# A Compact Numerical Scheme for the Heat Transfer of Mixed Convection Flow in Quantum Calculus
## 1 Introduction  
Techniques for solving non-linear equations and suggesting an appropriate solution have become a catchy area of research and engineering. Several scientists have used iterative approaches to solve non-linear equations, as evidenced by a literature survey [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], which indicated the evolution of iterative methods such as the homotopy perturbation technique, variational ieraFtive methods and the decomposition technique. [PERSON] [12] is the foremost scientist who initiated the study by introducing a basic quadratic convergent [PERSON] iterative method for solving non-linear equations. Later, his method was considered the most significant among the other ones. [PERSON] and [PERSON] [13], [PERSON] and [PERSON] [14], [PERSON] [15], [PERSON] and [PERSON] [16] and [PERSON] [17] put a step forth to modify the [PERSON] iterative method using quadrature rules, hence improving the efficiency and local order of convergence for [PERSON]'s method. Introducing promising variations of the Adomian decomposition method [18] by [PERSON] and [PERSON] [19] is a milestone in this field. They proposed a simple procedure that did not need the evaluation of the Adomian polynomial's derivative, demonstrating its superiority to the Adomian decomposition method. [PERSON] employed the Adomian decomposition method, while [PERSON] [20] and [PERSON] et al. [21, 22] proposed a family of iterative methods for solving non-linear equations that were more efficient and had a higher convergence order. This study is regarded as innovative since it establishes higher-order iterativemethods for solving non-linear equations toward the \(q\)-analogue of the iterative methods in the \(q\)-calculus. \(Q\)-calculus is the combination of mathematics and physics [23; 24; 25; 26; 27; 28; 29]. Due to its numerous applications in mathematics, including combinatorics, relativity theory, physics, number theory, and orthogonal polynomials, \(q\)-calculus became the most brain-dominating field in the last part of the twentieth century. As the inventor of the \(q\)-[PERSON]'s formula, [PERSON] [30] is regarded as a pioneer in this subject. They later used the \(q\)-differentiation approach to develop the \(q\)-[PERSON]'s formula and established results on the \(q\)-remainder in the \(q\)-[PERSON]'s formula by [PERSON] and [PERSON] [31]. There were four alternative \(q\)-[PERSON]'s formulas presented by [PERSON] [32] and the q integral remainder. Iterative methods, such as the \(q\)-analogue of the generalized [PERSON] method and the \(q\)-analogue of the [PERSON] method for solving algebraic transcendental equations, were studied by [PERSON] et al. [33] using \(q\)-[PERSON]'s formula. They compared the precision of the results obtained by the classical methods with those achieved using \(q\)-[PERSON]'s formula. It is possible to simulate a variety of linear and non-linear problems in everyday life and engineering problems using the \(q\)-differential equation. To solve the \(q\)-difference equations and develop convergence strategies, [PERSON] et al. [34] have accepted the Daftardar decomposition technique. In the literature, mathematical models for the flow over the surfaces have been constructed by various researchers using classical derivatives. Both ordinary and partial differential equations forms are considered in the literature to study the effect of involved dimensionless parameters on velocity, temperature and concentration profiles if the problem is related to the heat and mass transfer of some flows over the surfaces. A study was conducted of mixed convection, unsteady Darcy-Forchheimer flow over a stretching sheet [35]. The effects of Brownian motion and thermophoresis diffusion were also considered in the study. The obtained PDEs were tackled by employing the finite difference method. It was concluded that the velocity escalated by incrementing buoyancy and time and de-escalated by enhancing the porosity parameter and Forchheimer number.