Quantum calculus (also known as the q-calculus) is a technique that is similar to traditional calculus, but focuses on the concept of deriving q-analogous results without the use of the limits. In this paper, we suggest and analyze some new q-iterative methods by using the q-analogue of the Taylor’s series and the coupled system technique. In the domain of q-calculus, we determine the convergence of our proposed q-algorithms. Numerical examples demonstrate that the new q-iterative methods can generate solutions to the nonlinear equations with acceptable accuracy. These newly established methods also exhibit predictability. Furthermore, an analogy is settled between the well known classical methods and our proposed q-Iterative methods.

In computational mathematics, it is a matter of deep concern to recognize which of the given iteration schemes converges quickly with lesser error to the desired solution. Fixed point iterative schemes are constructed to be used for solving equations emerging in many fields of science and engineering. These schemes reformulate a nonlinear equation f(s) = 0 into a fixed point equation of the form s = g(s) ; such application determines the solution of the original equation via the support of fixed point iterative method and is subject to existence and uniqueness. In this manuscript, we introduce a new modified family of fixed point iterative schemes for solving nonlinear equations which generalize further recursive methods as particular cases. We also prove the convergence of our suggested schemes. We also consider some of the mathematical models which are categorically nonlinear in essence to authenticate the performance and effectiveness of these schemes which can be seen as an expansion and rationalization of some existing techniques.

The conventional infinitesimal calculus that concentrates on the idea of navigating the $ q- $symmetrical outcomes free from the limits is known as Quantum calculus (or $ q- $calculus). It focuses on the logical rationalization of differentiation and integration operations. Quantum calculus arouses interest in the modern era due to its broad range of applications in diversified disciplines of the mathematical sciences. In this paper, we instigate the analysis of Quantum calculus on the iterative methods for solving one-variable nonlinear equations. We introduce the new iterative methods called $ q- $iterative methods by employing the $ q- $analogue of Taylor's series together with the inclusion of an auxiliary function. We also investigate the convergence order of our newly suggested methods. Multiple numerical examples are utilized to demonstrate the performance of new methods with an acceptable accuracy. In addition, approximate solutions obtained are comparable to the analogous solutions in the classical calculus when the quantum parameter $ q $ tends to one. Furthermore, a potential correlation is established by uniting the $ q- $iterative methods and traditional iterative methods.