# A Compact Numerical Scheme for the Heat Transfer of Mixed Convection Flow in Quantum Calculus
## 5 Applications to Engineering Problems  
For the implementation of the proposed scheme, two examples are considered. The first example is a non-linear parabolic equation having a spatial \(q\)-derivative term. Numerical schemes will be tested on these two examples. **Example 1. **_Consider the following parabolic equation_  
\[\frac{\partial u}{\partial t}=\frac{\partial_{q}^{2}u}{\partial x^{2}}+\rho u -\rho u^{2} \tag{58}\]  
_subject to the boundary conditions_  
\[u(t,-0.2)=\frac{1}{\Big{(}1+e^{\sqrt{\frac{2}{5}}(-0.2)}-\frac{5\rho}{6}t \Big{)}^{2}}\ \&\ u(t,0.8)=\frac{1}{\Big{(}1+e^{\sqrt{\frac{2}{5}}(0.8)}-\frac{5\rho}{6}t \Big{)}^{2}}, \tag{59}\]  
_with an initial condition_  
\[u(0,x)=\frac{1}{\Big{(}1+e^{\sqrt{\frac{2}{5}}x}\Big{)}^{2}}. \tag{60}\]  
_Since Equation (58) is non-linear, and for applying the proposed scheme using the Gauss-Seidel iterative method, it must be linearized. For doing so, Equation (58) is expressed as_  
\[\frac{\partial u}{\partial t}=\frac{\partial_{q}^{2}u}{\partial x^{2}}+\rho u (1-\overline{u}) \tag{61}\]  
_where \(\overline{u}\) is considered as a fixed quantity, but it will be evaluated at each grid point \"\(i\)\" and at time level \"\(n\)\"._  
The proposed compact scheme is employed to solve Equation (61) using Equations (59) and (60). For time discretization, two different second-order schemes are employed. One of the adopted schemes is Crank-Nicolson that is second-order accurate an implicit scheme, and the second employed scheme is the second-order Runge-Kutta explicit scheme. However, Runge-Kutta explicit scheme with the proposed compact discretization in Quantum Calculus becomes implicit. Figures 1 and 2 show the comparison of the two schemes. Figure 1 shows the relative error produced by both schemes using the same time and space step sizes. This figure shows that the proposed scheme in the spatial direction with Runge-Kutta in the time direction converges but it does not produce much accurate solution over the whole considered domain. Additionally, it is observed that the error given by the scheme using the Runge-Kutta time discretization scheme will decay if the final time is chosen to be very small for this particular example. Figure 1: Comparison of two numerical schemes using \(N_{x}=30\), \(N_{t}=40\), \(\rho=100\), \(t_{f}=0.00001\). Figure 2 shows the norm of relative error over the consumed iterations. Since the Runge-Kutta technique yields less accurate solutions than one obtained by the Crank-Nicolson approach, the Runge-Kutta method's norm of relative error does not tend to zero over the consumed number of iterations. Figure 3 shows the variation of the \(q\)-derivative on the solution of Equations (59)-(61). From this Figure 3, it can be seen that solution decreases by enhancing \(q\). **Example 2. **_Flow over flat and oscillator sheet._  
Consider a laminar two-dimensional, unsteady, incompressible mixed convection flow over the flat and oscillatory sheets. Let the \(x^{*}\)-axis be kept with along the plate and the \(y^{*}\)-axis be taken perpendicular to the \(x^{*}\)-axis. The flow is generated due to the stretching of the plate.