A New Jarratt-Type Fourth-Order Method for Solving System of Nonlinear Equations and Applications

Abstract

Solving systems of nonlinear equations plays a major role in engineering problems. We present a new family of optimal fourthorder Jarratt-type methods for solving nonlinear equations and extend these methods to solve system of nonlinear equations. Convergence analysis is given for both cases to show that the order of the new methods is four. Cost of computations, numerical tests, and basins of attraction are presented which illustrate the new methods as better alternates to previous methods. We also give an application of the proposed methods to well-known Burger's equation.

Ball Comparison for Some Efficient Fourth Order Iterative Methods Under Weak Conditions

“…The solution of this problem is s * = 1 π . The results in Nedzhibov [1], Hueso et al [2], Junjua et al [3], and Behl et al [4] cannot be utilized. In particular, conditions on the 5th derivative of λ or may be even higher are considered there to obtain the convergence of these methods.…”

“…x j +y j 2 , B j = λ (y j ) −1 λ (x j ), and [•, •; λ] : D × D → LB(E 1 , E 1 ) is a first order divided difference. These methods specialize to the corresponding ones (when E 1 = E 2 = R i , i is a natural number) studied by Nedzhibov [1], Hueso et al [2], Junjua et al [3], and Behl et al [4], respectively. The 4-order convergence of them was established by Taylor series and conditions on the derivatives up to order five.…”

Local Convergence and Attraction Basins of Higher Order, Jarratt-Like Iterations

“…Suppose that the hypotheses of Theorem 1 are satisfied but r is defined by (28) and the "g 3 " function replaces the "g 1 " function. Then, the conclusions of Theorem 1 hold with Method-III replacing Method-I.…”

“…There is another important class of multistep methods based on Jarratt methods or Jarratt-type methods [20][21][22]. Such methods have been extensively studied in the literature; see [23][24][25][26][27][28] and references therein. In particular, Alzahrani et al [23] have recently proposed a class of sixth order methods for approximating solution of H(x) = 0 using a Jarratt-like composite scheme.…”

New Iterative Methods for Solving Nonlinear Problems with One and Several Unknowns

“…It is straightforward to say from the fourth-order methods for scalar equations and systems available in the literature [14][15][16][17][18][19][20][21][22][23][24] that they present free parameters only at the second step, in order to obtain new iterative methods. In this paper, we explore the idea of including free parameters or weight functions also in the first step.…”