64 research outputs found
New solitary wave solutions to nonlinear evolution equations by the Exp-function method
No full textAbstractA new application of the Exp-function method in combination with the dependent variable transformation from singularity analysis is proposed for constructing new generalized solitary wave solutions and periodic wave solutions for nonlinear evolution equations. The Korteweg–de Vries equation is chosen to illustrate the validity and applicability of the suggested approach
A geometric construction of iterative functions of order three to solve nonlinear equations
No full textAbstractIn this paper we consider a geometric construction of iteration functions of order three to develop cubically convergent iterative methods for solving nonlinear equations. This construction can be applied to any iteration function of order two to develop an iteration function of order three. Some examples are given of deriving several third-order iteration methods, and several numerical results follow to illustrate the performance of the derived methods
A geometric construction of iterative formulas of order three
No full textAbstractIn this paper, we consider a geometric construction for improving the order of convergence of iterative formulas of order two. Using this approach, new third-order modifications of Newton’s method are derived. A comparison with other existing methods is given
A simply constructed third-order modifications of Newton's method
No full textAbstractIn this paper, we present a simple and easily applicable approach to construct some third-order modifications of Newton's method for solving nonlinear equations. It is shown by way of illustration that existing third-order methods can be employed to construct new third-order iterative methods. The proposed approach is applied to the classical Chebyshev–Halley methods to derive their second-derivative-free variants. Numerical examples are given to support that the methods thus obtained can compete with known third-order methods
Iterative methods improving newton's method by the decomposition method
No full textAbstractIn 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
No full textA 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.</p
A method for obtaining iterative formulas of order three
No full textAbstractAn improved method for the order of convergence of iterative formulas of order two is given. Using this method, new third-order modifications of Newton’s method are derived. A comparison with other methods is given
Iterative Methods for Nonlinear Equations or Systems and Their Applications
No full textThe guest editors would like to express their gratitude to all those who submitted papers for publication and to the many reviewers whose reports were essential for us. We would also like to thank the editorial board members of this journal. Alicia Cordero and Juan R. Torregrosa were partially supported by Ministerio de Ciencia y Tecnologa MTM2011-28636-C02-02.Torregrosa Sánchez, JR.; Argyros, IK.; Chun, C.; Cordero Barbero, A.; Soleymani, F. (2013). Iterative methods for nonlinear equations or systems and their applications. Journal of Applied Mathematics. 2013. https://doi.org/10.1155/2013/656953S201
An algorithm for finding all solutions of a nonlinear system
No full textAbstractLet f:X→Rk be a Lipschitz continuous function on a compact subset X⊂Rd. Subdivision algorithms are described that can be used to find all solutions of the equation f(x)=0 that lie in X. Convergence is shown and numerical examples are presented. Modifications of the basic algorithm which speed convergence are given for the case of nondegenerate zeros of a vector field
Basins of attraction for several optimal fourth order methods for multiple roots
No full textThere are very few optimal fourth order methods for solving nonlinear algebraic equations having roots of multiplicity m. Here we compare 5 such methods, two of which require the evaluation of the (m--1)st root. The methods are usually compared by evaluating the computational e ciency and the e ciency index. In this paper all the methods have the same e ciency, since they are of the same order and use the same information. Frequently, comparisons of the various schemes are based on the number of iterations required for convergence, number of function evaluations, and/or amount of CPU time. If a particular algorithm does not converge or if it converges to a di erent solution, then that particular algorithm is thought to be inferior to the others. The primary aw in this type of comparison is that the starting point represents only one of an in nite number of other choices. Here we use the basin of attraction idea to recommend the best fourth order method. The basin of attraction is a method to visually comprehend how an algorithm behaves as a function of the various starting points
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