# A Compact Numerical Scheme for the Heat Transfer of Mixed Convection Flow in Quantum Calculus
## 1 Introduction
** Consider the \(q\)-derivative Taylor series expansions for \(u_{i+1}\) and \(u_{i-1}\) as  
\[u_{i+1}=u_{i}+\frac{\Delta x}{[1]_{q}!}u_{q,i}^{\prime}+\frac{\left(\Delta x \right)^{2}}{[2]_{q}!}u_{q,i}^{\prime\prime}+\frac{\left(\Delta x\right)^{3} }{[3]_{q}!}u_{q,i}^{\prime\prime\prime}+\frac{\left(\Delta x\right)^{4}}{[4]_{ q}!}u_{q,i}^{\prime\prime}+\ldots \tag{5}\]  
\[u_{i-1}=u_{i}-\frac{\Delta x}{[1]_{q}!}u_{q,i}^{\prime}+\frac{\left(\Delta x \right)^{2}}{[2]_{q}!}u_{q,i}^{\prime\prime}-\frac{\left(\Delta x\right)^{3} }{[3]_{q}!}u_{q,i}^{\prime\prime\prime}+\frac{\left(\Delta x\right)^{4}}{[4]_{ q}!}u_{q,i}^{\prime\prime}+\ldots \tag{6}\]  
Adding (5) and (6) results in  
\[u_{i+1}+u_{i-1}=2u_{i}+\frac{2(\Delta x)^{2}}{[2]_{q}!}u_{q,i}^{\prime\prime}+ O(\left(\Delta x\right)^{4})\]  
This implies  
\[u_{q,i}^{\prime\prime}=\frac{\left(u_{i-1}-2u_{i}+u_{i+1}\right)}{2\left( \Delta x\right)^{2}}[2]_{q}!+O(\left(\Delta x\right)^{2})=\delta_{x}^{2}u_{i} \bigg{(}\frac{1+q}{2}\bigg{)}+O(\left(\Delta x\right)^{2}) \tag{7}\]By taking second order \(q\)-derivative of Equations (5) and (6) yields  
\[u^{\prime\prime}_{q,i+1}=u^{\prime\prime}_{q,i}+\frac{\Delta x}{\left[1\right]_{q }!}u^{\prime\prime\prime}_{q,i}+\frac{(\Delta x)^{2}}{\left[2\right]_{q}!}u^{iv }_{q,i}+\ldots \tag{8}\]  
\[u^{\prime\prime}_{q,i-1}=u^{\prime\prime}_{q,i}-\frac{\Delta x}{\left[1\right] _{q}!}u^{\prime\prime\prime}_{q,i}+\frac{(\Delta x)^{2}}{\left[2\right]_{q}!}u^ {iv}_{q,i}+\ldots \tag{9}\]  
Adding Equations (8) and (9), it produces  
\[f(x)=\sum_{j=0}^{\infty}\frac{f_{q}^{j}(x_{0})}{[j]_{q}!}\left(x-x_{\circ} \right)^{j} \tag{2}\]0  
Adding \(bu^{\prime\prime}_{q,i}\) on both sides of Equation (10) yields  
\[f(x)=\sum_{j=0}^{\infty}\frac{f_{q}^{j}(x_{0})}{[j]_{q}!}\left(x-x_{\circ} \right)^{j} \tag{2}\]1  
Re-write Equation (11) as  
\[f(x)=\sum_{j=0}^{\infty}\frac{f_{q}^{j}(x_{0})}{[j]_{q}!}\left(x-x_{\circ} \right)^{j} \tag{2}\]2  
Adding Equations (5) and (6) and consider the remainder fourth order \(q\)-derivative term, it is obtained that  
\[f(x)=\sum_{j=0}^{\infty}\frac{f_{q}^{j}(x_{0})}{[j]_{q}!}\left(x-x_{\circ} \right)^{j} \tag{2}\]3  
Re-write Equation (13) in the form of  
\[f(x)=\sum_{j=0}^{\infty}\frac{f_{q}^{j}(x_{0})}{[j]_{q}!}\left(x-x_{\circ} \right)^{j} \tag{2}\]4  
Substituting Equation (14) into Equation (12), it is obtained that  
\[f(x)=\sum_{j=0}^{\infty}\frac{f_{q}^{j}(x_{0})}{[j]_{q}!}\left(x-x_{\circ} \right)^{j} \tag{2}\]5  
\[f(x)=\sum_{j=0}^{\infty}\frac{f_{q}^{j}(x_{0})}{[j]_{q}!}\left(x-x_{\circ} \right)^{j} \tag{2}\]6  
subject to the boundary and initial conditions,  
\[f(x)=\sum_{j=0}^{\infty}\frac{f_{q}^{j}(x_{0})}{[j]_{q}!}\left(x-x_{\circ} \right)^{j} \tag{2}\]7  
where \(\overline{\alpha}\) and \(\beta\) are constants. Applying operator \(\mathcal{A}\) on both sides of the Equation (17),  
\[f(x)=\sum_{j=0}^{\infty}\frac{f_{q}^{j}(x_{0})}{[j]_{q}!}\left(x-x_{\circ} \right)^{j} \tag{2}\]8  
where \(f(u)\) is linearized as  
\[f(x)=\sum_{j=0}^{\infty}\frac{f_{q}^{j}(x_{0})}{[j]_{q}!}\left(x-x_{\circ} \right)^{j} \tag{2}\]9  
Equation (19) with the effect of operator \(\mathcal{A}\) is given as  
\[\mathcal{A}u_{i}=\frac{u_{i-1}+bu_{i}+u_{i+1}}{b+2} \tag{3}\]0  
Equation (21) is a semi-discretized scheme in which a fourth-order \(q\)-compact scheme is employed in a spatial direction. For time direction, a second-order Runge-Kutta scheme is employed in Equation (21), that yields  
\[\mathcal{A}u_{i}=\frac{u_{i-1}+bu_{i}+u_{i+1}}{b+2} \tag{3}\]1  
Equation (22) represents the first stage of the Runge-Kutta scheme, and for its second stage, Equation (21) is fully discretized as  
\[\mathcal{A}u_{i}=\frac{u_{i-1}+bu_{i}+u_{i+1}}{b+2} \tag{3}\]2  
So the scheme represented in Equations (22) and (23) gives fourth-order accuracy in space and second-order accuracy in time. The time accuracy can be enhanced by employing a fourth-order Runge-Kutta scheme. Additionally, a second-order Crank-Nicolson scheme can be considered.