Bifurcation from double eigenvalue for nonlinear equation with third-order nondegenerate singularity

2013 ◽  
Vol 220 ◽  
pp. 549-559
Author(s):  
Qiang Zhang ◽  
Dongming Yan ◽  
Zhigang Pan
Keyword(s):  
Nonlinear Equation ◽  
Third Order ◽  
Double Eigenvalue

scholarly journals Higher-Order Equations of the KdV Type are Integrable

2010 ◽  
Vol 2010 ◽  
pp. 1-5 ◽  
Author(s):  
V. Marinakis
Keyword(s):  
Nonlinear Equation ◽  
Small Amplitude ◽  
Order Approximation ◽  
Higher Order ◽  
Integrable Equation ◽  
Incompressible Fluids ◽  
Third Order ◽  
Long Wavelength ◽  
Third Order Approximation

We show that a nonlinear equation that represents third-order approximation of long wavelength, small amplitude waves of inviscid and incompressible fluids is integrable for a particular choice of its parameters, since in this case it is equivalent with an integrable equation which has recently appeared in the literature. We also discuss the integrability of both second- and third-order approximations of additional cases.

scholarly journals Ball convergence theorem for a Steffensen-type third-order method

2017 ◽  
Vol 51 (1) ◽  
pp. 1-14
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Nonlinear Equation ◽  
Error Bounds ◽  
Order Method ◽  
Third Order ◽  
Numerical Examples ◽  
First Derivative ◽  
Fourth Derivative ◽  
Ball Convergence ◽  
Computable Error Bounds ◽  
Local Convergence Analysis

We present a local convergence analysis for a family of Steffensen-type third-order methods in order to approximate a solution of a nonlinear equation. We use hypothesis up to the first derivative in contrast to earlier studies such as [2, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 16, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28] using hypotheses up to the fourth derivative. This way the applicability of these methods is extended under weaker hypothesis. Moreover the radius of convergence and computable error bounds on the distances involved are also given in this study. Numerical examples are also presented in this study.

scholarly journals Further Acceleration of the Simpson Method for Solving Nonlinear Equations

2018 ◽  
Vol 14 (2) ◽  
pp. 7631-7639
Author(s):  
Rajinder Thukral
Keyword(s):  
Nonlinear Equation ◽  
Fourth Order ◽  
Efficiency Index ◽  
New Method ◽  
Order Method ◽  
Third Order ◽  
Type Method ◽  
New Formula ◽  
Order Four ◽  
Better Than

There are two aims of this paper, firstly, we present an improvement of the classical Simpson third-order method for finding zeros a nonlinear equation and secondly, we introduce a new formula for approximating second-order derivative. The new Simpson-type method is shown to converge of the order four.  Per iteration the new method requires same amount of evaluations of the function and therefore the new method has an efficiency index better than the classical Simpson method.  We examine the effectiveness of the new fourth-order Simpson-type method by approximating the simple root of a given nonlinear equation. Numerical comparisons is made with classical Simpson method to show the performance of the presented method.

Approximate periodic solution for nonlinear jerk equation as a third-order nonlinear equation via modified differential transform method

2014 ◽  
Vol 31 (4) ◽  
pp. 622-633 ◽  
Author(s):  
Alborz Mirzabeigy ◽  
Ahmet Yildirim
Keyword(s):  
Periodic Solution ◽  
Nonlinear Equation ◽  
Numerical Solutions ◽  
Nonlinear Differential Equations ◽  
Differential Transform Method ◽  
Third Order ◽  
Transform Method ◽  
Content Type ◽  
Differential Transform ◽  
Order Nonlinear Equation

Purpose – The nonlinear jerk equation is a third-order nonlinear equation that describes some physical phenomena and in general form is given by: x = J (x, x, x). The purpose of this paper is to employ the modified (MDTM) differential transform method (DTM) to obtain approximate periodic solutions of two cases of nonlinear jerk equation. Design/methodology/approach – The approach is based on MDTM that is developed by combining DTM, Laplace transform and Padé approximant. Findings – Comparison of results obtained by MDTM with those obtained by numerical solutions indicates the excellent accuracy of solution. Originality/value – The MDTM is extended to determining approximate periodic solution of third-order nonlinear differential equations.

scholarly journals On generalized Halley-like methods for solving nonlinear equations

2019 ◽  
Vol 13 (2) ◽  
pp. 399-422
Author(s):  
Miodrag Petkovic ◽  
Ljiljana Petkovic ◽  
Beny Neta
Keyword(s):  
Computer Graphics ◽  
Nonlinear Equation ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
Numerical Experiments ◽  
Multiple Root ◽  
Basins Of Attraction ◽  
Third Order ◽  
Numerical Examples ◽  
Theoretical Results

Generalized Halley-like one-parameter families of order three and four for finding multiple root of a nonlinear equation are constructed and studied. This presentation is, actually, a mixture of theoretical results, algorithmic aspects, numerical experiments, and computer graphics. Starting from the proposed class of third order methods and using an accelerating procedure, we construct a new fourth order family of Halley's type. To analyze convergence behavior of two presented families, we have used two methodologies: (i) testing by numerical examples and (ii) dynamic study using basins of attraction.

scholarly journals Some Third Order Methods for Solving Nonlinear Equations

2018 ◽  
Vol 13 (1) ◽  
pp. 169-174
Author(s):  
Jivandhar Jnawali
Keyword(s):  
Numerical Methods ◽  
Nonlinear Equation ◽  
Numerical Experiment ◽  
Nonlinear Equations ◽  
Efficiency Index ◽  
Subject Classification ◽  
Numerical Comparison ◽  
Single Variable ◽  
Third Order ◽  
Matlab Software

 There are several third order numerical methods having same efficiency index appeared in literature for solving nonlinear equations of a single variable. Practically, if we apply these methods in different nonlinear equations, we can observe that all methods are not performed equally for given nonlinear equations. The main objective of this paper is to show through numerical experiment that the performance of some third order methods having the same efficiency index does not perform equally for particular nonlinear equation. For the numerical comparison, we use Matlab software.2010 AMS Subject Classification: 65H05 Journal of the Institute of Engineering, 2017, 13(1): 169-174

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