Some improvements of Jarratt’s method with sixth-order convergence

A system of coupled nonlinear partial differential equations with convective and dispersive terms was modified from Boussinesq-type equations. Through a special formulation, a system of nonlinear partial differential equations was solved alternately and explicitly in time without linearizing the nonlinearity. Coupled compact schemes of sixth order accuracy in space were developed to obtain numerical solutions. Within couple compact schemes, variables and their first and second derivatives were solved altogether. The sixth order accuracy in space is achieved with a memory-saving arrangement of state variables so that the linear system is banded instead of blocked. This facilitates solving very large systems. The efficiency, simplicity, and accuracy make this coupled compact method viable as variational and weighted residual methods. Results were compared with exact solutions which were obtained via devised forcing terms. Error analyses were carried out, and the sixth order convergence in space and second order convergence in time were demonstrated. Long time integration was also studied to show stability and error convergence rates.

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Efficient three-step iterative methods with sixth order convergence for nonlinear equations

Numerical Algorithms ◽

10.1007/s11075-009-9315-y ◽

2009 ◽

Vol 53 (4) ◽

pp. 485-495 ◽

Cited By ~ 10

Author(s):

Alicia Cordero ◽

José L. Hueso ◽

Eulalia Martínez ◽

Juan R. Torregrosa

Keyword(s):

Iterative Methods ◽

Nonlinear Equations ◽

Sixth Order ◽

Order Convergence

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Some novel Newton-type methods for solving nonlinear equations

Boletim da Sociedade Paranaense de Matemática ◽

10.5269/bspm.v38i3.37351 ◽

2019 ◽

Vol 38 (3) ◽

pp. 111-123

Author(s):

Morteza Bisheh-Niasar ◽

Abbas Saadatmandi

Keyword(s):

Iterative Method ◽

Nonlinear Equations ◽

Basins Of Attraction ◽

Sixth Order ◽

New Method ◽

Order Convergence ◽

Cubic Convergence ◽

Numerical Examples ◽

Newton Iterative Method

The aim of this paper is to present a new nonstandard Newton iterative method for solving nonlinear equations. The convergence of the proposed method is proved and it is shown that the new method has cubic convergence. Furthermore, two new multi-point methods with sixth-order convergence, based on the introduced method, are presented. Also, we describe the basins of attraction for these methods. Finally, some numerical examples are given to show the performance of our methods by comparing with some other methods available in the literature

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A Modification of Newton-Type Method with Sixth-Order Convergence for Solving Nonlinear Equations

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.490-495.1839 ◽

2012 ◽

Vol 490-495 ◽

pp. 1839-1843

Author(s):

Rui Chen ◽

Liang Fang

Keyword(s):

Nonlinear Equations ◽

Newton’S Method ◽

Newton's Method ◽

Efficiency Index ◽

Numerical Results ◽

Sixth Order ◽

Order Convergence ◽

Type Method ◽

Second Derivatives ◽

Better Than

In this paper, we present and analyze a modified Newton-type method with oder of convergence six for solving nonlinear equations. The method is free from second derivatives. It requires three evaluations of the functions and two evaluations of derivatives in each step. Therefore the efficiency index of the presented method is 1.431 which is better than that of classical Newton’s method 1.414. Some numerical results illustrate that the proposed method is more efficient and performs better than classical Newton's method

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