Development of a New Multi‐step Iteration Scheme for Solving Non‐Linear Models with Complex Polynomiography

Abstract

The appearance of nonlinear equations in science, engineering, economics, and medicine cannot be denied. Solving such equations requires numerical methods having higher-order convergence with cost-effectiveness, for the equations do not have exact solutions. In the pursuit of efficient numerical methods, an attempt is made to devise a modified strategy for approximating the solution of nonlinear models in either scalar or vector versions. Two numerical methods of second-and sixth-order convergence are carefully merged to obtain a hybrid multi-step numerical method with twelfth-order convergence while using seven function evaluations per iteration, resulting in the efficiency index of about 1.4262. The convergence is also ascertained theoretically, and the asymptotic error constant is computed. Furthermore, various numerical methods of varying orders are used to compare the numerical results obtained with the proposed hybrid method in approximate solution, number of iterations, absolute error, absolute functional value, and the machine time measured in seconds. The obtained results outperformed the chosen methods when applied models arising from physical and natural fields were solved. Finally, to observe the convergence graphically, some complex polynomials are plotted as polynomiographs, wherein the rapid convergence of the proposed method is guaranteed.

An Improved Blended Numerical Root-Solver for Nonlinear Equations

“…Nonlinear equations are commonly found in real-world problems, and as such, they play a crucial role in computational mathematics [24]. Root-finding methods are crucial tools in the progress of research and engineering [23]. They enable scientists and engineers to efficiently determine the solutions to equations, which in turn helps them comprehend and forecast the behavior of intricate systems, optimize processes, and create new technologies [26].…”

Dynamical rational solutions and their interaction phenomena for an extended nonlinear equation

“…The quest for analytical solutions to nonlinear partial differential equations is essential in scientific and engineering applications since it provides a wealth of information on the mechanisms of complicated physical phenomena. Numerous effective methods have been devised to seek exact solutions for NPDEs in mathematical physics, such as the Burgan et al method [2], the similarity transformations [3], the parabolic equation method [4], the new modified unified auxiliary equation method [5], the G 1 ( ) ¢expansion method [6,7], Jacobi elliptic function expansion (JEFE) method [8], the simplified Hirotaʼs method [9], the Kudrayshov approach and its modified version [10,11], the modified auxiliary expansion method [12], and the generalized exponential rational function method and its modified version [13,14], as well as some numerical methods [15][16][17][18]. Each of these methods has its characteristics, and the simplified Hirota method is commonly used owing to its efficiency and directness.…”