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.
Citation: Gul Sana, Muhmmad Aslam Noor, Dumitru Baleanu. Novel higher order iterative schemes based on the $ q- $Calculus for solving nonlinear equations[J]. AIMS Mathematics, 2022, 7(3): 3524-3553. doi: 10.3934/math.2022196
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.
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Gul Sana, Muhmmad Aslam Noor, Dumitru Baleanu. Novel higher order iterative schemes based on the $ q- $Calculus for solving nonlinear equations[J]. AIMS Mathematics, 2022, 7(3): 3524-3553. doi: 10.3934/math.2022196
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