Publication

Abstract : Numerous real life models in applied sciences and engineering commonly deal with nonlinear equations that have to be solved using reliable numerical methods. The computational science research is continuously growing, which is a marked by either introduction of new iteration algorithms or improvement of the existing ones. Nevertheless, these numerical methods may have high computational cost, but they do have a faster rate of convergence. Hence, the aim of this paper is create new two-step fourth order iteration algorithms for finding simple roots of nonlinear equations. To achieve this goal we have taken Steffensen’s method in the first sub-step and a general quadratic interpolation polynomial in the second sub-step of a two-step iterative scheme. The main theorem demonstrates the fourth-order convergence of the proposed scheme. The proposed scheme is derivative free and achieves optimal fourth order convergence by using only three function evaluations per iteration satisfying the conjecture of Kung and Traub. To validate the theoretical results and to demonstrate the effectiveness of the presented methods, we explore some nonlinear models in medical sciences such as the law of blood flow, blood rheology, fluid permeability in biogels, and thermal regulation of the human body. The numerical results of the proposed scheme are presented in terms of number of iterations, errors in consecutive iterations, computational convergence order (CCO) and CPU time (sec). Several researchers have investigated basins of attraction on simple polynomials of the form \(z^n-1\) in the complex plan. However, we have studied the fractal behavior of different methods on real world problems stated above in terms of basins of attraction for comparison of their convergence regions. The proposed scheme generate the basins of attraction in less time with wider regions of convergence in comparison with the existing methods of similar kind. Dynamical analysis of the proposed derivative free scheme illustrate its superiority in comparison with some existing optimal fourth order methods involving derivatives.