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

likelihood analysis of non gaussian measurement time series summary in this paper we provide methods for estimating non gaussian time series models these techniques rely on markov chain monte carlo to carry out simulation smoothing and bayesian posterior analysis of parameters and on importance sampling to estimate the likelihood function for classical inference the time series structure of the models is used to ensure that our simulation algorithms are efficient

a review of the decomposition method in applied mathematics abstract the decomposition method can be an effective procedure for analytical solution of a wide class of dynamical systems without linearization or weak nonlinearity assumptions closure approximations perturbation theory or restrictive assumptions on stochasticitiy

a study on a boundary layer equation arising in an incompressible fluid in this paper a differential equation of mixing layer that arises in viscous incompressible fluid will be investigated the series solution method will be effectively implemented to a related integro differential equation the condition at infinity will be applied to a related pade approximants of the obtained series solution

a note on the adomian decomposition method for the generalized huxley equation in a recent paper ismail et al adomian decomposition method for burgers huxley and burgers fisher equations appl math comput 159 2004 291 301 employed the adomian decomposition method adm to solve in particular a generalized nonlinear huxley equation the purpose of this note is to correct ismail et al s numerical solutions of the generalized huxley equation in addition a convergence proof of the adm applied to the generalized huxley equation is also given

non gaussian state space modeling of nonstationary time series abstract a non gaussian state space approach to the modeling of nonstationary time series is shown the model is expressed in state space form where the system noise and the observational noise are not necessarily gaussian recursive formulas of prediction filtering and smoothing for the state estimation and identification of the non gaussian state space model are given also given is a numerical method based on piecewise linear approximation to the density functions for realizing these formulas significant merits of non gaussian modeling and the wide range of applicability of the method are illustrated by some numerical examples a typical application of this non gaussian modeling is the smoothing of a time series that has mean value function with both abrupt and gradual changes simple gaussian state space modeling is not adequate for this situation here the model with small system noise variance cannot detect jump whereas the one with large system noise variance yields unfavorable wiggle to work

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

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

chaos of a markov operator and the fourth moment condition we analyze from the viewpoint of an abstract markov operator recent results by nualart and peccati and nourdin and peccati on the fourth moment as a condition on a wiener chaos to have a distribution close to gaussian in particular we are led to introduce a notion of chaos associated to a markov operator through its iterated gradients and present conditions on the pure point spectrum for a sequence of chaos eigenfunctions to converge to a gaussian distribution convergence to gamma distributions may be examined similarly

smoothing and interpolation with the state space model abstract a new result dealing with smoothing and interpolation in the state space model is developed and explored the result simplifies the derivation of existing smoothing algorithms and provides alternate forms that have analytic and practical computing advantages the connection to signal extraction and interpolation is explored and diffuse specifications are considered

a new approach to nonlinear partial differential equations abstract the author s decomposition method for the solution of operator equations which may be nonlinear and or stochastic is generalized to multidimensional cases

recurrence triangle for adomian polynomials in this paper a recurrence technique for calculating adomian polynomials is proposed the convergence of the series for the adomian polynomials is discussed and the dependence of the convergent domain of the solution s decomposition series n 0 u n on the initial component function u 0 is illustrated by introducing the index vectors of the adomian polynomials the recurrence relations of the index vectors are discovered and the recurrence triangle is given the method simplifies the computation of the adomian polynomials in order to obtain a solution s decomposition series with larger domain of convergence we illustrate by examples that the domain of convergence can be changed by choosing a different u 0 and a modified iteration

convergence of adomian s method applied to differential equations this paper deals with a new proof of convergence of adomian s method applied to differential equations we also give new formulae and properties and we suggest a simple computational form for adomian s polynomials

filtering and forecasting with misspecified arch models i getting the right variance with the wrong model this paper investigates the properties of the conditional covariance estimates generated by a misspecified arch model for example suppose that we observe a diffusion process at discrete time intervals of length h for each h we use a garch 1 1 model to estimate the instantaneous conditional covariance matrix of the diffusion under mild regularity conditions the difference between these conditional covariance estimates and the true conditional covariance converges to zero in probability as h 0 many other arch models for example exponential arch have similar consistency properties this may well account for the success of arch models in short term forecasting using high frequency data since even misspecified arch models can produce good estimates of volatility

arch models as diffusion approximations abstract this paper investigates the convergence of stochastic difference equations e g arch to stochastic differential equations as the length of the discrete time intervals between observations goes to zero these results are applied to the garch 1 1 model of bollerslev 1986 and to the ar 1 exponential arch model of nelson 1989 in their continuous time limits the conditional variance processes in these models have stationary distributions that are inverted gamma and lognormal respectively in addition a class of diffusion approximations based on the exponential arch model is developed

exponential family state space models based on a conjugate latent process summary this paper introduces non linear and non gaussian state space models with analytic updating recurnions for filtering and prediction this new class of models involves some well known results in the theory of exponential models and of exponential dispersion models and the latent process is defined in such a way that both the filtering and the prediction distributions turn out to be conjugate to the observation distribution at each step the corresponding analytic and inferential properties are investigated and some simple examples are presented

generalized autoregressive conditional heteroskedasticity a natural generalization of the arch autoregressive conditional heteroskedastic process introduced in engle 1982 to allow for past conditional variances in the current conditional variance equation is proposed stationarity conditions and autocorrelation structure for this new class of parametric models are derived maximum likelihood estimation and testing are also considered finally an empirical example relating to the uncertainty of the inflation rate is presented

a convenient computational form for the adomian polynomials abstract recent important generalizations by g adomian stochastic systems academic press 1983 have extended the scope of his decomposition method for nonlinear stochastic operator equations see also iterative method inverse operator method symmetrized method or stochastic green s function method very considerably so that they are now applicable to differential partial differential delay and coupled equations which may be strongly nonlinear and or strongly stochastic or linear or deterministic as subcases thus for equations modeling physical problems solutions are obtained rapidly easily and accurately the methodology involves an analytic parametrization in which certain polynomials an dependent on the nonlinearity are derived this paper establishes simple symmetry rules which yield adomian s polynomials quickly to high orders

the decomposition method for approximate solution of the goursat problem abstract in this paper the adomian s decomposition method is effectively implemented to establish an analytic solution and a reliable numerical approximation to the goursat partial differential equation linear and nonlinear goursat examples are examined and the method supplied quantitatively reliable results for these types of problems the accuracy level of the results obtained indicates the superiority of the decomposition method over existing numerical methods that were applied to the goursat problem

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

symbolic implementation of the algorithm for calculating adomian polynomials in this paper a symbolic implementation code is developed of a technique proposed by wazwaz appl math comput 111 2000 53 for calculating adomian polynomials for nonlinear operators the algorithm proposed by him appl math comput 111 2000 53 offers a promising approach for calculating adomian polynomials for all forms of nonlinearity but it is not easy to implement due to its huge size of algebraic calculations complicated trigonometric terms and unique summation rules it is well known that the algebraic manipulation language such as mathematica is useful to facilitate such a hard computational work pattern matching capabilities peculiar feature of mathematica are used in index regrouping which is a key role in constructing adomian polynomials the computer algebra software mathematica is used to collect terms to their order and to simplify the terms the symbolic implementation code author developed appearing at appendix has the flexibility that may easily cover any length of adomian polynomial for many forms of nonlinear cases a nonlinear evolution equation is investigated in order to justify the availability of symbolic implementation code

local scale models state space alternative to integrated garch processes state space alternative to autoregressive conditional heteroskedasticity models are proposed the initial model which is labelled the gaussian local scale model has a measurement density which is gaussian conditional on the unobservable precision the precision is assumed to be a gamma variable which evolves by being scaled by a beta variable the resulting forecast is a student s t random variable with a scale which is approximately an exponentially weighted moving average ewma of the squares of the past observations the degrees of freedom of the student s t distribution is controlled by the size of the discount parameter of the ewma procedure the gaussianity of the measurement density is potentially inadequate when the model is applied to heavy tailed finance data instead this assumption can be replaced by an exponential power density which allows the observed excess kurtosis to be modelled the choice of the exponential power means that the model still maintains conjugacy so allowing the derivation of an exact filter and likelihood function this model is called the generalised local scale model it has been used to model two exchange rate series

solving a system of nonlinear fractional differential equations using adomian decomposition adomian decomposition method has been employed to obtain solutions of a system of nonlinear fractional differential equations dxiyi x ni x y1 yn yi k 0 cki 0 k i 1 i n and dxi denotes caputo fractional derivative some examples are solved as illustrations using symbolic computation

a non gaussian state space model and application to prediction of records on developpe une classe de modeles qui generalisent le travail de smith 1979 et bathes 1975 pour des donnees censurees

convenient analytic recurrence algorithms for the adomian polynomials abstract in this article we present four analytic recurrence algorithms for the multivariable adomian polynomials as special cases we deduce the four simplified results for the one variable adomian polynomials these algorithms are comprised of simple orderly and analytic recurrence formulas which do not require time intensive operations such as expanding regrouping parametrization and so on they are straightforward to implement in any symbolic software and are shown to be very efficient by our verification using mathematica 7 0 we emphasize that from the summation expressions a n k 1 n u n k for the multivariable adomian polynomials and an k 1 n f k u 0 c n k for the one variable adomian polynomials we obtain the recurrence formulas for the u n k and the c n k these provide a theoretical basis for developing new algorithmic approaches such as for parallel computing in particular the recurrence process of one particular algorithm for the one variable adomian polynomials does not involve the differentiation operation but significantly only the arithmetic operations of multiplication and addition are involved precisely c n 1 u n n 1 and c n k 1 n j 0 n k j 1 u j 1 c n 1 j k 1 2 k n we also discuss several other algorithms previously reported in the literature including the adomian rach recurrence algorithm 1 and this author s index recurrence algorithm 23 36