YoonMee Ham

Some sixth-order variants of Ostrowski root-finding methods

Abstract In this paper, we present some sixth-order class of modified Ostrowski’s methods for solving nonlinear equations. Per iteration each class member requires three function and one first derivative evaluations, and is shown to be at least sixth-order convergent. Several numerical examples are given to illustrate the performance of some of the presented methods.

Some fourth-order modifications of Newton’s method

Abstract In this paper, we construct some fourth-order modifications of Newton’s method for solving nonlinear equations. Any two existing fourth-order methods can be effectively used to give rise to new fourth-order methods. Per iteration the new methods require two evaluations of the function and one of its first-derivative. Numerical examples are given to show the performance of the presented methods.

A one-parameter fourth-order family of iterative methods for nonlinear equations

Abstract In this paper, we present a new one-parameter fourth-order family of iterative methods for solving nonlinear equations. The new family requires two evaluations of the given function and one of its first derivative per iteration. The well-known Traub–Ostrowski’s fourth-order method is shown to be part of the family. Several numerical examples are given to illustrate the efficiency and performance of the presented methods.

Some second-derivative-free variants of super-Halley method with fourth-order convergence

In this paper, we present some new variants of super-Halley method for solving nonlinear equations. These methods are free from second derivatives and require one function and two first derivative evaluations per iteration. Analysis of convergence shows that the methods are fourth-order. Several numerical examples are given to illustrate the performance of the presented methods.

A fifth-order iterative method for solving nonlinear equations

In this paper, we present a new fifth-order method for solving nonlinear equations. Per iteration the new method requires two function and two first derivative evaluations. It is shown that the new method is fifth-order convergent. Several numerical examples are given to illustrate the performance of the presented method.

Internal layer oscillations in Fitzhugh-Nagumo equation

A controller-propagator system with a FitzHugh-Nagumo equation can be reduced to a free boundary problem when a layer parameter e is equal to zero. We shall show the existence of solutions and the occurence of a Hopf bifurcation for this free boundary problem as the controlling parameter τ varies.

A HOPF BIFURCATION IN A FREE BOUNDARY PROBLEM DEPENDING ON THE SPATIAL AVERAGE OF AN ACTIVATOR

We shall consider an activator–inhibitor system proposed by Radehaus [1990]. In this system, the activator is inhibited by not only the inhibitor but also its own spatial average. The purpose of this paper is to analyze the dynamics of interfaces in an interfacial problem which is reduced from the system in order to examine how this problem is different from an activator–inhibitor system [Ham-Lee et al., 1994].

A Hopf Bifurcation in a Three-Component Reaction-Diffusion System with a Chemoattraction

We consider a three-component reaction-diffusion system with a chemoattraction. The purpose of this work is to analyze the chemotactic effects due to the gradient of the chemotactic sensitivity and the shape of the interface. Conditions for existence of stationary solutions and the Hopf bifurcation in the interfacial problem as the bifurcation parameters vary are obtained analytically.

Hopf Bifurcation with the Spatial Average of an Activator in a Radially Symmetric Free Boundary Problem

An interface problem derived by a bistable reaction-diffusion system with the spatial average of an activator is studied on an -dimensional ball. We analyze the existence of the radially symmetric solutions and the occurrence of Hopf bifurcation as a parameter varies in two and three-dimensional spaces.