frequency analysis of strongly nonlinear generalized duffing oscillators using he s frequency amplitude formulation and he s energy balance method in this paper he s frequency amplitude formulation hfaf and he s energy balance method hebm are employed to solve the generalized duffing equation in the form of x x a 3x 3 a 5x 5 a 7x 7 a nx n 0 for any arbitrary power of n a frequency analysis is carried out and the relationship between the natural frequency and the initial amplitude is obtained in analytical form accuracy and validity of the proposed techniques are then verified by comparing the numerical results obtained based on the hfaf hebm and exact integration method numerical simulations are extended for even very strong nonlinearities and very good correlations are achieved between the numerical results

nonparametric estimation of successive duration times under dependent censoring summary this paper considers nonparametric inference for duration times of two successive events since the second duration process becomes observable only if the first event has occurred the length of the first duration affects the probability of the second duration being censored dependent censoring arises if the two duration times are correlated which is often the case standard approaches to this problem fail because of dependent censoring mechanism a new product limit estimator for the second duration variable and a pathdependent joint survival function estimator are proposed both modified for the dependent censoring properties of the estimators are discussed an example from lawless 1982 is studied for illustrative purposes as well as a simulation study

application of he s amplitude frequency formulation to a nonlinear oscillator with discontinuity in this paper he s amplitude frequency formulation is used to obtain a periodic solution of a nonlinear oscillator with discontinuity the accuracy arrives at 0 73 even when the amplitude tends to infinity

nonparametric tests for the gap time distributions of serial events based on censored data summary this article deals with the problem of comparing two populations with respect to the distribution of the gap time between two successive events when each subject can experience a series of events and when the event times are potentially right censored several families of nonparametric tests are developed all of which allow arbitrary distributions and dependence structures for the serial events the asymptotic and small sample properties of the proposed tests are investigated an illustration with data taken from a colon cancer study is provided the related problem of testing the independence of two successive gap times is also studied

the max min approach to a relativistic equation this paper applies the max min approach proposed by ji huan he to the relativistic oscillator comparison with the exact solution shows that the method is very effective and convenient for solving nonlinear equations

on a class of noninterpolating solutions of the many anyon problem in the many anyon problem in two space dimensions irregular but square integrable solutions of the schrodinger equation may exist a class of such solutions is constructed for anyons confined in a harmonic oscillator it is shown that these may have lower energies than the usual regular solutions but they do not exist throughout the range between the bosonic and fermionic limits and as such do not interpolate continuously

solution of the epidemic model by homotopy perturbation method in this article the problem of the spread of a non fatal disease in a population which is assumed to have constant size over the period of the epidemic is considered he s homotopy perturbation method is employed to compute an approximation to the solution of the system of nonlinear ordinary differential equations governing on the problem the results are compared with the results obtained by adomian decomposition method some plots are presented to show the reliability and simplicity of the method

numerical solutions of the nonlinear integro differential equations wavelet galerkin method and homotopy perturbation method this paper compares the homotopy perturbation method with the wavelet galerkin method for solving nonlinear integro differential equations from the computational viewpoint the homotopy perturbation method is more efficient and easy to use

he s frequency formulation for nonlinear oscillators using a golden mean location he s frequency formulation derived on the basis of an ancient chinese mathematical method is an effective method for treating nonlinear oscillators there are different approaches for choosing location points in the formulation this paper suggests another location point an example is given and the accuracy is improved

investigation of classical and fractional bose einstein condensation for harmonic potential abstract in this study classical and fractional gross pitaevskii gp equations were solved for harmonic potential and repulsive interactions between the boson particles using the homotopy perturbation method hpm to investigate the ground state dynamics of bose einstein condensation bec the purpose of writing fractional gp equations is to consider the system in a more realistic manner the memory effects of non markovian processes involving long range interactions between bosons with the restriction of the ergodic hypothesis and the effect of non gaussian distributions of bosons in the condensation can be taken into account with time fractional and space fractional gp equations respectively the obtained results of the fractional gp equations differ from the results of the classical one while the gauss distribution describing the homogeneous reversible and unitary system is obtained from the classical gp equation the probability density of the solution function of fractional gp equations is non conserved this situation describes the inhomogeneous irreversible and non unitary systems

application of he chengtian s interpolation to bethe equation in this work he chengtian s interpolation which has millennial history is applied to the bethe equation the result shows that the present technology is very convenient and effective

homotopy perturbation method for solving hyperbolic partial differential equations this paper applies the homotopy perturbation method proposed by ji huan he to obtain approximate analytic solutions of hyperbolic partial differential equations the procedure of the method is systematically illustrated to give an extensive account of the method some examples are provided the results derived by this method will be compared with the results of characteristics method the results of homotopy perturbation method are of high accuracy verifying that the method is very effective and promising

application of the harmonic balance method to a nonlinear oscillator typified by a mass attached to a stretched wire abstract the first order harmonic balance method via the first fourier coefficient is used to construct two approximate frequency amplitude relations for a conservative nonlinear oscillatory system in which the restoring force has an irrational form this system corresponds to the motion of a mass attached to a stretched wire two procedures are used to approximately solve the nonlinear differential equation in the first the differential equation is rewritten in a form that does not contain the square root expression while in the second the differential equation is solved directly the approximate frequency obtained using the second procedure is more accurate than the frequency obtained with the first due to the fact that in the second procedure application of the harmonic balance method produces an infinite set of harmonics while in the first procedure only two harmonics are produced both approximate frequencies are valid for the complete range of oscillation amplitudes and excellent agreement of the approximate frequencies with the exact one are demonstrated and discussed the discrepancy between the second approximate frequency and the exact one never exceeds 2 2

weighted kaplan meier statistics a class of distance tests for censored survival data a class of statistics based on the integrated weighted difference in kaplan meier estimators is introduced for the two sample censored data problem with positive weight functions these statistics are intuitive for and sensitive against the alternative of stochastic ordering the standard weighted log rank statistics are not always sensitive against this alternative particularly if the hazard functions cross qualitative comparisons are made between the weighted log rank statistics and these weighted kaplan meier wkm statistics a statement of null asymptotic distribution theory is given and the choice of weight function is discussed in some detail results from small sample simulation studies indicate that these statistics compare favorably with the log rank procedure even under the proportional hazards alternative and may perform better than it under the crossing hazards alternative

regression analysis of multivariate incomplete failure time data by modeling marginal distributions abstract many survival studies record the times to two or more distinct failures on each subject the failures may be events of different natures or may be repetitions of the same kind of event in this article we consider the regression analysis of such multivariate failure time observations each marginal distribution of the failure times is formulated by a cox proportional hazards model no specific structure of dependence among the distinct failure times on each subject is imposed the regression parameters in the cox models are estimated by maximizing the failure specific partial likelihoods the resulting estimators are shown to be asymptotically jointly normal with a covariance matrix that can be consistently estimated simultaneous inferential procedures are then proposed extensive monte carlo studies indicate that the normal approximation is adequate for practical use the new methods allow time dependent covariates missing observations and arbitrary patterns of censorship they are illustrat

application of the modified frequency formulation to a nonlinear oscillator he s frequency formulation is used to obtain the relationship between the frequency and amplitude of a nonlinear oscillator the general approach is to choose two linear oscillators in this paper however one linear oscillator and the duffing oscillator are chosen as trial equations the solution procedure is of utter simplicity while the result is of high accuracy

comparison of homotopy perturbation method and homotopy analysis method comparison of homotopy perturbation method hpm and homotopy analysis method is made revealing that the former is more powerful than the later furthermore the hpm is further developed in this paper by applying the modern perturbation methods

the homotopy perturbation method and analytical solution of the problem of flow past a rotating disk analytical solutions have been obtained for the problem of steady laminar flow of a viscous incompressible fluid past a rotating disk using the homotopy perturbation method hpm the single parameter hpm is attempted first since it does not give adequately accurate results the extended hpm is invoked next in which the independent variable is scaled and the scale factor is expanded in a power series in p the homotopy parameter the coefficients in this power series are calculated by ensuring that the resulting solution is free of secular terms it is shown that the extended hpm solution converges to the true solution and that just eight terms in the perturbation expansion are sufficient to produce a highly accurate yet fully analytical solution

checking the cox model with cumulative sums of martingale based residuals summary this paper presents a new class of graphical and numerical methods for checking the adequacy of the cox regression model the procedures are derived from cumulative sums of martingale based residuals over follow up time and or covariate values the distributions of these stochastic processes under the assumed model can be approximated by zero mean gaussian processes each observed process can then be compared both visually and analytically with a number of simulated realizations from the approximate null distribution these comparisons enable the data analyst to assess objectively how unusual the observed residual patterns are special attention is given to checking the functional form of a covariate the form of the link function and the validity of the proportional hazards assumption an omnibus test consistent against any model misspecification is also studied the proposed techniques are illustrated with two real data sets

nonparametric estimation of the gap time distributions for serial events with censored data summary in many follow up studies each subject can potentially experience a series of events which may be repetitions of essentially the same event or may be events of entirely diverent natures this paper provides a simple nonparametric estimator for the multivariate distribution function of the gap times between successive events when the follow up time is subject to right censoring the estimator is consistent and upon proper normalisation converges weakly to a zero mean gaussian process with an easily estimated covariance function numerical studies demonstrate that both the distribution function estimator and its covariance function estimator perform well for practical sample sizes an application to a colon cancer study is presented

an ancient chinese mathematical algorithm and its application to nonlinear oscillators an ancient chinese mathematical method is briefly introduced and its application to nonlinear oscillators is elucidated where he s amplitude frequency formulation is outlined three examples are given to show the extremely simple solution procedure and remarkably accurate solutions

nonparametric analysis of bivariate gap time with competing risks this article considers nonparametric methods for studying recurrent disease and death with competing risks we first point out that comparisons based on the well known cumulative incidence function can be confounded by different prevalence rates of the competing events and that comparisons of the conditional distribution of the survival time given the failure event type are more relevant for investigating the prognosis of different patterns of recurrence disease we then propose nonparametric estimators for the conditional cumulative incidence function as well as the conditional bivariate cumulative incidence function for the bivariate gap times that is the time to disease recurrence and the residual lifetime after recurrence to quantify the association between the two gap times in the competing risks setting a modified kendall s tau statistic is proposed the proposed estimators for the conditional bivariate cumulative incidence distribution and the association measure account for the induced dependent censoring for the second gap time uniform consistency and weak convergence of the proposed estimators are established hypothesis testing procedures for two sample comparisons are discussed numerical simulation studies with practical sample sizes are conducted to evaluate the performance of the proposed nonparametric estimators and tests an application to data from a pancreatic cancer study is presented to illustrate the methods developed in this article

approximate solutions for nonlinear oscillation of a mass attached to a stretched elastic wire in this paper the approximate solutions of the mathematical model of a mass attached to a stretched elastic wire are presented at the beginning of the study the equation of motion is derived in a detailed way he s max min approach he s frequency amplitude method and the parameter expansion method are implemented to solve the established model the numerical results are further compared with the approximate analytical solutions for both a small and large amplitude of oscillations and a very good agreement is observed the relative errors are computed to illustrate the strength of agreement between the numerical and approximate analytical results

homotopy perturbation technique abstract the homotopy perturbation technique does not depend upon a small parameter in the equation by the homotopy technique in topology a homotopy is constructed with an imbedding parameter p 0 1 which is considered as a small parameter some examples are given the approximations obtained by the proposed method are uniformly valid not only for small parameters but also for very large parameters

solution of nonlinear equations by an ancient chinese algorithm in this paper we re study an ancient chinese algorithm the ying buzu shu in about second century bc known as the rule of double false position in west after 1202 ad the method can powerfully applied to solve nonlinear equations some examples are given

on the homotopy perturbation method for boussinesq like equations abstract we comment on some analytical solutions to a class of boussinesq like equations derived recently by means of the homotopy perturbation method hpm we show that one may obtain exactly the same result by means of the taylor series in the time variable we derive more general results by means of traveling waves and argue that a curious superposition principle proposed by the author may not be of any mathematical or physical significance

an improvement to homotopy perturbation method for solving system of linear equations in this paper we present an efficient numerical algorithm to find exact solutions for the system of linear equations based on homotopy perturbation method hpm a reliable modification is proposed and the modified method is employed to solve the system of linear equations the results are compared with those obtained by the original homotopy perturbation method two examples are given to illustrate the ability and reliability of the modified method the results reveal that the modified method is very simple and effective

a model of the calogero type in the d dimensional space in a previous paper we have proposed a new integrable hamiltonian describing two interacting particles in a harmonic mean field in d 1 dimensional space here we generalize this hamiltonian to the d 2 dimensional space we show that the system is exactly solvable for a certain domain of the interaction constant and that the size of this domain increases with d we also show this d dimensional hamiltonian to be supersymmetric and shape invariant very much as in the d 1 case

the homotopy perturbation method for nonlinear oscillators with discontinuities in the paper the homotopy perturbation method is applied to the nonlinear oscillators with discontinuities only one iteration leads to high accuracy of the solutions

analysis of a nonlinear oscillator with discontinuity in this paper a nonlinear oscillator with discontinuity is analyzed using he s amplitude frequency formulation the natural frequency of the oscillator system can be obtained using concise and straightforward calculation steps good agreement between the results from the proposed and other classical approaches is reached

a new family of iterative methods for solving system of nonlinear algebric equations abstract homotopy perturbation method hpm is applied to construct a new iterative method for solving system of nonlinear algebric equations comparison of the result obtained by the present method with that obtained by revised adomian decomposition method hossein jafari varsha daftardar gejji appl math comput 175 2006 1 7 reveals that the accuracy and fast convergence of the new method

a class of superintegrable systems of calogero type we show that the three body calogero model with inverse square potentials can be interpreted as a maximally superintegrable and multiseparable system in euclidean three space as such it is a special case of a family of systems involving one arbitrary function of one variable

he s frequency amplitude formulation for the duffing harmonic oscillator he s frequency amplitude formulation is used to solve the duffing harmonic oscillator problem the solution procedure is simple and the result obtained is valid for the whole solution domain with high accuracy

a new model of the calogero type we propose a new integrable hamiltonian describing two interacting particles in a harmonic mean field in d 1 dimensional space this model is found to be both supersymmetric and shape invariant we show that for a given domain of the coupling constant the irregular solution is acceptable and contributes to the spectrum we also discuss two inequalities of the bertlmann martin type which link the ground state mean square radius to the lowest excitation energy

determination of natural frequencies of fluid conveying pipes using homotopy perturbation method in the present paper the natural frequencies of fluid conveying pipes with pinned pinned boundary condition are derived explicitly in a systematical and straightforward way with the help of homotopy perturbation method numerical results are presented for two cases and the effect of fluid flow velocity on the natural frequencies is discussed good agreement with their experimental and fem counterparts is found numerically over ranges of practical interest

travelling wave solutions of two nonlinear evolution equations by using the g g expansion method abstract by using the g g expansion method proposed recently we give the exact travelling wave solutions of two different types of nonlinear evolution equations in this paper the travelling wave solutions are expressed by three types of functions the computation for the method appears to be easier and faster by general mathematical software

he s frequency amplitude formulation for nonlinear oscillators with an irrational force in this paper he s frequency amplitude formulation is applied to determine the periodic solution for a nonlinear oscillator system with an irrational force comparison with the exact solution shows that the result obtained is of high accuracy

some algebraically solvable three body problems in one dimension the three body problem in one dimension with a repulsive inverse square potential between every pair was solved by calogero after mapping the three body problem to that of a particle on a plane the known results of supersymmetric quantum mechanics are used to solve this problem as well as a number of new ones algebraically this general technique is applicable when the potential is separable in the radial and angular variables on the plane and its supersymmetric partner is shape invariant after discussing one example in detail an exhaustive list of such exactly solvable potentials is given

the travelling wave solutions for non linear initial value problems using the homotopy perturbation method in this article we have used the homotopy perturbation method hpm to find the travelling wave solutions for some non linear initial value problems in the mathematical physics these problems consist of the burgers fisher equation the kuramoto sivashinsky equation the coupled schordinger kdv equations and the long short wave resonance equations together with initial conditions the results of these problems reveal that the hpm is very powerful effective convenient and quite accurate to the systems of non linear equations it is predicted that this method can be found widely applicable in engineering and physics

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 study of new solvable few body problems we study new solvable few body problems consisting of generalizations of the calogero and the calogero marchioro wolfes three body problems by introducing non translationally invariant three body potentials after separating the radial and angular variables by appropriate coordinate transformations we provide eigensolutions of the schrodinger equation with the corresponding energy spectrum we found a domain of the coupling constant for which the irregular solutions are square integrable

he s variational approach for nonlinear oscillators with high nonlinearity in this paper he s variational approach is applied to the strongly nonlinear oscillators with high nonlinearity some examples are given to illustrate the effectiveness and convenience of this method comparison with the exact solutions is also made revealing that the present method leads to accurate solutions

new applications of variational iteration method abstract the variational iteration method is used for solving three types of nonlinear partial differential equations such as coupled schrodinger kdv generalized kdv and shallow water equations the exact and numerical solutions obtained by variational iteration method are compared with that obtained using adomian decomposition method the comparison shows that the two solutions obtained are excellent agreement the method is made showing that the former is more effective than the latter in this paper he s variational iteration method is introduced to overcome the difficulty arising in calculating adomian polynomials

a new perturbative approach to nonlinear problems a recently proposed perturbative technique for quantum field theory consists of replacing nonlinear terms in the lagrangian such as 4 by 2 1 and then treating as a small parameter it is shown here that the same approach gives excellent results when applied to difficult nonlinear differential equations such as the lane emden thomas fermi blasius and duffing equations

application of he s methods to nonlinear chemistry problems in this paper he s variational iteration method vim and homotopy perturbation method hpm are implemented for solving analytically two systems of ordinary differential equations that often appear in chemical applications in both methods the initial approximations can be freely chosen with possible unknown constants which can be determined by imposing the boundary and initial conditions the solution procedure is of utter simplicity and the obtained solution is of high accuracy

global error minimization method for solving strongly nonlinear oscillator differential equations a modified variational approach called global error minimization gem method is developed for obtaining an approximate closed form analytical solution for nonlinear oscillator differential equations the proposed method converts the nonlinear differential equation to an equivalent minimization problem a trial solution is selected with unknown parameters next the gem method is used to solve the minimization problem and to obtain the unknown parameters this will yield the approximate analytical solution of the nonlinear ordinary differential equations odes this approach is simple accurate and straightforward to use in identifying the solution to illustrate the effectiveness and convenience of the suggested procedure a cubic duffing equation with strong nonlinearity is considered comparisons are made between results obtained by the proposed gem method the exact solution and results from five recently published methods for addressing duffing oscillators the maximal relative error for the frequency obtained by the gem method compared with the exact solution is 0 0004 which indicates the remarkable precision of the gem method

nonparametric estimation of a recurrent survival function abstract recurrent event data are frequently encountered in studies with longitudinal designs let the recurrence time be the time between two successive recurrent events recurrence times can be treated as a type of correlated survival data in statistical analysis in general because of the ordinal nature of recurrence times statistical methods that are appropriate for standard correlated survival data in marginal models may not be applicable to recurrence time data specifically for estimating the marginal survival function the kaplan meier estimator derived from the pooled recurrence times serves as a consistent estimator for standard correlated survival data but not for recurrence time data in this article we consider the problem of how to estimate the marginal survival function in nonparametric models a class of nonparametric estimators is introduced the appropriateness of the estimators is confirmed by statistical theory and simulations simulation and analysis from schizophrenia data are pre

homotopy perturbation transform method for nonlinear equations using he s polynomials in this paper a combined form of the laplace transform method with the homotopy perturbation method is proposed to solve nonlinear equations this method is called the homotopy perturbation transform method hptm the nonlinear terms can be easily handled by the use of he s polynomials the proposed scheme finds the solution without any discretization or restrictive assumptions and avoids the round off errors the fact that the proposed technique solves nonlinear problems without using adomian s polynomials can be considered as a clear advantage of this algorithm over the decomposition method