Root-finding of non-linear equations is one of the most appearing problems in engineering sciences. Most of the complicated engineering problems can be modeled easily by means of non-linear functions. The role of iterative algorithms via computers for solving such functions is much important and cannot be denied in the modern age. In an iterative algorithm, the convergence order and the computational cost per iteration are the main characteristics that depict its efficiency and performance i.e., a method with higher-order and lower computational cost will be more efficient and vice versa. Keeping these facts into consideration, the main goal of this paper is to introduce a new derivative-free iterative method that performs better. We develop this algorithm by utilizing the forward- and finite-difference schemes on well-known Househ"older’s method, resulting in an efficient and derivative-free algorithm with a low per iteration computing cost. We also look at the developed algorithm’s convergence criterion and show that it is quartic-order convergent. We investigate nine test-examples and solve them to demonstrate its correctness, validity, and efficiency numerically. Some real-world engineering problems in civil and chemical engineering are also included in these examples. The numerical results of the test-examples reveal that the newly constructed method outperforms the existing similar algorithms found in the literature. We consider various different-degrees complex polynomials for the graphical analysis and used a computer tool to create the polynomiographs of the proposed quartic-order algorithm and compare it to other comparable existing approaches. The graphical findings show that the developed method has a faster convergence speed than the other comparable algorithms.

Solving non-linear equation is perhaps one of the most difficult problems in all of numerical computations, especially in a diverse range of engineering applications. The convergence and performance characteristics can be highly sensitive to the initial guess of the solution for most numerical methods such as Newton’s method. However, it is very difficult to select reasonable initial guess of the solution for most systems of non-linear equations. Besides, the computational efficiency is not high enough. Taking this into account, based on variational iteration technique, we develop some new iterative algorithms for solving one-dimensional non-linear equations. The convergence criteria of these iterative algorithms has also been discussed. The superiority of the proposed iterative algorithms is illustrated by solving some test examples and comparing them with other well-known existing iterative algorithms in literature. In the end, the graphical comparison of the proposed iterative algorithms with other well-known iterative algorithms have been made by means of polynomiographs of different complex polynomials which reflect the fractal behavior and dynamical aspects of the proposed iterative algorithms.

The task of root-finding of the non-linear equations is perhaps, one of the most complicated problems in applied mathematics especially in a diverse range of engineering applications. The characteristics of the root-finding methods such as convergence rate, performance, efficiency, etc., are directly relied upon the initial guess of the solution to execute the process in most of the systems of non-linear equations. Keeping these facts into mind, based on Taylor’s series expansion, we present some new modifications of Halley, Househölder and Golbabai and Javidi’s methods and then making them second derivative free by applying Taylor’s series. The convergence analysis of the suggested methods is discussed. It is established that the proposed methods possess convergence of orders five and six. Several numerical problems have been tested to demonstrate the validity and applicability of the proposed methods. These test examples also include some real-life problems associated with chemical and civil engineering such as open channel flow problem, the adiabatic flame temperature equation, conversion of nitrogen-hydrogen feed to ammonia and the van der Wall’s equation whose numerical results prove the better performance of the suggested methods as compared to other well-known existing methods of the same kind in the literature. Finally, the dynamics of the presented algorithms in the form of polynomiographs have been shown with the aid of computer program by considering some complex polynomials and compared them with the other well-known iterative algorithms that revealed the convergence speed and other dynamical aspects of the presented methods.

In this paper, we proposed and analyzed three new root-finding algorithms for solving nonlinear equations in one variable. We derive these algorithms with the help of variational iteration technique. We discuss the convergence criteria of these newly developed algorithms. The dominance of the proposed algorithms is illustrated by solving several test examples and comparing them with other well-known existing iterative methods in the literature. In the end, we present the basins of attraction using some complex polynomials of different degrees to observe the fractal behavior and dynamical aspects of the proposed algorithms.

The primary objective of this paper is to develop a new method for root-finding by combining forward and finite-difference techniques in order to provide an efficient, derivative-free algorithm with a lower processing cost per iteration. This will be accomplished by combining forward and finite-difference techniques. We also detail the convergence criterion that was devised for the root-finding approach, and we show that the method that was recommended is quintic-order convergent. We addressed a few engineering issues in order to illustrate the validity and application of the developed root-finding algorithm. The quantitative results justified the constructed root-finding algorithm’s robust performance in comparison to other quintic-order methods that can be found in the literature. For the graphical analysis, we make use of the newly discovered method to plot some novel polynomiographs that are attractive to the eye, and then we evaluate these new plots in relation to previously established quintic-order root-finding strategies. The graphic analysis demonstrates that the newly created method for root-finding has better convergence with the larger area than the other comparable methods do.