# A Compact Numerical Scheme for the Heat Transfer of Mixed Convection Flow in Quantum Calculus
## 1 Introduction
There also was a study on mixed convective Jeffery nanofluid stratified flow with the effect of magnetohydrodynamics [36]. The convective boundary condition was implemented, and effects of heat absorption and heat generation were also considered. On the non-linear convected flow of thiotropic non-Newtonian fluid [37], the dimensionless model was solved by an optimal homotopy analysis method. The method finds the solution in infinite series form. The main advantage of using this method is that it contains the parameter(s) that can be used to control the convergence of the solution. Radiative mixed convective flow over a vertical cylinder using porous media and irregular heat sink/source has been studied [38]. A MATLAB solver for solving boundary value problems has been employed to solve the transformed model, and solutions of the model with both of their branches were portrayed in graphs. There are also various analytical and numerical methods to solve these models. The accuracy of solutions of models depends on the used method. One of the ways to check the accuracy of numerical methods is to use the Taylor series. The accuracy may depend on the degree of the considered interpolation polynomial. Some methods have been constructed in the literature using polynomial interpolation. The idea for the interpolation of the polynomial has been used in literature to construct numerical schemes in fractional calculus. The present contribution is concerned with proposing a compact numerical scheme for solving differential equations having spatial second order \(q\)-derivatives. Two useful numerical approximations for discretizing \(q\)-derivative are given, and the proposed numerical scheme is based on these approximations. For handling time derivative terms, Crank-Nicolson and Runge-Kutta methods are employed. The difference equations obtained by applying Crank-Nicolson/Runge-Kutta schemes in time direction and proposed compact scheme in spatial direction to the considered equations are tackled by the Gauss-Seidel iterative method. The mathematical model for the heat transfer of a mixed convection boundary layer flow over stretching sheets with a spatial \(q\)-derivative diffusion term is given. Later on, it is solved by the proposed numerical schemes. A non-linear mathematical model of mixed convection flow is chosen for the present study. Two types of boundary conditions are chosen for the considered flow problem. A non-linear differential equation is linearized first and then discretized by a proposed compact computational scheme. **Definition 1** ([39]). : The \(q\)_-derivative for real-valued continuous function \(f(x)\) is expressed as_  
\[f_{q}^{\prime}(x)=\frac{f(qx)-f(x)}{qx-x},\;0<q<1 \tag{1}\]  
_when \(q\to 1\), \(f_{q}^{\prime}(x)\to f^{\prime}(x)\)._  
**Theorem 1** ([39]). : The \(q\)_-derivative Taylor series is expressed as_  
\[f(x)=\sum_{j=0}^{\infty}\frac{f_{q}^{j}(x_{0})}{[j]_{q}!}\left(x-x_{\circ} \right)^{j} \tag{2}\]  
_where \([j]_{q}=[j]_{q}[j-1]_{q}\ldots[2]_{q}[1]_{q}\) and \([j]_{q}=\frac{1-q^{j}}{1-q}\)._  
A useful result is proposed before starting the construction procedure of the proposed compact scheme. **Theorem 2**. : _Let \(\mathcal{A}\) be an operator defined as_  
\[\mathcal{A}u_{i}=\frac{u_{i-1}+bu_{i}+u_{i+1}}{b+2} \tag{3}\]  
_where \(b=2\big{(}1+q^{2}\big{)}\big{(}1+q+q^{2}\big{)}-2\), then \(u_{q,i}^{\prime\prime}=\left(\frac{1+q}{2}\right)\delta_{x}^{2}u_{i}+O\big{(}h ^{2}\big{)}\), and_  
\[\mathcal{A}u_{q,i}^{\prime\prime}=\frac{u_{q,i-1}^{\prime\prime}+bu_{q,i}^{ \prime\prime}+u_{q,i+1}^{\prime\prime}}{b+2}=\bigg{(}\frac{1+q}{2}\bigg{)} \delta_{x}^{2}u_{i}+O\big{(}h^{4}\big{)} \tag{4}\]  
_where \(\delta_{x}^{2}u_{i}=\frac{u_{i+1}-2u_{i}+u_{i-1}}{\left(\Delta x\right)^{2}}\)._  
**Proof.