The transformed algebraic equations are plotted graphically and numerically on the MATLAB software package. The behavior of physical quantities such as velocity graph, magnetic field graph, and temperature graph along with their slopes, that is skin friction, magnetic intensity, and heat transfer, under the effect of different parameters included in the flow model is discussed. The impact of physical parameters such as Prandtl number \(P_{r}\), mixed convective parameter \(\lambda\), stratification parameter \(St\), density number \(m\), magnetic-force number \(M\), and magnetic Prandtl \(P_{m}\) is interpreted graphically and numerically. Figure 2a-c are presented for the behavior of velocity graph, temperature plot, and magnetic field for different values of temperature dependent density \(m=0.0,\ 1.0,\ 2.0,\ \text{and}\ 3.0\) with some fixed parameters. It was deduced that the velocity profile was enhanced at maximum value of \(m=3.0\) but similar behavior wasnoted for other values of \(m\) in Figure 2a. This happens because an increasing value of density parameter tends to increase the density variation with temperature, which ultimately increases the buoyancy force. The magnetic field wasincreased at the lower value of \(m=0.0\) but decreased at the maximum value of \(m=3.0\) in Figure 3b. In Figure 3c, the temperature distribution wasincreased at lower value of \(m=0.0\) but decreased at maximum value of \(m=3.0\). The good variations in each profile werenoted for each value of temperature-density parameter \(m\). Hence, as \(m\) increased, the fluid was under two forces: the first force increased the velocity of the fluid due to increase in the buoyancy forces and the second force decreased the velocity of the fluid due to the decrease in temperature. Figure 3a-c are plotted against the various values of stratification parameter \(St=0.0,\ 0.1,\ 0.2,\ 0.3\) for velocity graph, temperature graph, and magnetic field. It was examined that the velocity profile increased at lower value of \(St=0.0\) but the minimum value wasnoted at the higher value of \(St=0.3\) in Figure 3a. It waseduced that the magnetic field wasenhanced at the large value of \(St=0.3\) but decreased at the small value of \(St=0.0\) in Figure 3b. This happens because magnetic diffusion is reduced by increasing the magnetic-Prandtl number which is responsible for above-said phenomena. This behavior of the aforementioned variables supports the physical hypothesis that as the magnetic field grows stronger, more resistance is formed inside the fluid flow domain, which reduces the velocity distribution and improves the temperature profile. The temperature distribution increased at the lower value of \(St=0.0\) but decreased at the maximum value of \(St=3.0\) in Figure 3c. Figure 4a-c depict the behavior of velocity \(f^{\prime}(\eta)\), temperature \(\theta(\eta)\), and magnetic field \(g^{\prime}(\eta)\) for different quantities of \(\lambda=0.1,\ 0.5,\ 1.5,\ \text{and}\ 2.0\) with some fixed parameters. It was deduced that the velocity profile wasenhanced at larger \(\lambda=2.0\) but decreased at small value of \(\lambda=0.1\) in Figure 4a. It was examined that the magnetic field increased at lower value of \(\lambda=0.1\) while minimum value wasnoted at higher value of \(\lambda=2.0\) in Figure 4b.