an efficient algorithm for the multivariable adomian polynomials abstract in this article the sum of the series of multivariable adomian polynomials is demonstrated to be identical to a rearrangement of the multivariable taylor expansion of an analytic function of the decomposition series of solutions u 1 u 2 u m about the initial solution components u 1 0 u 2 0 u m 0 of course the multivariable adomian polynomials were developed and are eminently practical for the solution of coupled nonlinear differential equations the index matrices and their simplified forms of the multivariable adomian polynomials are introduced we obtain the recurrence relations for the simplified index matrices which provide a convenient algorithm for rapid generation of the multivariable adomian polynomials another alternative algorithm for term recurrence is established in these algorithms recurrence processes do not require complicated operations such as parametrization expanding and regrouping derivatives etc as practiced in prior art the mathematica program generating the adomian polynomials based on the algorithm in this article is designed

estimating varying coefficients for partial differential equation models partial differential equations pdes are used to model complex dynamical systems in multiple dimensions and their parameters often have important scientific interpretations in some applications pde parameters are not constant but can change depending on the values of covariates a feature that we call varying coefficients we propose a parameter cascading method to estimate varying coefficients in pde models from noisy data our estimates of the varying coefficients are shown to be consistent and asymptotically normally distributed the performance of our method is evaluated by a simulation study and by an empirical study estimating three varying coefficients in a pde model arising from lidar data

parameter estimation for differential equations a generalized smoothing approach summary we propose a new method for estimating parameters in models that are defined by a system of non linear differential equations such equations represent changes in system outputs by linking the behaviour of derivatives of a process to the behaviour of the process itself current methods for estimating parameters in differential equations from noisy data are computationally intensive and often poorly suited to the realization of statistical objectives such as inference and interval estimation the paper describes a new method that uses noisy measurements on a subset of variables to estimate the parameters defining a system of non linear differential equations the approach is based on a modification of data smoothing methods along with a generalization of profiled estimation we derive estimates and confidence intervals and show that these have low bias and good coverage properties respectively for data that are simulated from models in chemical engineering and neurobiology the performance of the method is demonstrated by using real world data from chemistry and from the progress of the autoimmune disease lupus

the theoretical foundation of the adomian method abstract in this paper we work out the theoretical basis of the adomian method this powerful method that g adomian has developed in the beginning of the 1980 s supplies analytical approximations to the solution of many kinds of equations thanks to our decomposition theory we justify the practical method and we generalize the convergence results obtained by y cherruault afterward we explain how to solve rigorously equations with linear nonlinear or composed operator and also equations with several terms or with a second member

comparison between adomian s method and he s homotopy perturbation method in this paper it is revealed that modified form of he s homotopy perturbation method corresponds to adomian s decomposition method for certain nonlinear problems

the integration of stiff initial value problems in odes using modified extended backward differentiation formulae abstract a class of modified extended backward differentiation formulae mebdf suitable for the approximate numerical integration of first order systems of stiff ordinary differential equations is introduced the computational aspects of this new class of formulae are examined in detail in particular algorithms for varying both order and stepsize are given and this leads to a variable step variable order process using highly stable formulae of order 2 8 extensive numerical results for the well known detest set are given and on this basis a comparison is made between a code incorporating mebdf a code due to hindmarsh based on conventional backward differentiation formulae and a code due to skeel and kong based on blended linear multistep methods it is shown that on this test set the mebdf code is very reliable and is competitive with the other two codes on a significant class of problems a computer listing of the mebdf code used to obtain the results presented in this paper is available from the author

type insensitive ode codes based on implicit a stable formulas a special concept of stiffness is appropriate for implict a stable formulas it is possible to recognize this kind of stiffness economically and reliably using information readily available during the integration of an ode using this development a variety of effective ode solvers could be made insensitive to the type of problem i e the code would automatically recognize and alter automatically its algorithm at any step depending on whether the problem is stiff there

functional mixed effects models in this article a new class of functional models in which smoothing splines are used to model fixed effects as well as random effects is introduced the linear mixed effects models are extended to nonparametric mixed effects models by introducing functional random effects which are modeled as realizations of zero mean stochastic processes the fixed functional effects and the random functional effects are modeled in the same functional space which guarantee the population average and subject specific curves have the same smoothness property these models inherit the flexibility of the linear mixed effects models in handling complex designs and correlation structures can include continuous covariates as well as dummy factors in both the fixed or random design matrices and include the nested curves models as special cases two estimation procedures are proposed the first estimation procedure exploits the connection between linear mixed effects models and smoothing splines and can be fitted using existing software the second procedure is a sequential estimation procedure using kalman filtering this algorithm avoids inversion of large dimensional matrices and therefore can be applied to large data sets a generalized maximum likelihood gml ratio test is proposed for inference and model selection an application to comparison of cortisol profiles is used as an illustration

an analytic approximation to the solution of heat equation by adomian decomposition method and restrictions of the method adomian decomposition has been applied to solve many functional equations so far in this article we have used this method to solve the heat equation which governs on numerous scientific and engineering experimentations some special cases of the equation are solved as examples to illustrate ability and reliability of the method restrictions on applying adomian decomposition method for these equations are discussed

spectral methods for modeling supersonic chemically reacting flowfields on emploie une methode de runge kutta a deux stades pour integrer les equations en temps et une methode spectrale de chebyshev pour l integration des equations dans l espace comparaison avec une methode de differences finies implicites

numerical steady state and hopf bifurcation analysis on the diffusive nicholson s blowflies equation1research partially supported by natural sciences and engineering research council of canada 1 for the dirichlet boundary value problem of the diffusive nicholson s blowflies equation it was shown in ref 17 that in a certain range of the parameter space there is a unique positive steady state solution in this paper we propose a scheme to compute this steady state numerically in addition we describe an iterative procedure to locate the critical values of the delay where a hopf bifurcation of time periodic solutions takes place near the steady state some numerical simulations of both schemes are given

wavelet based functional mixed models increasingly scientific studies yield functional data in which the ideal units of observation are curves and the observed data consist of sets of curves that are sampled on a fine grid we present new methodology that generalizes the linear mixed model to the functional mixed model framework with model fitting done by using a bayesian wavelet based approach this method is flexible allowing functions of arbitrary form and the full range of fixed effects structures and between curve covariance structures that are available in the mixed model framework it yields nonparametric estimates of the fixed and random effects functions as well as the various between curve and within curve covariance matrices the functional fixed effects are adaptively regularized as a result of the non linear shrinkage prior that is imposed on the fixed effects wavelet coefficients and the random effect functions experience a form of adaptive regularization because of the separately estimated variance components for each wavelet coefficient because we have posterior samples for all model quantities we can perform pointwise or joint bayesian inference or prediction on the quantities of the model the adaptiveness of the method makes it especially appropriate for modelling irregular functional data that are characterized by numerous local features like peaks

a study of rosenbrock type methods of high order this paper deals with the solution of nonlinear stiff ordinary differential equations the methods derived here are of rosenbrock type this has the advantage that they area stable or stiffly stable and nevertheless do not require the solution of nonlinear systems of equations we derive methods of orders 5 and 6 which require one evaluation of the jacobian and onelu decomposition per step we have written programs for these methods which use richardson extrapolation for the step size control and give numerical results

remark on modifications of adomian decomposition method a vital exception to the superiority of the chebyshev polynomial modification of the adomian decomposition method to other recent modifications is shown in this article also a new modification of the decomposition method is proposed and shown to be reliable

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

a reliable symbolic implementation of algorithm for calculating adomian polynomials in this paper we introduce a reliable symbolic implementation of the algorithm for calculating adomian polynomials and present some examples to show the simplicity and efficiency of the new method

analytical solution of a time fractional navier stokes equation by adomian decomposition method abstract the aim of the present analysis is to apply adomian decomposition method for the solution of a time fractional navier stokes equation in a tube by using an initial value the explicit solution of the equation has been presented in the closed form and then its numerical solution has been represented graphically the present method performs extremely well in terms of efficiency and simplicity

numerical solution of differential equations in the deterministic limit of stochastic theory although the methodology of stochastic differential equations was developed to achieve statistical separability in the stochastic case the deterministic limit is investigated both as a verification of the theory and also for possible value as a deterministic technique

solution of systems of integral differential equations by adomian decomposition method in this article adomian decomposition method as a well known method for solving functional equations has been employed to solve systems of integral differential equations theoretical considerations are discussed and convergence of the method for these systems is addressed some examples are presented to show the ability of the method for such systems

an improvement to an alternate algorithm for computing adomian polynomials in special cases in a published article entitled an alternate algorithm for computing adomian polynomials in special cases by the first author and some of his collaborators the algorithm were applied in special cases which restricts the application of the algorithm in this article new ideas are suggested for applying the algorithm in any cases without any more assumptions some examples are presented to illustrate the applications of the algorithm in more comprehensive situations and also to show the ability and simplicity of the algorithm

the analysis of longitudinal data using mixed model l splines l splines are a large family of smoothing splines defined in terms of a linear differential operator this article develops l splines within the context of linear mixed models and uses the resulting mixed model l spline to analyze longitudinal data from a grassland experiment in the spirit of time series analysis a periodic mixed model l spline is developed which partitions data into a smooth periodic component plus smooth long term trend

linear mixed effects modeling by parameter cascading a linear mixed effects model lme is a familiar example of a multilevel parameter structure involving nuisance and structural parameters as well as parameters that essentially control the model s complexity marginalization over nuisance parameters such as the restricted maximization likelihood method has been the usual estimation strategy but it can involve onerous and complex algorithms to achieve the integrations involved parameter cascading is described as a multicriterion optimization algorithm that is relatively simple to program and leads to fast and stable computation the method is applied to lme where well developed marginalization methods are already available our results suggest that parameter cascading is at least as good as if not better than the available methods we also extend the lme model to multicurve data smoothing by introducing a basis partitioning scheme and defining roughness penalty terms for both functional fixed effect and random effects the results are substantially

a comparison of gcv and gml for choosing the smoothing parameter in the generalized spline smoothing problem the partially improper prior behind the smoothing spline model is used to obtain a generalization of the maximum likelihood gml estimate for the smoothing parameter then this estimate is compared with the generalized cross validation gcv estimate both analytically and by monte carlo methods the comparison is based on a predictive mean square error criteria it is shown that if the true unknown function being estimated is smooth in a sense to be defined then the gml estimate undersmooths relative to the gcv estimate and the predictive mean square error using the gml estimate goes to zero at a slower rate than the mean square error using the gcv estimate if the true function is rough then the gcv and gml estimates have asymptotically similar behavior a monte carlo experiment was designed to see if the asymptotic results in the smooth case were evident in small sample sizes mixed results were obtained for n 32 gcv was somewhat better than gml for n 64 and gcv was decidedly superior for n 128 in the n 32 case gcv was better for smaller sigma 2 and the comparison close for larger sigma 2 the theoretical results are shown to extend to the generalized spline smoothing model which includes the estimate of functions given noisy values of various integrals of them

extension of the exp function method for systems of two dimensional burgers equations the exp function method has been applied to solve many functional equations so far but it has not been used for systems of equations directly in this paper the exp function method is applied to obtain generalized solitonary solutions of a system of two dimensional burgers equations stdbe it has been shown that the exp function method with the help of symbolic computation provides a very effective and powerful mathematical tool for solving a system of functional equations

an alternate algorithm for computing adomian polynomials in special cases in this article we introduce an alternate algorithm for computing adomian polynomials and present some examples to show the simplicity and efficiency of the new method

the decomposition method for initial value problems abstract in this paper initial value problems for the heat wave and plate equations in which the vectorial space variable varies in the whole r d will be handled using the reliable decomposition method the solutions are obtained in the form of rapidly convergent power series with elegantly computable terms showing that the new technique is reliable powerful and promising

numerical solution of duffing equation by the laplace decomposition algorithm the laplace decomposition algorithm lda is a numerical algorithm can be adapted to solve duffing equations this paper both describes the principle of lda and discusses its advantages and drawbacks concrete example are also studied to show with numerical results how the lda works efficiently

smoothing spline models for the analysis of nested and crossed samples of curves abstract we introduce a class of models for an additive decomposition of groups of curves stratified by crossed and nested factors generalizing smoothing splines to such samples by associating them with a corresponding mixed effects model the models are also useful for imputation of missing data and exploratory analysis of variance we prove that the best linear unbiased predictors blups from the extended mixed effects model correspond to solutions of a generalized penalized regression where smoothing parameters are directly related to variance components and we show that these solutions are natural cubic splines the model parameters are estimated using a highly efficient implementation of the em algorithm for restricted maximum likelihood reml estimation based on a preliminary eigenvector decomposition variability of computed estimates can be assessed with asymptotic techniques or with a novel hierarchical bootstrap resampling scheme for nested mixed effects models our methods are applied to me

a new algorithm for calculating adomian polynomials for nonlinear operators in this paper a reliable technique for calculating adomian polynomials for nonlinear operators will be developed the new algorithm offers a promising approach for calculating adomian polynomials for all forms of nonlinearity the algorithm will be illustrated by studying suitable forms of nonlinearity a nonlinear evolution model will be investigated

solution of nonlinear equations by modified adomian decomposition method in this paper we modify the standard adomian method for solution of the nonlinear equation f x 0 four examples are presented and compared using standard and modified adomian methods

a computational method for solution of the prey and predator problem in this article a mathematical model of the problem of prey and predator being presented and adomian decomposition method is employed to compute an approximation to the solution of the system of nonlinear volterra differential equations governing on the problem some plots for the population of the prey and predator versus time are presented to illustrate the solution

recent advances in methods for numerical solution of o d e initial value problems abstract this is a review paper which describes recent advances in numerical methods and computer codes for solving initial value problems of ordinary differential equations particular emphasis is placed upon stiff systems

nonparametric mixed effects models for unequally sampled noisy curves we propose a method of analyzing collections of related curves in which the individual curves are modeled as spline functions with random coefficients the method is applicable when the individual curves are sampled at variable and irregularly spaced points this produces a low rank low frequency approximation to the covariance structure which can be estimated naturally by the em algorithm smooth curves for individual trajectories are constructed as best linear unbiased predictor blup estimates combining data from that individual and the entire collection this framework leads naturally to methods for examining the effects of covariates on the shapes of the curves we use model selection techniques akaike information criterion aic bayesian information criterion bic and cross validation to select the number of breakpoints for the spline approximation we believe that the methodology we propose provides a simple flexible and computationally efficient means of functional data analysis