Construction of a Class of Higher-Order Iterative Techniques and its Convergence in Banach Spaces

We propose a novel sixth-order convergence scheme for solving scalar equations, based on the weight function approach. This approach provides us with the flexibility to construct new iterative techniques with the same level of convergence. In addition, we extend the same idea to nonlinear systems of equations with the help of Banach space operators. Further, a challenging semi-local convergence analysis is conducted to establish the theoretical foundation of the scheme. We also demonstrate the applicability and efficiency of the scheme by applying it to three problems in applied science: an integral equation, a boundary value problem (BVP), and the two-dimensional Burger’s equation. Our scheme not only achieves smaller absolute residual errors, reduced differences between successive iterations, and requires fewer iterations to reach the desired accuracy compared to existing methods, but it also demonstrates lower CPU time consumption and stable convergence order. Finally, we conclude that our scheme exhibits superior efficiency and compatibility compared to existing methods of the same convergence order.