Iterative methods improving newton's method by the decomposition method

AbstractIn this paper, we present a sequence of iterative methods improving Newton's method for solving nonlinear equations. The Adomian decomposition method is applied to an equivalent coupled system to construct the sequence of the methods whose order of convergence increases as it progresses. The orders of convergence are derived analytically, and then rederived by applying symbolic computation of Maple. Some numerical illustrations are given

Error estimates for the bifurcation function for semilinear elliptic boundary value problems

A semilinear elliptic boundary value problem of the form Lu+gx,u,l=0 and a corresponding discrete problem based on the finite element method are considered. The method of alternative problems is used to reduce the boundary value problem to an equivalent finite-dimensional problem Bc,l=0 . The bifurcation function Bc,l is a vector field on Rd for fixed l . The solutions of the reduced problem are in a one-to-one correspondence with the solutions of the boundary value problem. The method of alternative problems is also applied to reduce the discrete problem to an equivalent lower-dimensional problem. An approximate bifurcation function Bhc,l for the lower-dimensional problem is also defined as a vector field on Rd , whose zeros are in a one-to-one correspondence with the solutions of the discrete problem. Estimates of the differences Bc,l-Bh c,l,Bc c,l-B hcc,l , and Ec,l-E hc,l are derived. Here, Ec,l (resp. Ehc,l ) denotes the value of energy functional associated with the boundary value problem (resp. the discrete problem). Morse decompositions are computed for some classical examples, and their bifurcation diagrams are presented. Results from numerical experiments on the orders of convergence for the difference Bc,l-Bh c,l are presented

Basins of attraction for several third order methods to find multiple roots of nonlinear equations

The article of record as published may be found at http://dx.doi.org/10.1016/j.amc.2015.06.068There are several third order methods for solving a nonlinear algebraic equation having roots of a given multiplicity m. Here we compare a recent family of methods of order three to Euler-Cauchy's method which is found to be the best in the previous work. There are fewer fourth order methods for multiple roots but we will not include them here.Basic Science Research Program through the National Reserach Foundation of Korea (NRF)Ministry of Education (NRF-2013R1A1A2005012

Basins of attraction for several methods to find simple roots of nonlinear equations

The article of record as published may be located at http://dx.doi.org/10.1016/j.amc.2010.04.017There are many methods for solving a nonlinear algebraic equation. The methods are clas- sified by the order, informational efficiency and efficiency index. Here we consider other criteria, namely the basin of attraction of the method and its dependence on the order. We discuss several third and fourth order methods to find simple zeros. The relationship between the basins of attraction and the corresponding conjugacy maps will be discussed in numerical experiments. The effect of the extraneous roots on the basins is also discussed

Recurrence relations for a third-order family of methods in Banach Spaces

Recently, PArida and Gupta (J. Comp. Appl.Math. 2-6 (2007), 873-877) used Rall's recurrence relations approach (from 1961) to approximate roots of nonlinear equations, by developing several methods, the latest of which is free of second derivative and it is of third order. In this paper, we use an idea of Kou and Li (appl. Math. Comp. 187 (2007), 1027-1032) and modify the approach of Parida and Gupta, obtaining yet another third-order method to approximate a solution of a non-linear equation in a Banach space. We give several applications to our method

Analytical solution for nonlinear Gas Dynamic equation by Homotopy Analysis Method

In this paper, the Homotopy Analysis Method (HAM) is used to implement the homogeneous gas dynamic equation. The analytical solution of this equation is calculated in form of a series with easily computable components

New Exact Solutions to the KdV-Burgers-Kuramoto Equation with the Exp-Function Method

Based on the characteristics of the truncated Painlevé expansion method and the Exp-function method, new generalized solitary wave solutions are constructed for the KdV-Burgers-Kuramoto equation, which cannot be directly constructed from the Exp-function method. This work highlights the power of the Exp-function method in providing generalized solitary wave solutions of different physical structures

Iterative methods for nonlinear equations or systems and their applications 2014

Alicia Cordero and Juan R. Torregrosa were partially supported by Ministerio de Ciencia y Tecnología MTM2011-28636-C02-02.Torregrosa Sánchez, JR.; Argyros, IK.; Chun, C.; Cordero Barbero, A.; Soleymani, F. (2014). Iterative methods for nonlinear equations or systems and their applications 2014. Journal of Applied Mathematics. 2014. https://doi.org/10.1155/2014/293263S201

The basins of attraction of Murakami's fifth order family of methods

The article of record as published may be found at http://dx.doi.org/10.1016/j.apnum.2016.07.012In this paper we analyze Murakami’s family of fifth order methods for the solution of nonlinear equations. We show how to find the best performer by using a measure of closeness of the extraneous fixed points to the imaginary axis. We demonstrate the performance of these members as compared to the two members originally suggested by Murakami. We found several members for which the extraneous fixed points are on the imaginary axis, only one of these has 6 such points (compared to 8 for the other members). We show that this member is the best performer.Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2013R1A1A2005012)Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2013R1A1A2005012

A New Trigonometrically-Fitted Method for Second Order Initial Value Problems

Funded by Naval Postgraduate SchoolNumerical schemes approximating the solution of ordinary initial value problems interpolate polynomial up to a certain degree. Brock and Murray have suggested in 1952 to interpolate exponential functions when the solution is of exponential type. In 1961 Gautschi has suggested the use of complex exponential functions of the frequency and multiples of it. Later others developed methods based on combination of both. The basis of all these methods are the well-known linear multistep (including Obrechkoff schemes) and RungeKutta (RKN) schemes. We develop methods for the solution of first and second order systems having periodic solution with approximately known period. Our methods based on Obrechkoff schemes. We compare our new methods to existing ones.Naval Postgraduate SchoolApproved for public release; distribution is unlimited