distinct variants of the kdv equation with compact and noncompact structures in this paper we examine three distinct variants of the generalized kdv equation both the compact and the noncompact structures are treated for these variants we also emphasize the different physical structures of the two classes the focusing branch and the defocusing branch

a renewal theorem and its applications to some sequential procedures a renewal theorem of the elementary type for some stopping times which arise from some statistical estimation problems has been established it is applied to prove the asymptotic risk efficiency for the problem of estimating the mean when the loss function is a weighted sum of the squared error and the sample size and the variance is unknown it is also applied to verify a conjecture of robbins and siegmund 1974 on the evaluation of the variance of the estimator of the logarithm of the odds ratio for a sequential procedure

the tanh and the sine cosine methods for the complex modified k dv and the generalized k dv equations the complex modified k dv cmk dv equation and the generalized k dv equation are investigated by using the tanh method and the sine cosine method a variety of exact travelling wave solutions with compact and noncompact structures are formally obtained for each equation the study reveals the power of the two schemes where each method complements the other

testing violations of the exponential assumption in cancer clinical trials with survival endpoints personalized cancer therapy requires clinical trials with smaller sample sizes compared to trials involving unselected populations that have not been divided into biomarker subgroups the use of exponential survival modeling for survival endpoints has the potential of gaining 35 efficiency or saving 28 required sample size miller 1983 making personalized therapy trials more feasible however the use of exponential survival has not been fully accepted in cancer research practice due to uncertainty about whether or not the exponential assumption holds we propose a test for identifying violations of the exponential assumption using a reduced piecewise exponential approach compared with an alternative goodness of fit test which suffers from inflation of type i error rate under various censoring mechanisms the proposed test maintains the correct type i error rate we conduct power analysis using simulated data based on different types of cancer survival distribution in the seer registry database and demonstrate the implementation of this approach in existing cancer clinical trials

sequential determination of estimator as well as sample size on considere certains problemes d estimation ponctuelle ou par intervalles et on propose des methodes sequentielles pour le choix d un estimateur et de la taille de l echantillon sur lequel sera base l estimateur

a nonlinear renewal theory soit t le premier instant auquel une marche aleatoire perturbee traverse une frontiere non lineaire on considere les approximations de la distribution de l exces au dela de la frontiere de l esperance du temps d arret et de la variance du temps d arret

sizing clinical trials when comparing bivariate time to event outcomes clinical trials with multiple primary time to event outcomes are common use of multiple endpoints creates challenges in the evaluation of power and the calculation of sample size during trial design particularly for time to event outcomes we present methods for calculating the power and sample size for randomized superiority clinical trials with two correlated time to event outcomes we do this for independent and dependent censoring for three censoring scenarios i the two events are non fatal ii one event is fatal semi competing risk and iii both are fatal competing risk we derive the bivariate log rank test in all three censoring scenarios and investigate the behavior of power and the required sample sizes separate evaluations are conducted for two inferential goals evaluation of whether the test intervention is superior to the control on 1 all of the endpoints multiple co primary or 2 at least one endpoint multiple primary copyright 2017 john wiley sons ltd

a note on the variance of a stopping time let sn e xi n 0 be a random walk with positive drift t ex1 0 and finite variance a2 varxl let t b inf n 2 1 sn b rb st b b m minn osn t t o and h s lai and siegmund show that var t b ba2 1t3 k m2 o 1 as b oo but give an unpleasant expression for the constant k using the identity jeh r y dp m y et h h et the expression for k can be simplified to a form that depends only on moments of ladder variables

estimation in the general linear model when the accuracy is specified before data collection an estimator of is accurate with accuracy a and confidence ay o y 1 if p bfl ea y for all 3 given a sequence y1 y2 of independent vector valued homoscedastic normally distributed random variables generated via the general linear model yi xi e the k dimensional parameter d3 is accurately estimated using a sequential version of the maximum probability estimator developed by l weiss and j wolfowitz the procedure given also generalizes c stein s fixed width confidence sets to several dimensions

asymptotic optimality of invariant sequential probability ratio tests it is well known that wald s sprt for testing simple hypotheses based on i i d observations minimizes the expected sample size both under the null and under the alternative hypotheses among all tests with the same or smaller error probabilities and with finite expected sample sizes under the two hypotheses in this paper it is shown that this optimum property can be extended at least asymptotically as the error probabilities tend to 0 to invariant sprts like the sequential t test the savage sethuraman sequential rank order test etc in fact not only do these invariant sprts asymptotically minimize the expected sample size but they also asymptotically minimize all the moments of the sample size distribution among all invariant tests with the same or smaller error probabilities modifications of these invariant sprts to asymptotically minimize the moments of the sample size at an intermediate parameter are also considered

small sample properties of censored data rank tests small sample properties of censored data rank tests such as mantel s logrank test the breslow test or the general dass of tests proposed by peto and peto 1972 journal oj the royal statistical society series a 135 185 206 are examined it is found that the latter dass of tests in particular can be substantially nonconservative when applied to small sam pies the most serious inaccuracies occur in unbalanced trials when higher event rates are inferred in the smaller sampie

sequential estimation for the autocorrelations of linear processes this paper considers sequential point estimation of the autocorrelations of stationary linear processes within the framework of the sequential procedure initiated by robbins the sequential estimator proposed here is based on the usual sample autocorrelations and is shown to be risk efficient in the sense of starr as the cost per observation approaches zero to achieve the asymptotic risk efficiency we are led to study the uniform integrability and random central limit theorem of the sample autocorrelations some moment conditions are provided for the errors of the linear processes to establish the uniform integrability and random central limit theorem

modal hermite spectral collocation method for solving multi dimensional hyperbolic telegraph equations abstract the present research is contemplated proposing a numerical solution of multi dimensional hyperbolic telegraph equations with appropriate initial time and boundary space conditions the truncated hermite series with unknown coefficients are used for approximating the solution in both of the spatial and temporal variables the basic idea for discretizing the considered one dimensional 1d two dimensional 2d and three dimensional 3d telegraph equations is based on the collocation method together with the hermite operational matrices of derivatives the resulted systems of linear algebraic equations are solved by some efficient methods such as lu factorization the solution of the algebraic system contains the coefficients of the truncated hermite series numerical experiments are provided to illustrate the accuracy and efficiency of the presented numerical scheme comparisons of numerical results associated to the proposed method with some of the existing numerical methods confirm that the method is accurate and fast experimentally

homotopy perturbation method a new nonlinear analytical technique in this paper a new perturbation method is proposed in contrast to the traditional perturbation methods this technique does not require a small parameter in an equation in this method according to the homotopy technique a homotopy with an imbedding parameter email protected 0 1 is constructed and the imbedding parameter is considered as a small parameter so the method is called the homotopy perturbation method which can take the full advantages of the traditional perturbation methods and homotopy techniques to illustrate its effectiveness and its convenience a duffing equation with high order of nonlinearity is used the result reveals that its first order of approximation obtained by the proposed method is valid uniformly even for very large parameter and is more accurate than the perturbation solutions

sequential estimation results for a two parameter exponential family of distributions we consider the problem of sequentially estimating one parameter in a class of two parameter exponential family of distributions we assume a weighted squared error loss with a fixed cost of estimation error the stopping rule based on the maximum likelihood estimate of the nuisance parameter is shown to be independent of the terminal estimate the asymptotic normality of the stopping variable is established and approximations to its mean and to the regret associated with it are also provided the general results are exemplified by the normal gamma and the inverse gaussian densities

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

negative regret optional stopping and the elimination of outliers abstract this article examines the phenomenon of negative regret in sequential point estimation in which a stopping rule used in ignorance of the variance of the distribution sometimes performs better than the best fixed sample size when the variance is known an explanation in terms of a certain robustifying effect of the stopping rule is given results of simulation studies illustrating both negative regret and its explanation are also given suppose x 1 x 2 are iid with mean and variance 2 one may stop observing the sequence of x i s after any number of observations n and estimate by the sample mean subject to the loss if is known then the best fixed i e nonrandom sample size in the sense of minimum risk can be used if is unknown the best fixed sample size is unknown necessitating a sequential procedure for a sequential procedure of the type first proposed by robbins 1959 the regret expected additional loss from not knowing and using the sequential procedure instead

second order approximation to the risk of a sequential procedure given x 1 x 2 cdots i i d with mean mu and variance sigma 2 suppose that at stage n one wishes to estimate mu by the sample mean bar x n subject to the loss function l n a bar x n mu 2 n a 0 if sigma is known the optimal fixed sample size n 0 approx a 1 2 sigma can be used with corresponding risk r n 0 but if sigma is unknown there is no fixed sample size procedure that will achieve the risk r n 0 for the sequential estimation procedure with stopping rule t inf n geq n a n 1 sum n 1 x i bar x n 2 leq a 1 n 2 the second order approximation of woodroofe 1977 to the risk r t for normal x i is extended to the distribution free case specifically if the x i have finite moments of order greater than eight and are non lattice under certain conditions on the delay n a it is shown that the regret r t r n 0 c o 1 as a rightarrow infty where c depends on the first four moments of the distribution of the x i for the lattice case bounds of the form c 1 o 1 leq r t r n 0 leq c 2 o 1 are obtained where the c j are c pm 3 it follows from these approximations that the regret can take arbitrarily large negative values as the distribution of the x i varies in contrast to previous results for normal and gamma cases

exact log rank tests for unequal follow up les tests asymptotiques du log rank et le test de wilcoxon generalise sont des procedures standards pour comparer des distributions de survie en presence de censures pour la comparaison d echantillons de tailles tres differentes un test exact base sur l ensemble des permutations du log rank ou des scores de wilcoxon est disponible alors que les tests asymptotiques ne conservent pas leur risque nominal lorsque les echantillons sont de tailles tres differentes le test exact base sur les permutations exige cependant un suivi identique dans les echantillons c est pourquoi nous avons developpe deux nouveaux tests exacts adaptes a des suivis inegaux le premier est exactement equivalent au log rank test asymptotique conditionnellement a l ensemble des risques observes alors que la deuxieme approche permute les temps de survie conditionnellement au suivi observe dans chaque groupe dans une etude empirique nous comparons ces nouvelles procedures au log rank test asymptotique au test exact de permutation complete ainsi qu a une approche proposee anterieurement qui egalise les distributions de suivi en les censurant artificiellement les resultats confirment les performances tres satisfaisantes du test exact conditionnellement au suivi observe particulierement dans le cas de suivis inegaux l avantage du nouveau test sur les autres possibilites d analyse est finalement illustre par l analyse d une etude portant sur le cancer du sein

convergence analysis of an iterative scheme for solving initial value problem for multidimensional partial differential equations existence and uniqueness of solutions of initial value problem for certain classes of multidimensional linear and nonlinear partial differential equations are proved using banach fixed point theorem an iterative scheme is derived and rigorous convergence analysis of this scheme and an error estimate are included as well several numerical examples for high dimensional initial value problem for heat and wave type partial differential equations are presented to demonstrate reliability and performance of proposed iterative scheme

variational iteration method new development and applications variational iteration method has been favourably applied to various kinds of nonlinear problems the main property of the method is in its flexibility and ability to solve nonlinear equations accurately and conveniently in this paper recent trends and developments in the use of the method are reviewed major applications to nonlinear wave equation nonlinear fractional differential equations nonlinear oscillations and nonlinear problems arising in various engineering applications are surveyed the confluence of modern mathematics and symbol computation has posed a challenge to developing technologies capable of handling strongly nonlinear equations which cannot be successfully dealt with by classical methods variational iteration method is uniquely qualified to address this challenge the flexibility and adaptation provided by the method have made the method a strong candidate for approximate analytical solutions this paper outlines the basic conceptual framework of variational iteration technique with application to nonlinear problems both achievements and limitations are discussed with direct reference to approximate solutions for nonlinear equations a new iteration formulation is suggested to overcome the shortcoming a very useful formulation for determining approximately the period of a nonlinear oscillator is suggested examples are given to illustrate the solution procedure

a renewal theory with varying drift let r be the excess over the boundary in renewal theory it is well known that er has a limit r when the drift of the random walk 1 0 we study renewal theorems with varying u conditions are given under which the tail er r is uniformly dominated by a decreasing integrable function for u in a compact interval in 0 oc conditions are also given under which the derivative of the tail d ldl er r is uniformly dominated by a directly riemann integrable function

lie symmetry analysis and conservation law of variable coefficient davey stewartson equation abstract a variable coefficient davey stewartson vcds equation is investigated in this paper infinitesimal generators and symmetry groups are presented by the lie group method and the optimal system is presented with adjoint representation based on the optimal system similarity reductions to partial differential equations pdes are obtained then some pdes are reduced to ordinary differential equations odes by two dimensional subalgebras and the similarity solutions are provided including periodic solutions and elliptic function solutions with lagrangian it is shown that vcds is nonlinearly self adjoint furthermore based on nonlinear self adjointness conservation laws for vcds equation are derived

solitary wave solutions of nonlinear evolution and wave equations using a direct method and macsyma the direct algebraic method for constructing travelling wave solutions on nonlinear evolution and wave equations has been generalized and systematized the class of solitary wave solutions is extended to analytic rather than rational functions of the real exponential solutions of the linearized equation expanding the solutions in an infinite series in these real exponentials an exact solution of the nonlinear pde is obtained whenever the series can be summed methods for solving the nonlinear recursion relation for the coefficients of the series and for summing the series in closed form are discussed the algorithm is now suited to solving nonlinear equations by any symbolic manipulation program this direct method is illustrated by constructing exact solutions of a generalized kdv equation the kuramoto sivashinski equation and a generalized fisher equation

sequentially deciding between two experiments for estimating a common success probability abstract to estimate a success probability p two experiments are available individual bernoulli p trials or the product of individual bernoulli p trials this problem has its roots in reliability where either single components can be tested or a system of identical components can be tested a total of n experiments can be performed and the problem is to sequentially select some combination allocation of these two experiments along with an estimator of p to achieve low mean squared error mse of the final estimate this scenario is similar to that of the better known group testing problem but here the goal is to estimate failure rates rather than to identify defective units the problem also arises in epidemiological applications such as estimating disease prevalence information maximization considerations and analysis of the asymptotic mse of several estimators lead to the following adaptive procedure use the maximum likelihood estimator mle to estimate p and if this estimator is b

sequential estimation of the mean of a first order stationary autoregressive process on considere un estimateur ponctuel sequentiel base sur une moyenne d echantillon et on montre qu il est asymptotiquement efficace en risque lorsque le cout par observation tend vers zero

an approximation to the solution of hyperbolic equations by adomian decomposition method and comparison with characteristics method adomian decomposition method has been applied to solve many functional equations so far in this work this method is applied to solve hyperbolic partial differential equations and results will be compared with characteristics method

withdrawal censoring and the sequential logrank procedure the effects of withdrawal censoring on the power of the group sequential logrank procedure are examined in the context of survival studies special attention is given to an assumption that the withdrawal censoring mechanism is only conditionally independent of survival time simulation studies of two sample fixed duration trials under random right censoring staggered entry and exponential life times reveal a general decrease in statistical power that is most pronounced when the withdrawals are unbalanced the majority occurring in the group with the higher hazard rate generally this decrease in power is not of practical concern under typical withdrawal rates except when conditional independence is present in which case large fluctuations in power and type i error rates can occur

a sequential procedure with asymptotically negative regret for estimating a normal mean rn e ln acr2 n n is minimized by taking n n ao 2 1 2 and the corresponding minimum fixed sample size risk is rn 2n since u2 is unknown robbins 1959 proposed the following type of sequential procedure let 1 1 t inf n m n avn n 1 2 i where m is the initial sample size and vn e n xn 2 and estimate u by xt woodroofe 1977 showed that if m 4 the regret of the procedure rt rn 1 2 o 1 as a oc the purpose of this article is to show that there does exist a sequential procedure for which the regret is negative at u 0 and 1 2 at u 0 asymptotically instead of vn n as an estimator of j 2 consider an estimator o n of the form

fixed accuracy estimation of an autoregressive parameter abstract for a first order non explosive autoregressive process with unknown parameter beta epsilon 1 1 it is shown that if data are collected according to a particular stopping rule the least squares estimator of beta is asymptotically normally distributed uniformly in beta in the case of normal residuals the stopping rule may be interpreted as sampling until the observed fisher information reaches a preassigned level the situation is contrasted with the fixed sample size case where the estimator has a non normal limiting distribution when beta 1 author

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

non parametric analysis of treatment covariate interaction in the presence of censoring the demonstration of varying treatment efficacy among different subsets of patients is an important part of the analysis of clinical trials the paper commences by clarifying the meaning of interaction and by reviewing valid procedures of analysis in the presence of interaction suitable descriptive measures of treatment covariate interactions are ratios of hazard ratios to which generalized patel hoel tests for qualitative or ordinal categorical covariates are related both the hazard ratios and the tests are given in a formulation for use with different scoring systems such as u or e furthermore the procedures are related to common k sample tests and thus are suitable for proportional and for converging hazards a worked example is included

a two sample censored data rank test for acceleration a score test for the null hypothesis of proportional hazards against rank regression alternatives is proposed as a complement to the logrank test for comparing censored survival curves the test statistic has an asymptotic normal distribution that is independent of the logrank distribution under the null hypothesis and its power is good against acceleration alternatives i e with crossing hazards where the logrank test fails monte carlo studies indicate that its small sample properties are comparable to those of the logrank and ranksum procedures

on guaranteed estimation of the mean of an autoregressive process this paper considers the problem of sequential point estimation of the mean of a stable autoregressive process with unknown scale and autoregressive parameters the construction ofa sequential procedure makes use of special stopping rules and some modifications of least squares estimates the procedure enables estimating the mean with prescribed mean square accuracy uniformly in nuisance parameters the uniform asymptotic normality and the asymptotic minimaxity of the sequential estimate are established the asymptotic formula for the mean sample size is obtained

statistics of natural populations iii sequential sampling plans for the estimation of gene frequencies in previous work several models have been developed for genetic surveys of natural populations parents of unknown genotype are collected from a natural population polymorphic at a single genetic locus from each of these n cryptic parents a number of offspring are identified for their genotype our problem is to select an efficient offspring sampling plan for estimating the frequency of an allele in the cryptic adult population based on the n family profiles of juvenile genotypes a criterion called the information per unit cost of observation is introduced to evaluate sequential sampling plans in which the number of offspring per family examined is random some simple practical schemes for stopping the sampling of offspring from a collected parent are introduced one example is stopping when i the offspring are definitive about the parental genotype s for the first time ii a fixed number of one genotype only is seen or iii a fixed maximum feasible number of offspring have been genotyped this sampling scheme is recommended for each sampling scheme the best linear unbiased estimator and the sequential maximum likelihood estimator of the allele frequency are characterized from the moments of these estimators it is then possible to tabulate efficient sequential sampling plans which are better in the sense of information per unit cost just as simple and less costly than corresponding fixed sampling plans in use

time sequential point estimation through estimating equations time sequential point estimation is studied in the model of fully parametric censored data and cox s regression model both are investigated in the context of counting processes through estimating equations defined by martingales the concept of information and a related inequality developed in estimating function theory by godambe are adapted to these models these suggest some optimality criteria for choosing stopping times as well as estimators these lead naturally to some sequential procedures which are shown to be asymptotically efficient with respect to the above criteria

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

an algorithm for solving multi term diffusion wave equations of fractional order in this paper an algorithm based on a new modified homotopy perturbation method mhpm is presented to obtain approximate solutions of multi term diffusion wave equations of fractional order to illustrate the method some examples are provided the results show the simplicity and the efficiency of the algorithm

a monte carlo study of some two sample rank tests with censored data abstract the powers of several nonparametric tests for the two sample problem with censored data are compared by simulation the tests studied include gehan s efron s and peto s and prentice s generalized wilcoxon tests along with the logrank test the test with the greatest power changes with the sample sizes censoring mechanism and distribution of the random variables the test that performed the best overall was the peto prentice generalized wilcoxon statistic with an asymptotic variance estimate

permutational distribution of the log rank statistic under random censorship with applications to carcinogenicity assays in the random censorship model the log rank test is often used for comparing a control group with different dose groups if the number of tumors is small so called exact methods are often applied for computing critical values from a permutational distribution two of these exact methods are discussed and shown to be incorrect the correct permutational distribution is derived and studied with respect to its behavior under unequal censoring in the light of recent results proving that the permutational version and the unconditional version of the log rank test are asymptotically equivalent even under unequal censoring the log rank test is studied by simulations of a realistic scenario from a bioassay with small numbers of tumors

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

on sequential estimation of parameters in semimartingale regression models with continuous time parameter we consider the problem of parameter estimation for multidimensional continuous time linear stochastic regression models with an arbitrary finite number of unknown parameters and with martingale noise the main result of the paper claims that the unknown parameters can be estimated with prescribed mean square precision in this general model providing a unified description of both discrete and continuous time process among the conditions on the regressors there is one bounding the growth of the maximal eigenvalue of the design matrix with respect to its minimal eigenvalue this condition is slightly stronger as compared with the corresponding conditions usually imposed on the regressors in asymptotic investigations but still it enables one to consider models with different behavior of the eigenvalues the construction makes use of a two step procedure based on the modified least squares estimators and special stopping rules

multihypothesis sequential probability ratio tests ii accurate asymptotic expansions for the expected sample size for pt i see ibid vol 45 p 2448 61 1999 we proved in pt i that two specific constructions of multihypothesis sequential tests which we refer to as multihypothesis sequential probability ratio tests msprts are asymptotically optimal as the decision risks or error probabilities go to zero the msprts asymptotically minimize not only the expected sample size but also any positive moment of the stopping time distribution under very general statistical models for the observations in this paper based on nonlinear renewal theory we find accurate asymptotic approximations up to a vanishing term for the expected sample size that take into account the overshoot over the boundaries of decision statistics the approximations are derived for the scenario where the hypotheses are simple the observations are independent and identically distributed i i d according to one of the underlying distributions and the decision risks go to zero simulation results for practical examples show that these approximations are fairly accurate not only for large but also for moderate sample sizes the asymptotic results given here complete the analysis initiated by baum and veeravalli 1994 where first order asymptotics were obtained for the expected sample size under a specific restriction on the kullback leibler distances between the hypotheses

variants of the generalized kdv equation with compact and noncompact structures abstract in this paper nonlinear dispersive variants of the generalized kdv equation are investigated the compact and the noncompact physical structures are studied two general formulas for the focusing branch and the defocusing branch that are of substantial interest are formally derived

general compactons solutions for the focusing branch of the nonlinear dispersive k n n equations in higher dimensional spaces in this paper we will study the focusing branch of the genuinely nonlinear dispersive k n n equation that exhibits compactons solitons with finite wavelengths the equation will be studied in one two and three dimensional spaces general formulas for compactons solutions are developed for all positive integers n n 1 two sets of entirely new formulas one set produces compactons for even integers n whereas the second set produces compactons and anticompactons for odd integers n n 1 are established

exact propagating dromion like localized wave solutions of generalized 2 1 dimensional davey stewartson equations the capacity of the double exponential exp function method as an alternative approach to obtain the analytic solutions of higher dimensional coupled nonlinear partial differential equations in engineering mathematics has been revealed we employ the double exp method aided with symbolic computation to construct dromion like localized solutions for a 2 1 dimensional generalized davey stewartson 2dgds system of equations governing the dynamics of weakly nonlinear modulation of a lattice wave packet in a multi dimensional lattice the multidromion like solutions obtained by the present approach are more general than those obtained earlier

study of convergence of homotopy perturbation method for systems of partial differential equations the aim of this paper is convergence study of homotopy perturbation method for systems of nonlinear partial differential equations the sufficient condition for convergence of the method is addressed since mathematical modeling of numerous scientific and engineering experiments lead to brusselator and burgers system of equations it is worth trying new methods to solve these systems we construct a new efficient recurrent relation to solve nonlinear burgers and brusselator systems of equations comparison of the results obtained by homotopy perturbation method with those of adomian s decomposition method and dual reciprocity boundary element method leads to significant consequences two standard problems are used to validate the method

application of the homotopy perturbation method to zakharov kuznetsov equations the aim of this paper is to apply the homotopy perturbation method hpm to solve the zakharov kuznetsov zk m n k equations the two special cases zk 2 2 2 and zk 3 3 3 are chosen to show the ability of the method general formulas for the solutions of zk m n k are established the results reveal that the method is very effective and simple

system of linear ordinary differential and differential algebraic equations and pseudo spectral method in this paper first we introduce briefly pseudo spectral method to solve linear ordinary differential equations odes and then extend it to solve a system of linear odes and then differential algebraic equations daes furthermore because of appropriate choice of chebyshev gauss raudo points we will show that this method can be used to solve a dae whenever some coefficient functions in constraint are not analytic by providing some numerical examples

asymptotic operating characteristics of an optimal change point detection in hidden markov models let 0 1 1 be observations from the hidden markov model with probability distribution p 0 and let 1 be observations from the hidden markov model with probability distribution p 1 the parameters 0 and 1 are given while the change point is unknown the problem is to raise an alarm as soon as possible after the distribution changes from p 0 to p 1 but to avoid false alarms specifically we seek a stopping rule n which allows us to observe the s sequentially such that e n is large and subject to this constraint sup k e k n k n k is as small as possible here e k denotes expectation under the change point k and e denotes expectation under the hypothesis of no change whatever in this paper we investigate the performance of the shiryayev roberts pollak srp rule for change point detection in the dynamic system of hidden markov models by making use of markov chain representation for the likelihood function the structure of asymptotically minimax policy and of the bayes rule and sequential hypothesis testing theory for markov random walks we show that the srp procedure is asymptotically minimax in the sense of pollak ann statist 13 1985 206 227 next we present a second order asymptotic approximation for the expected stopping time of such a stopping scheme when w 1 motivated by the sequential analysis in hidden markov models a nonlinear renewal theory for markov random walks is also given

sequential detection of targets in multichannel systems it is supposed that there is a multichannel sensor system which performs sequential detection of a target sequential detection is done by implementing a generalized wald s sequential probability ratio test which is based on the maximum likelihood ratio statistic and allows one to fix the false alarm rate and the rate of missed detections at specified levels we present the asymptotic performance of this sequential detection procedure and show that it is asymptotically optimal in the sense of minimizing the expected sample size when the probabilities of erroneous decisions are small we do not assume that the observations are independent and identically distributed i i d the first order asymptotic optimality result holds for general statistical models that are not confined to the restrictive i i d assumption however for i i d and quasi i i d cases where the log likelihood ratios can be represented in the form of sums of random walks and slowly changing sequences we obtain much stronger results specifically using the nonlinear renewal theory we are able to obtain both tight expressions for the error probabilities and higher order approximations for the average sample size up to a vanishing term the performance of the multichannel sequential detection algorithm is illustrated by an example of detection of a deterministic signal in correlated colored gaussian noise in this example we provide both the results of theoretical analysis and the results of a monte carlo experiment these results allow us to conclude that the use of the sequential detection algorithm substantially reduces the required resources of the system compared to the best nonsequential algorithm

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

convergence and error estimation of homotopy perturbation method for fredholm and volterra integral equations abstract in this paper an application of the homotopy perturbation method for solving fredholm and volterra integral equations of the second kind is presented discussed method consists in constructing the functional series sum of which determines the function giving the solution of considered problem conditions under which the constructed series is convergent are formulated and proved in the paper in general the composed series is rather fast convergent thanks to which calculation of few first terms ensures a very good approximation of the sought solution estimation of errors of approximate solution obtained by taking the partial sum of the series is also elaborated in the paper

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

a frequentistic approach to sequential estimation in the general linear model abstract independent observations y 1 y 2 are generated via the general linear model yi xi i x i being either a random or a sequentially designed matrix the parameter is estimated sequentially using the least squares estimator subject to a loss structure that is the sum of a cost due to sampling and a loss due to estimation error if 1 2 are iid n 0 2 i with unknown no fixed sample procedure will minimize the risk for all 0 the sequential procedure presented here performs nearly as well as the constant risk best fixed sample procedure when 2 is known its risk and the regret in not knowing 2 are calculated up to terms that are o c as the cost of one observation c 0 or as the asymptotic moments of the stopping time are given and the normalized stopping time s limiting distribution is shown to be normal its first two moments are calculated up to second order terms

the tanh method for traveling wave solutions of nonlinear equations in this work we employ the tanh method for traveling wave solutions of nonlinear equations the study is extended to equations that do not have tanh polynomial solutions the efficiency of the method is demonstrated by applying it for a variety of selected equations

sequential confidence bands for densities this paper proposes a fully sequential procedure for constructing a fixed width confidence band for an unknown density on a finite interval and shows the procedure has the desired coverage probability asymptotically as the width of the band approaches zero the procedure is based on a result of bickel and rosenblatt its implementation in the sequential setting cannot be obtained using anscombe s theorem because the normalized maximal deviations between the kernel estimate and the true density are not uniformly continuous in probability u c i p instead we obtain a slightly weaker version of the u c i p property and a correspondingly stronger convergence property of the stopping rule these together yield the desired results