comparative study of methods of various orders for finding repeated roots of nonlinear equations abstract in this paper we are considering 20 families of methods for finding repeated roots of a nonlinear equation the methods are of order up to 8 we use the idea of basin of attraction to compare the methods we found that 4 methods performed best based on 3 quantitative criteria

two new families of sixth order methods for solving non linear equations in this paper we developed two new families of sixth order methods for solving simple roots of non linear equations per iteration these methods require two evaluations of the function and two evaluations of the first order derivatives which implies that the efficiency indexes of our methods are 1 565 these methods have more advantages than newton s method and other methods with the same convergence order as shown in the illustration examples finally using the developing methodology described in this paper two new families of improvements of jarratt method with sixth order convergence are derived in a straightforward manner notice that kou s method in jisheng kou yitian li an improvement of the jarratt method appl math comput 189 2007 1816 1821 and wang s method in xiuhua wang jisheng kou yitian li a variant of jarratt method with sixth order convergence appl math comput 204 2008 14 19 are the special cases of the new improvements

higher order efficient class of chebyshev halley type methods construction of two point sixth order methods for simple root is an ambitious and challenging task in numerical analysis therefore the main aim of this paper is to introduce a new highly efficient two point sixth order class of chebyshev halley type methods free from second order derivative for the first time each member of the proposed class requires only four functional evaluations viz two evaluations of function f and two of first order derivative f per full iteration a variety of concrete numerical examples illustrate that our proposed methods are more efficient and perform better than existing two point three point sixth order methods available in the literature from their dynamical study it has been observed that our proposed methods have better stability and robustness as compared to the other existing methods

he s variational iteration method for treating nonlinear singular boundary value problems this paper applies he s variational iteration method for solving nonlinear singular boundary value problems the solution process is illustrated and various physically relevant results are obtained comparison of the obtained results with exact solutions shows that the method used is an effective and highly promising method for treating various classes of both linear and nonlinear singular boundary value problems

he s variational iteration method for solving linear and non linear systems of ordinary differential equations abstract in this article he s variational iteration method vim is employed to solve a system of differential equations of first order since every ordinary differential equations of higher order can be converted into a system of differential of the first order this method can be used for solving most differential equations some examples are presented to show the ability of the method for linear and non linear systems of differential equations the results reveal that the method is very effective and simple

variational iteration method for autonomous ordinary differential systems in this paper a new iteration technique is proposed to solve autonomous ordinary differential systems in this method general lagrange multipliers are introduced to construct correction functionals for the systems the multipliers in the functionals can be identified by the variational theory the initial approximations can be freely chosen with possible unknown constants which can be determined by imposing the boundary initial conditions some examples are given the results reveal that the method is very effective and convenient

a class of two point sixth order multiple zero finders of modified double newton type and their dynamics under the assumption of the known multiplicity of zeros of nonlinear equations a class of two point sextic order multiple zero finders and their dynamics are investigated in this paper by means of extensive analysis of modified double newton type of methods with the introduction of a bivariate weight function dependent on function to function and derivative to derivative ratios higher order convergence is obtained additional investigation is carried out for extraneous fixed points of the iterative maps associated with the proposed methods along with a comparison with typically selected cases through a variety of test equations numerical experiments strongly support the theory developed in this paper in addition relevant dynamics of the proposed methods is successfully explored for various polynomials with a number of illustrative basins of attraction

numerical solution for an inverse heat source problem by an iterative method abstract in this paper we consider a typical inverse heat source problem that is we determine two separable source terms in a heat equation from the initial and boundary data along with two additional measurements by a simple transformation the original problem is changed into a nonlinear problem and then we use an iterative method to solve it after giving an algorithm we prove some holder convergence rates for both the reconstructed heat source terms and the temperature distribution subject to certain bounds of the data numerical results show that our method is accurate and efficient

solving the inverse problem of identifying an unknown source term in a parabolic equation the inverse problem of determining an unknown source term in a parabolic equation with the leading coefficient depending on time and space variables under a final overdetermination condition is considered the series pattern solution of the problem is obtained by using the weighted homotopy analysis method wham a description of the method for solving the problem and identifying the unknown source term is derived finally several examples are presented to illustrate this method

a biparametric extension of king s fourth order methods and their dynamics a class of two point quartic order simple zero finders and their dynamics are investigated in this paper by extending king s fourth order family of methods with the introduction of an error corrector having a weight function dependent on a function to function ratio higher order convergence is obtained through a variety of test equations numerical experiments strongly support the theory developed in this paper in addition relevant dynamics of the proposed methods is successfully explored for a prototype quadratic polynomial as well as parameter spaces and dynamical planes

fourth order iterative methods free from second derivative in this paper we present a family of new iterative methods with order of convergence four for solving non linear equations per iteration these methods require one evaluation of the function and two of its first derivative analysis of efficiency in term of function evaluations shows that this family of methods has great superiority which is also demonstrated by numerical examples

source term identification in 1 d ihcp we propose stable numerical solutions for the simultaneous identification of temperature temperature gradient and general source terms in the one dimensional inverse heat conduction problem ihcp the numerical solution consists of a regularization procedure based on the mollification method and a marching scheme for the solution of the stabilized problem the stability error analysis and implementation of the algorithm are presented together with a set of numerical results

variational iteration method for solving burger s and coupled burger s equations by means of variational iteration method the solutions of burger s equation and coupled burger s equations are exactly obtained comparison with the adomian decomposition method is made showing that the former is more effective than the later in this paper he s variational iteration method is introduced to overcome the difficulty arising in calculating adomian polynomials

basins of attraction for optimal eighth order methods to find simple roots of nonlinear equations several optimal eighth order methods to obtain simple roots are analyzed the methods are based on two step fourth order optimal methods and a third step of modified newton the modification is performed by taking an interpolating polynomial to replace either f z n or f z n in six of the eight methods we have used a hermite interpolating polynomial the other two schemes use inverse interpolation we discovered that the eighth order methods based on jarratt s optimal fourth order methods perform well and those based on king s or kung traub s methods do not in all cases tested the replacement of f z by hermite interpolation is better than the replacement of the derivative f z

numerical solution of the inverse problem of determining an unknown source term in a two dimensional heat equation a numerical procedure for an inverse problem of determination of unknown source term in two dimensional parabolic equation is presented the approach of the proposed method is to approximate unknown function by a piecewise linear function whose coefficients are determined from the solution of minimisation problem based on the overspecified data some numerical examples are presented

application of the variational iteration method to inverse heat source problems this paper investigates the inverse problem of determining a heat source in the parabolic heat equation using the usual conditions the numerical solution is developed by using the variational iteration method this method is based on the use of lagrange multipliers for the identification of optimal values of parameters in a functional using this method a rapid convergent sequence can be obtained which tends to the exact solution of the problem furthermore the variational iteration method does not require the discretization of the problem thus the variational iteration method is suitable for finding the approximation of the solution without discretization of the problem two numerical examples are presented to illustrate the strength of the method

basins of attraction for several methods to find simple roots of nonlinear equations there are many methods for solving a nonlinear algebraic equation the methods are classified 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

on developing a higher order family of double newton methods with a bivariate weighting function a high order family of two point methods costing two derivatives and two functions are developed by introducing a two variable weighting function in the second step of the classical double newton method their theoretical and computational properties are fully investigated along with a main theorem describing the order of convergence and the asymptotic error constant as well as proper choices of special cases a variety of concrete numerical examples and relevant results are extensively treated to verify the underlying theoretical development in addition this paper investigates the dynamics of rational iterative maps associated with the proposed method and an existing method based on illustrated description of basins of attraction for various polynomials

compensatability and optimal compensation under randomly varying distributed delays this paper establishes necessary and sufficient conditions for existence uniqueness and global optimality of the linear quadratic coupled delay compensator lqcdc which is designed to circumvent the detrimental effects of randomly varying delays from sensor to controller and from controller to actuator as well as the time skew caused by missynchronization of sensor and controller sampling instants these conditions are derived on the basis of the concepts of stabilizability detectability and compensatability in the mean square sense in the absence of random delays from sensor to controller and from controller to actuator it has been shown that lqcdc problems reduce to the classical linear quadratic gaussian lqg

choosing weight functions in iterative methods for simple roots weight functions with a parameter are introduced into an iteration process to increase the order of the convergence and enhance the behavior of the iteration process the parameter can be chosen to restrict extraneous fixed points to the imaginary axis and provide the best basin of attraction the process is demonstrated on several examples

extraneous fixed points basin boundaries and chaotic dynamics for schro der and ko nig rational iteration functions the schroder and konig iteration schemes to find the zeros of a polynomial functiong z represent generalizations of newton s method in both schemes iteration functionsf m z are constructed so that sequencesz n 1 f m z n converge locally to a rootz ofg z aso z n z m it is well known that attractive cycles other than the zerosz may exist for newton s method m 2 asm increases the iteration functions add extraneous fixed points and cycles whether attractive or repulsive they affect the julia set basin boundaries the konig functionsk m z appear to minimize such perturbations in the case of two roots e g g z z 2 1 cayley s classical result for the basins of attraction of newton s method is extended for allk m z the existence of chaotic z n sequences is also demonstrated for these iteration methods

variational iteration method for solving two point boundary value problems variational iteration method is introduced to solve two point boundary value problems numerical results demonstrate that the method is promising and may overcome the difficulty arising in adomian decomposition method

nonhomogeneous heat equation identification and regularization for the inhomogeneous term we study the nonhomogeneous heat equation under the form ut uxx t f x where the unknown is the pair of functions u f under various assumptions about the function and the final value in t 1 i e g x we propose different regularizations on this ill posed problem based on the fourier transform associated with a lebesgue measure for 0 the solution is unique

the jarratt method in banach space setting abstract in this note we extend the jarratt method of order four into banach spaces we also establish a kantorovich type convergence theorem see the references and give an explicit expression for the error bound of the method jarratt 1966 1969

a new family of eighth order iterative methods for solving nonlinear equations a family of eighth order iterative methods with four evaluations for the solution of nonlinear equations is presented kung and traub conjectured that an iteration method without memory based on n evaluations could achieve optimal convergence order 2 n 1 the new family of eighth order methods agrees with the conjecture of kung traub for the case n 4 therefore this family of methods has efficiency index equal to 1 682 numerical comparisons are made with several other existing methods to show the performance of the presented methods

some modifications of newton s method with higher order convergence for solving nonlinear equations in yoonmee ham etal some higher order modifications of newton s method for solving nonlinear equations j comput appl math 222 2008 477 486 some higher order modifications of newton s method for solving nonlinear equations are constructed but if p 2 then their main theorem did not hold in this paper we first give an example to show that yoonmee ham etal s methods are not always correct in the case p 2 then we present the condition that h x y should satisfy such that the order of convergence increases three or four or five units per iteration they only need two additional function evaluations to increase the order based on this and multi step newton s scheme we give further modifications of the method to obtain higher order convergent iterative methods finally several examples are given to demonstrate the efficiency and performance of our modified methods and compare them with some other methods

a super fast tri parametric iterative method with memory for the first time a tri parametric method with memory for solving nonlinear equations is constructed at which a super fast convergence rate is achieved by using only three functional evaluations subsequently a very high computational efficiency index nearly 2 is attained finally to support the theory some numerical experiments are reported

some improvements of jarratt s method with sixth order convergence abstract in this paper we present a one parameter family of variants of jarratt s fourth order method for solving nonlinear equations it is shown that the order of convergence of each family member is improved from four to six even though it adds one evaluation of the function at the point iterated by jarratt s method per iteration several numerical examples are given to illustrate the performance of the presented methods

variational iteration method some recent results and new interpretations this paper is an elementary introduction to the concepts of variational iteration method first the main concepts in variational iteration method such as general lagrange multiplier restricted variation correction functional are explained heuristically subsequently the solution procedure is systematically addressed in particular for nonlinear oscillators particular attention is paid throughout the paper to give an intuitive grasp for the method the main motivation is to put things together in a convenient form for later reference and systematic use

a fourth order derivative free algorithm for nonlinear equations a two step derivative free iterative algorithm is presented for solving nonlinear equations error analysis shows that the algorithm is fourth order with efficiency index equal to 1 5874 a lot of numerical results show that the algorithm is effective and is preferable to some existing derivative free methods in terms of computation cost

comparison of several families of optimal eighth order methods several families of optimal eighth order methods to find simple roots are compared to the best known eighth order method due to wang and liu 2010 we have tried to improve their performance by choosing the free parameters of each family using two different criteria

different anomalies in a jarratt family of iterative root finding methods abstract in this paper the behavior of a jarratt family of iterative methods applied to quadratic polynomials is studied some anomalies are found in this family be means of studying the dynamical behavior of this fourth order family of methods parameter spaces are shown and the study of the stability of all the fixed points is presented dynamical planes for members with good and bad dynamical behavior are also provided

on the dynamics of a triparametric family of optimal fourth order multiple zero finders with a weight function of the principal mth root of a function to function ratio abstract under the assumption of known root multiplicity m n a triparametric family of two point optimal quartic order methods locating multiple zeros are investigated in this paper by introducing a weight function dependent on a function to function ratio special cases of weight functions with selected free parameters are considered and studied through various test equations and numerical experiments to support the theory developed in this paper in addition we explore the relevant dynamics of proposed methods via mobius conjugacy map when applied to a prototype polynomial z a m z b m the results of such dynamics are visually illustrated through a variety of parameter spaces as well as dynamical planes

numerical solutions of nonlinear evolution equations using variational iteration method the variational iteration method is used to solve three kinds of nonlinear partial differential equations coupled nonlinear reaction diffusion equations hirota satsuma coupled kdv system and drinefel d sokolov wilson equations numerical solutions obtained by the variational iteration method are compared with the exact solutions revealing that the obtained solutions are of high accuracy he s variational iteration method is introduced to overcome the difficulty arising in calculating adomian polynomial in adomian method the method is straightforward and concise and it can also be applied to other nonlinear evolution equations in mathematical physics

some higher order iteration functions for solving nonlinear models abstract in this paper we present a new efficient sixth order family of jarratt type methods for solving scalar equations then we extend this family to the multidimensional case preserving the same order of convergence we also discuss the theoretical convergence properties of the proposed scheme in the case of scalar as well as multidimensional case the derivation of these schemes are based on weight function approach and free disposable parameters we also demonstrate the applicability of them on total six number of problems first five are real life problems namely continuous stirred tank reactor cstr chemical engineering the trajectory of an electron in the air gap between two parallel plates hammerstein integration and boundary value problems last one is the standard academic test problem in addition numerical comparisons are made to show the performance of the proposed iterative techniques with the existing techniques of the same order in the scalar as well as multi dimensional case finally on the basis of numerical results we conclude that our techniques perform better in terms of residual error error between the two consecutive iterations asymptotic error constant term and approximated root as compared to the existing ones of same order in scalar as well as multidimensional case

fourth order integro differential equations using variational iteration method in this paper the variational iteration method proposed by ji huan he is applied to solve both linear and nonlinear boundary value problems for fourth order integro differential equations the numerical results obtained with minimum amount of computation are compared with the exact solutions to show the efficiency of the method the results show that the variational iteration method is of high accuracy more convenient and efficient for solving integro differential equations

a sixth order method for nonlinear equations abstract in this paper a sixth order method is developed by extending a third order method of weerakoon and fernando s weerakoon t g i fernando a variant of newton s method with accelerated third order convergence appl math lett 13 2000 87 93 for finding the real roots of nonlinear equations in r starting with a suitably chosen x 0 the method generates a sequence of iterates converging to the root the convergence analysis is provided to establish its sixth order of convergence in terms of computational cost it requires evaluations of only two functions and two first derivatives per iteration this implies that efficiency index of our method is 1 565 our method is comparable with the methods of neta b neta a sixth order family of methods for nonlinear equations intern j computer math 7 1979 157 161 and that of kou and li jisheng kou yitian li an improvement of jarratt method appl math comput 189 2007 1816 1821 it does not require the evaluation of the second order derivative of the given function as required in the family of chebyshev halley type methods jisheng kou xiuhua wang sixth order variants of chebyshev halley methods for solving non linear equations appl math comput 190 2007 1839 1843 jisheng kou on chebyshev halley methods with sixth order convergence for solving non linear equations appl math comput 190 2007 126 131 the efficacy of the method is tested on a number of numerical examples it is observed that our method takes less number of iterations than newton s method and the method of weerakoon and fernando on comparison with the other sixth order methods it behaves either similarly or better for the examples considered

a class of optimal eighth order derivative free methods for solving the danchick gauss problem abstract a derivative free optimal eighth order family of iterative methods for solving nonlinear equations is constructed using weight functions approach with divided first order differences its performance along with several other derivative free methods is studied on the specific problem of danchick s reformulation of gauss method of preliminary orbit determination numerical experiments show that such derivative free high order methods offer significant advantages over both the classical and danchick s newton approach