Derivative-Free Method for Nonlinear Systems with Some Real-Life Applications (European Journal of Pure and Applied Mathematics, 2025)
This study implements a very effective derivative-free approach for root determination in the context of nonlinear equations. The use of Homeier’s third-order technique, which is based on the application of Newton’s theorem with regard to the inverse function as well as the development of a novel category of Newton-type procedures that exhibit cubic convergence, is the basis of this study. A variety of nonlinear problems, such as nonlinear equations and nonlinear systems of equations, have been examined to examine the proposed method’s efficacy in contrast to other widely used derivative-free approaches. The results suggest that the proposed method outperforms the currently examined techniques published. The scheme given in this study demonstrates a reduced number of iterations acquired in attaining the intended result. Exhibiting a convergence order of around 2.5. This convergence order surpasses the convergence order observed in comparable approaches. Moreover, by following the prescribed procedure of Broyden’s technique while employing the recommended approach with regards to solving systems of nonlinear equations, the issue pertaining to the Jacobian may be circumvented. Hence, the technique suggested may be regarded as a prominent approach that achieves rapid convergence in determining the roots of nonlinear equations without the use of derivatives. In practice, we employ the novel approach to address practical challenges, such as the mixed Hammerstein integral equation, discretized Poisson equation, and Weierstrass method. Through comparisons and illustrative examples, it is evident that the proposed method proves to be effective and on par with existing techniques of equivalent order.
Keywords: Homeier method; Nonlinear equations; Derivativefree methods