mean square power series solution of random linear differential equations this paper deals with the construction of random power series solutions of linear differential equations containing uncertainty through the diffusion coefficient the source term as well as the initial condition under appropriate hypotheses on the data we establish that the constructed random power series solution is mean square convergent in a certain interval whose length depends on the mean square norm of the random variable coefficient also the main statistical functions of the approximating stochastic process solution generated by truncation of the exact power series solution are given finally we apply the proposed technique to several illustrative examples

on errors in variables for binary regression models summary we consider binary regression models when some of the predictors are measured with error for normal measurement errors structural maximum likelihood estimates are considered we show that if the measurement error is large the usual estimate of the probability of the event in question can be substantially in error especially for high risk groups in the situation of large measurement error we investigate a conditional maximum likelihood estimator and its properties

multi element generalized polynomial chaos for arbitrary probability measures we develop a multi element generalized polynomial chaos me gpc method for arbitrary probability measures and apply it to solve ordinary and partial differential equations with stochastic inputs given a stochastic input with an arbitrary probability measure its random space is decomposed into smaller elements subsequently in each element a new random variable with respect to a conditional probability density function pdf is defined and a set of orthogonal polynomials in terms of this random variable is constructed numerically then the generalized polynomial chaos gpc method is implemented element by element numerical experiments show that the cost for the construction of orthogonal polynomials is negligible compared to the total time cost efficiency and convergence of me gpc are studied numerically by considering some commonly used random variables me gpc provides an efficient and flexible approach to solving differential equations with random inputs especially for problems related to long term integration large perturbation and stochastic discontinuities

second order linear differential equations with analytic uncertainties stochastic analysis via the computation of the probability density function abstract this paper concerns the analysis of random second order linear differential equations usually solving these equations consists of computing the first statistics of the response process and that task has been an essential goal in the literature a more ambitious objective is the computation of the solution probability density function we present advances on these two aspects in the case of general random non autonomous second order linear differential equations with analytic data processes the frobenius method is employed to obtain the stochastic solution in the form of a mean square convergent power series we demonstrate that the convergence requires the boundedness of the random input coefficients further the mean square error of the frobenius method is proved to decrease exponentially with the number of terms in the series although not uniformly in time regarding the probability density function of the solution at a given time which is the focus of the paper we rely on the law of total probability to express it in closed form as an expectation for the computation of this expectation a sequence of approximating density functions is constructed by reducing the dimensionality of the problem using the truncated power series of the fundamental set we prove several theoretical results regarding the pointwise convergence of the sequence of density functions and the convergence in total variation the pointwise convergence turns out to be exponential under a lipschitz hypothesis as the density functions are expressed in terms of expectations we propose a crude monte carlo sampling algorithm for their estimation this algorithm is implemented and applied on several numerical examples designed to illustrate the theoretical findings of the paper after that the efficiency of the algorithm is improved by employing the control variates method numerical examples corroborate the variance reduction of the monte carlo approach

solution of the stochastic radiative transfer equation with rayleigh scattering using rvt technique abstract this paper considers the solution of the radiative transfer equation in a semi infinite continuous stochastic medium with rayleigh scattering the available solutions in literature for this stochastic integro differential equation side are represented only by the ensemble averaged solutions here the deterministic solution of the problem is obtained at first then the random variable transformation rvt technique together with an integral transformation to the input stochastic process is implemented to get the complete stochastic solution of the problem this solution is represented by the probability density function of the solution process moreover the nth statistical moment of the solution is derived numerical results are given for exponential and gaussian distributions of the input stochastic process also these results are given for different boundary conditions

modeling conception as an aggregated bernoulli outcome with latent variables via the em algorithm we describe a general method for analyzing aggregated bernoulli outcomes the research is motivated by an epidemiological reproductive study where the outcome was whether pregnancy was detected in a woman s particular menstrual cycle and the bernoulli trials corresponded to days with intercourse during the cycle each cycle is either viable or not i e is or is not susceptible to conception we develop an em algorithm approach to maximizing the observed data pseudo likelihood based on a set of unobservable latent outcomes linked to the specific days with intercourse this method is flexible in that it allows one to model effect of covariates on the susceptibility factor and on the latent outcomes application of the method to fertility studies enables one to investigate covariates with a long term or transient effect on the daily conception probability a complication is that most couples contribute more than one cycle in a prospective study a generalized estimating equation approach adjusts for the dependency among outcomes within individual couples the method can be applied in any setting where dependency among bernoulli trials is induced by a susceptibility factor and the trial outcomes are only observable in the aggregate

a random laplace transform method for solving random mixed parabolic differential problems this paper deals with the explicit solution of random mixed parabolic equations in unbounded domains by using the random laplace transform to second order stochastic processes the mean square random laplace operational calculus is stated and its application to the random parabolic equation together with previous results of the underlying random ordinary differential equations allow us to obtain an explicit solution of the problem a numerical example which includes simulations illustrates the developed method

markov regression models for time series a quasi likelihood approach this paper discusses a quasi likelihood ql approach to regression analysis with time series data we consider a class of markov models referred to by cox 1981 scandinavian journal of statistics 8 93 115 as observation driven models in which the conditional means and variances given the past are explicit functions of past outcomes the class includes autoregressive and markov chain models for continuous and categorical observations as well as models for counts e g poisson and continuous outcomes with constant coefficient of variation e g gamma we focus on poisson and gamma data for illustration analogous to ql for independent observations large sample properties of the regression coefficients depend only on correct specification of the first conditional moment

sobolev type fractional abstract evolution equations with nonlocal conditions and optimal multi controls this paper investigates the existence and uniqueness of mild solutions for a class of sobolev type fractional nonlocal abstract evolution equations in banach spaces we use fractional calculus semigroup theory a singular version of gronwall inequality and leray schauder fixed point theorem for the main results a new kind of sobolev type appears in terms of two linear operators is introduced to extend previous works in the field an existence result of optimal multi control pairs governed by the presented system is proved finally an example is also given to illustrate the obtained theory

the damped pendulum random differential equation a comprehensive stochastic analysis via the computation of the probability density function this work has been supported by the spanish ministerio de economia y competitividad grant mtm2017 89664 p marc jornet acknowledges the doctorate scholarship granted by programa de ayudas de investigacion y desarrollo paid universitat politecnica de valencia the authors are grateful for the valuable comments raised by the reviewers that have improved the final version of the paper

a developed solution of the stochastic milne problem using probabilistic transformations abstract this paper considers the solution of the milne problem of radiative transfer with isotropic scattering in a continuous stochastic medium properties of the medium are assumed to be continuous random functions of the spatial dimensions the available solutions in literature for this stochastic integro differential equation side are represented only by the ensemble average of the radiant energy density in this paper a developed algorithm based on the implementation of the random variable transformation technique together with an integral transformation to the stochastic properties is introduced a complete stochastic solution represented by the probability density function p d f of the radiant energy density is obtained using the closed form of the p d f the nth moment of the stochastic solution is evaluated in realization of this work exponential and gaussian statistics for the medium properties are assumed results are physically acceptable and found to be compatible with those in the literature

analytic and numerical solutions of a riccati differential equation with random coefficients in this paper an analytic mean square solution of a riccati equation with randomness in the coefficients and initial condition is given this analytic solution can be expressed in an explicit form by using a general theorem for the chain rule for stochastic processes that can be written as a composition of a c 1 function and a stochastic process belonging to the banach space l p p 1 moreover the exact mean and variance functions of the riccati equation are computed and they are compared to those obtained by monte carlo differential transform and generalized chaos polynomial methods advantages and disadvantages of these methods are discussed for this equation

laplace transforms for approximation of highly oscillatory volterra integral equations of the first kind abstract this paper focuses on laplace and inverse laplace transforms for approximation of volterra integral equations of the first kind with highly oscillatory bessel kernels where the explicit formulae for the solution of the first kind integral equations are derived from which the integral equations can also be efficiently calculated by the clenshaw curtis filon type methods furthermore by applying the asymptotics of the solution some simpler formulas for approximating the solution for large values of the parameters are deduced preliminary numerical results are presented based on the approximate formulae and the explicit formulae which are compared with the convolution quadrature and numerical inverse laplace transform methods all these methods share that the costs the same independent of large values of frequencies

polynomial chaos for random fractional order differential equations fractional order models provide a powerful instrument for description of memory and hereditary properties of systems in comparison to integer order models where such effects are difficult to incorporate and often neglected also many physical real world problems that involve uncertainties and errors can be best modeled with random differential equations thus to be able to deal with many real life problems it is important to develop mathematical methodologies to solve systems that include both memory effects and uncertainty the aim of this paper is to study the application of the generalized polynomial chaos gpc to random fractional ordinary differential equations the method of polynomial chaos has played an increasingly important role when dealing with uncertainties the main idea of the method is the projection of the random parameters and stochastic processes in the system onto the space of polynomial chaoses however to apply polynomial chaos to random fractional differential equations requires careful attention due to memory effects and the increasing of the computation time in respect to the classic random differential equations in order to avoid more complex numerical computations and obtain accurate solutions we rely on richardson extrapolation it is shown that the application of generalized polynomial chaos method in conjunction with richardson extrapolation is a reliable and accurate method to numerically solve random fractional ordinary differential equations

solving second order linear differential equations with random analytic coefficients about ordinary points a full probabilistic solution by the first probability density function abstract this paper deals with the approximate computation of the first probability density function of the solution stochastic process to second order linear differential equations with random analytic coefficients about ordinary points under very general hypotheses our approach is based on considering approximations of the solution stochastic process via truncated power series solution obtained from the random regular power series method together with the so called random variable transformation technique the validity of the proposed method is shown through several illustrative examples

analysis of repeated ordered categorical outcomes with possibly missing observations and time dependent covariates abstract this article describes a method for comparing responses in two groups of subjects observed repeatedly at a common set of observation times when the response is an ordered categorical outcome the method which allows both time dependent covariates and missing observations consists of two analytic steps in the first step the data are analyzed separately at each occasion using a regression model chosen from the class of models for ordinal data proposed by mccullagh 1980 the joint asymptotic distribution of the estimates of these occasion specific regression coefficients and a consistent estimator of their asymptotic covariance matrix are obtained without imposing any parametric model of dependence on the repeated observations in the second step this asymptotic distribution together with appropriate simultaneous inference procedures is used to characterize the overall difference between groups and the variation in group differences over time the missing data process may differ between gro

bayesian modeling of the level and duration of fertility in the menstrual cycle time to pregnancy studies that identify ovulation days and collect daily intercourse data can be used to estimate the day specific probabilities of conception given intercourse on a single day relative to ovulation in this article a bayesian semiparametric model is described for flexibly characterizing covariate effects and heterogeneity among couples in daily fecundability the proposed model is characterized by the timing of the most fertile day of the cycle relative to ovulation by the probability of conception due to intercourse on the most fertile day and by the ratios of the daily conception probabilities for other days of the cycle relative to this peak probability the ratios are assumed to be increasing in time to the peak and decreasing thereafter generalized linear mixed models are used to incorporate covariate and couple specific effects on the peak probability and on the day specific ratios a markov chain monte carlo algorithm is described for posterior estimation and the methods are illustrated through application to caffeine data from a north carolina pregnancy study

z differential equations this paper is devoted to make a framework for studying a class of uncertain differential equations called z differential equations in order to achieve the purpose we first introduce four basic operations on z numbers based on semigranular function then the limit and continuity concepts of a z number valued function are given under a definition of a metric on the space of z numbers moreover the concepts of z differentiability z integral and z laplace transform of a z number valued function are introduced in addition by giving some theories proved in this paper a basis for calculus z calculus is established we further give theories based on which existence and uniqueness of z differential equations are investigated a conceptual unity between z differential equations and z numbers is also shown the conceptual unity demonstrates that a z differential equation may be expressed as a bimodal differential equation combining a fuzzy differential equation fde and a random differential equation moreover the concept of a bimodal cut called s mu cut is introduced and its relation to other new concepts such as acceptable time and acceptable information area is explained using an example the application of z differential equations in medicine is clarified it is demonstrated that z differential equations outperform fdes in making a decision under uncertainty

analytic numerical approximating processes of diffusion equation with data uncertainty this paper deals with the construction of approximating processes for solving mixed problems related to the diffusion equation where the initial condition is assumed to be a stochastic process and the diffusion coefficient is a random variable expectation and standard deviation of the approximation processes are given

the analysis of binomial data by a generalized linear mixed model summary methods for generalized linear models are extended to provide estimates of location and variance parameters for mixed models fitted to binomial data formed by classifying samples from an underlying normal distribution the method estimates the parameters directly on the underlying scale for a balanced one way random effects model the variance estimator simplifies to the usual analysis of variance one the estimation of variances and the prediction of random effects for binomial traits is required by animal breeders the predictors given are analogous to best linear unbiased predictors henderson 1973 but differ from those presented by harville mee 1984

solving linear and quadratic random matrix differential equations using a mean square approach the non autonomous case abstract this paper is aimed to extend the non autonomous case the results recently given in the paper casaban et al 2016 for solving autonomous linear and quadratic random matrix differential equations with this goal important deterministic results like the abel liouville jacobi s formula are extended to the random scenario using the so called l p random matrix calculus in a first step random time dependent matrix linear differential equations are studied and in a second step random non autonomous riccati matrix differential equations are solved using the hamiltonian approach based on dealing with the extended underlying linear system illustrative numerical examples are also included

multilevel monte carlo path simulation we show that multigrid ideas can be used to reduce the computational complexity of estimating an expected value arising from a stochastic differential equation using monte carlo path simulations in the simplest case of a lipschitz payoff and a euler discretisation the computational cost to achieve an accuracy of o e is reduced from o e 3 to o e 2 log e 2 the analysis is supported by numerical results showing significant computational savings

on the generalization of probabilistic transformation method abstract the probabilistic transformation method with the finite element analysis is a new technique to solve random differential equation the advantage of this technique is finding the exact expression of the probability density function of the solution when the probability density function of the input is known however the disadvantage is due to the characteristics geometrics and materials of the analyzed structure included in the random differential equation in this paper a developed formula is used to generalize this technique by obtaining the exact joint probability density function of the solution in any situations as well as the proposed technique for the non linear case

transient effective hydraulic conductivities under slowly and rapidly varying mean gradients in bounded three dimensional random media we have shown elsewhere tartakovsky and neuman this issue a that in randomly heterogeneous media the ensemble mean transient flux is generally nonlocal in space time and therefore non darcian we have also shown tartakovsky and neuman this issue b that there are special situations in which this flux can be localized so as to render it darcian in real or transformed domains each such situation gives rise to an effective hydraulic conductivity which relates mean gradient to mean flux at any point in real or transformed space time in this paper we develop first order analytical expressions for effective hydraulic conductivity under three dimensional transient flow through a boxshaped domain due to a mean hydraulic gradient that varies slowly in space and time when the mean gradient varies rapidly in time the laplace transform of the mean flux is local but its real time equivalent includes a temporal convolution integral we develop analytical expressions for the real time kernel of this convolution integral the box is embedded within a statistically homogeneous natural log hydraulic conductivity field that is gaussian and exhibits an anisotropic exponential spatial correlation structure by the effective hydraulic conductivity of a finite box in such a field we imply the ensemble mean expected value of all random equivalent conductivities that one could associate with the box under these conditions we explore the influence of domain size time and statistical anisotropy on effective conductivity and include a simple new formula for its variation with statistical anisotropy ratio in an infinite domain under steady state appendixes c e are available with entire article on microfiche order by mail from american geophysical union 2000 florida avenue n w washington dc 20009 or by phone at 800 966 2481 2 50 document w97 002 payment must accompany order

numerical results on two dimensional dendritic solidification abstract we consider a recently derived non local equation for two dimensional dendritic growth at small undercooling a linear expansion is performed which is shown to be an efficient starting point for the numerical solution at zero anisotropy the absence of eigensolutions is confirmed in the opposite case our results suggest an infinite number of solutions

a system of convection diffusion equations with small diffusion coefficient arising in semiconductor physics abstract we consider a system of two convection diffusion equations with a small diffusion parameter in one space dimension subject to dirichlet boundary conditions the system governs the evolution of the flow of electrons and holes in semiconductor devices on the dielectrical relaxation time scale the equations are coupled by nonlinear nonlocal electric field driven convection terms we prove the convergence in suitable topologies of the solutions of the diffusion convection problem to the unique solution of the convective limit problem subject to inflow boundary conditions as the diffusion coefficient tends to zero

ordered shapes in nonequilibrium growth abstract patterns observed during nonequilibrium growth display complex ordering on many length scales we focus on ordered patterns which reflect the interplay of microscopic and macroscopic dynamics the fundamental morphologies which result and which are the building blocks of more complex patterns include dendritic and tip splitting growth the latter gives rise to the two dimensional dense branching morphology dbm we review the current understanding of how dendritic growth and the dbm arise from the microscopic dynamics of surface tension and surface kinetics we emphasize the open questions with particular attention to the question of developing theory for morphology selection and transitions between dendritic and dense branching growth in this context we review our hypotheses of the selection of the fastest growing morphology and the existence of first and second order like morphology transitions theoretical issues are illustrated using the hele shaw and electrodeposition experiments

a numerical scheme for the variance of the solution of the random transport equation abstract we present a numerical scheme based on godunov s method rea algorithm for the variance of the solution of the 1d random linear transport equation with homogeneous random velocity and stochastic initial condition we obtain the stability conditions of the method and we also show its consistency with a deterministic nonhomogeneous advective diffusive equation which means convergency numerical results are considered to validate our scheme

on the random gamma function theory and computing abstract this paper deals with the extension in the mean square sense of the deterministic gamma function to the random framework in a first step we provide such extension to a by assuming that the parameter a is a positive random variable satisfying certain conditions related to its exponential moments as a particular case we show that every positive random variable satisfies such conditions if it is bounded and bounded away from zero in a second step we establish the formula a 1 a a that allows us to extend the random gamma function to a class of random variables whose supports lie over the real line with the exception of small neighborhoods of zero and of the negative integers this retains the classical definition of the gamma function when a becomes a deterministic parameter the study is based on the l p stochastic calculus with p 2 and 4 usually referred to as mean square and mean fourth stochastic calculus respectively next we compute the mean and the variance of the random gamma function including several illustrative examples finally with the aid of the random gamma function we define the random bessel function and compute reliable approximations of its mean and variance

solving the random legendre differential equation mean square power series solution and its statistical functions in this paper we construct by means of random power series the solution of second order linear differential equations of legendre type containing uncertainty through its coefficients and initial conditions by assuming appropriate hypotheses on the data we prove that the constructed random power series solution is mean square convergent in addition the main statistical functions of the approximate solution stochastic process generated by truncation of the exact power series solution are given finally we apply the proposed method to some illustrative examples to compare the numerical results for the average and the variance with respect to those obtained by the monte carlo approach

random effects models for longitudinal data models for the analysis of longitudinal data must recognize the relationship between serial observations on the same unit multivariate models with general covariance structure are often difficult to apply to highly unbalanced data whereas two stage random effects models can be used easily in two stage models the probability distributions for the response vectors of different individuals belong to a single family but some random effects parameters vary across individuals with a distribution specified at the second stage a general family of models is discussed which includes both growth models and repeated measures models as special cases a unified approach to fitting these models based on a combination of empirical bayes and maximum likelihood estimation of model parameters and using the em algorithm is discussed two examples are taken from a current epidemiological study of the health effects of air pollution

erratum transient flow in bounded randomly heterogeneous domains 1 exact conditional moment equations and recursive approximations water resources research 1998 34 1 1 12

stochastic transforms for jump diffusion processes combined with related backward stochastic differential equations abstract this study considers three stochastic transforms for jump diffusion processes the stochastic laplace transform stochastic fourier transform and stochastic wavelet transform first we introduce the stochastic laplace transform for processes adapted by brownian filtration as a solution for complex number valued backward stochastic differential equations bsdes this transform can be explained well by the girsanov theorem which also allows us to define the stochastic laplace transform for jump diffusion processes directly based on this perspective we give natural definitions of the stochastic fourier transform and stochastic wavelet transform the advantages of these stochastic transforms are all related to the uniqueness of the processes compared with the classical transforms the newly introduced parameters guarantee the uniqueness of the stochastic transforms for the adapted processes while they also agree with the corresponding parameters in the classical transforms which can represent the frequency property of the processes in addition these three stochastic transforms can also be regarded as the solutions of related bsdes with jumps

transient flow in bounded randomly heterogeneous domains 1 exact conditional moment equations and recursive approximations in a recent paper we developed exact nonlocal conditional moment equations for transient flow in bounded domains driven by random forcing terms sources initial head and boundary conditions whereas our conditional mean equations took into account the randomness of forcing terms our conditional second moment equations did not we do so in this brief addendum

a new approach to modeling daily probabilities of conception summary standard conception probabilities models assume that different acts of intercourse make independent contributions to the probability of conception in viable cycles we propose an alternative approximate model based on the assumption that the act of intercourse closest to the time of maximum fertility is the one most likely to have caused conception we describe an adaptive algorithm the most fertile intercourse day mfid algorithm that estimates the most fertile intercourse day in each cycle the approach is easily extended to include covariates and random between couple differences in fecundability that affect the probability of conception in a given cycle reanalyses of two data sets reported in the literature are presented estimates of the probability of conception during the most fertile period of the cycle and of the effects of covariates are similar to estimates found using standard models

model for the analysis of binary longitudinal pain data subject to informative dropout through remedication abstract we address the problem of accounting for informative dropout in the form of rescue medication when comparing pain relievers with respect to longitudinal binary pain relief outcomes we present a selection model approach for binary longitudinal data that accommodates informative dropout the relationship between dropout or remedication and the binary pain relief response is assumed to be characterized by a random effect that is conditional on this random effect response and dropout are independent unlike previous approaches to this problem which rely on numerical or approximation methods we obtain a closed form expression for the marginal log likelihood of response and dropout by specifying a complementary log log link function for both components and a conjugate log gamma random effect distribution a data analysis supported by simulation results suggest that the model fits reasonably well results are compared to those obtained from conventional but somewhat inappropriate analyses

variance component models with binary response interviewer variability summary interviewer variability in a binary response is an example of a problem requiring variance component estimation in a non normal family the maximum likelihood estimation procedure is derived and used to examine some binary items on a large questionnaire this raises some interesting questions about the use of unbalanced ano va methods with these data two interviewers asking the same question of the same respondent can obtain different responses such interviewer effects create problems in the design and analysis of survey questionnaires collins 1980 gives a review of these problems together with a variety of partial remedies these include pilot studies and careful design to eliminate difficulties with wording of questions and instructions on the questionnaires in spite of these precautions interviewer effects remain questionnaire items are usually categorical and occasionally binary but usually for analysis the questionnaire item is reduced to a binary response it is the variability of the population of interviewers that is of interest and not just the individual interviewers thus this variability is best modelled as a random effect rather than a fixed effect the number of interviews completed by each interviewer varies so that the design is unbalanced thus we have an unbalanced variance component model with a categorical or binary response the intraclass correlation the ratio of the interviewer variance to the sum of all variances in the model is of great interest as a measure of interviewer variability in estimating the variance components the categorical or binary nature of the response is usually ignored and the analysis carried out using analysis of variance this is not maximum likelihood but analysis of variance does have some desirable properties when the error variables are non normal the rule of thumb generally applied is that the analysis of variance is reasonably accurate as long as the proportions in each of the categories of the response are between 0 1 and 0 9 scheffe 1959 p 343 discusses the effects of non normality in a one way balanced model very little theoretical work has been

probabilistic solution of the homogeneous riccati differential equation this paper deals with the determination of the first probability density function of the solution stochastic process to the homogeneous riccati differential equation taking advantage of both linearization and random variable transformation techniques the study is split in all possible casuistries regarding the deterministic random character of the involved input parameters an illustrative example is provided for each one of the considered cases

computing probabilistic solutions of the bernoulli random differential equation the random variable transformation technique is a powerful method to determine the probabilistic solution for random differential equations represented by the first probability density function of the solution stochastic process in this paper that technique is applied to construct a closed form expression of the solution for the bernoulli random differential equation in order to account for the general scenario all the input parameters coefficients and initial condition are assumed to be absolutely continuous random variables with an arbitrary joint probability density function the analysis is split into two cases for which an illustrative example is provided finally a fish weight growth model is considered to illustrate the usefulness of the theoretical results previously established using real data

random hermite differential equations mean square power series solutions and statistical properties abstract this paper deals with the construction of random power series solution of second order linear differential equations of hermite containing uncertainty through its coefficients and initial conditions under appropriate hypotheses on the data we establish that the constructed random power series solution is mean square convergent we provide conditions in order to obtain random polynomial solutions and as a consequence random hermite polynomial are introduced also the main statistical functions of the approximate stochastic process solution generated by truncation of the exact power series solution are given finally we apply the proposed technique to several illustrative examples comparing the numerical results with respect to those provided by other available approaches including monte carlo simulation

random differential operational calculus theory and applications in this article we obtain a product rule and a chain rule for mean square derivatives an application of the chain rule to the mean square solution of random differential equations is shown however to achieve such mean square differentiation rules fourth order properties were needed and therefore we first studied a mean fourth order differential and integral calculus results are applied to solve random linear variable coefficient differential problems

a proposed technique of sfem on solving ordinary random differential equation the stochastic finite element sfem is a new method to solve random differential equation using the known deterministic finite element technique fem adapted to stochastic techniques for solving stochastic differential equations in this paper a proposed technique of fem and random variable transformation rvt is used to solve a differential equation with random excitation the technique shows high accuracy when solving a case study compared with the exact solution

the probability density function to the random linear transport equation we present a formula to calculate the probability density function of the solution of the random linear transport equation in terms of the density functions of the velocity and the initial condition we also present an expression for the joint probability density function of the solution in two different points our results have shown good agreement with monte carlo simulations

random fractional generalized airy differential equations a probabilistic analysis using mean square calculus abstract the aim of this paper is to study a generalization of fractional airy differential equations whose input data coefficient and initial conditions are random variables under appropriate hypotheses assumed upon the input data we construct a random generalized power series solution of the problem and then we prove its convergence in the mean square stochastic sense afterwards we provide reliable explicit approximations for the main statistical information of the solution process mean variance and covariance further we show a set of numerical examples where our obtained theory is illustrated more precisely we show that our results for the random fractional airy equation are in full agreement with the corresponding to classical random airy differential equation available in the extant literature finally we illustrate how to construct reliable approximations of the probability density function of the solution stochastic process to the random fractional airy differential equation by combining the knowledge of the mean and the variance and the principle of maximum entropy

qualitatively stable finite difference schemes for advection reaction equations a systematic procedure is proposed and implemented for the design of nonstandard finite difference methods as reliable numerical simulations that preserve significant properties inherent to the solutions of advection reaction equations in the case of hyperbolic fixed points a renormalization of the denominators of the discrete derivatives is performed for the numerical solutions to display the linear stability properties of the exact solutions non hyperbolic fixed points are described with the help of two new monotonic properties the construction of schemes which preserve these properties being done by nonlocal approximation of nonlinear terms in the reaction terms

solving random mean square fractional linear differential equations by generalized power series analysis and computing abstract this paper deals with solving the general random caputo fractional linear differential equation under general assumptions on random input data initial condition forcing term and diffusion coefficient our contribution extends in two directions the results presented in a recent contribution by the authors in that paper a mean square random generalized power series solution has been constructed in the case that the fractional order say of the caputo derivative lies on the interval 0 1 and assuming that the diffusion coefficient belongs to a class of random variables that contains all bounded random variables however significant families of unbounded random variables such as gaussian and exponential for example do not fall into class now in this contribution we first enlarge the class of random variables to which the diffusion coefficient belongs and we prove that the constructed random generalized power series solution is mean square convergent too we show that any bounded random variable and important unbounded random variables including gaussian and exponential ones are allowed to play the role of the diffusion coefficient as well secondly we construct a mean square random generalized power series solution in the case that parameter lies on the larger interval 0 2 as a consequence the results established in our previous contribution are fairly generalized it is particularly enlightening the numerical study of the convergence of the approximations to the mean and the standard deviation of the solution stochastic process in terms of parameter and on the type of the probability distribution chosen for the diffusion coefficient

mean square solution of bessel differential equation with uncertainties this paper deals with the study of a bessel type differential equation where input parameters coefficient and initial conditions are assumed to be random variables using the so called l p random calculus and assuming moment conditions on the random variables in the equation a mean square convergent generalized power series solution is constructed as a result of this convergence the sequences of the mean and standard deviation obtained from the truncated power series solution are convergent as well the results obtained in the random framework extend their deterministic counterpart the theory is illustrated in two examples in which several distributions on the random inputs are assumed finally we show through examples that the proposed method is computationally faster than monte carlo method

the beta geometric distribution applied to comparative fecundability studies a convenient measure of fecundability is time number of menstrual cycles required to achieve pregnancy couples attempting pregnancy are heterogeneous in their per cycle probability of success if success probabilities vary among couples according to a beta distribution then cycles to pregnancy will have a beta geometric distribution the likelihood ratio test can thus be employed in studies of exposures that may impair fecundability covariates are incorporated in a natural way the model is illustrated by applying it to data on cycles to pregnancy in smokers and nonsmokers with adjustment for covariates summary in fre excerpt

using fem rvt technique for solving a randomly excited ordinary differential equation with a random operator the technique of stochastic finite element sfem which is the finite element technique fem adapted to stochastic problems can be re described to use random variable transformation technique rvt a new fem rvt technique was successfully used in solving stochastic problems with random excitation m el tawil w el tahhan a hussein a proposed technique of sfem on solving ordinary random differential equation j appl math comput 161 2005 35 47 in this paper the technique is adapted to solve a randomly excited differential equation with a random operator the technique shows high accuracy when solving a case study compared with the exact solution finally a problem with unknown exact solution is solved using this technique

solving initial and two point boundary value linear random differential equations a mean square approach abstract this paper deals with the construction of mean square real valued solutions to both initial and boundary value problems of linear differential equations whose coefficients are assumed to be stochastic processes and initial and boundary conditions are random variables a key result to conduct our study is the extension of the leibniz integral rule to the random framework taking advantage of the so called random fourth calculus exact expressions for the main statistical functions average and variance associated to the solutions to both problems are also provided illustrative examples computing the average and standard deviation are included

the combined laplace transform adomian decomposition method for handling nonlinear volterra integro differential equations in this work a combined form of the laplace transform method with the adomian decomposition method is developed for analytic treatment of the nonlinear volterra integro differential equations the combined method is capable of handling both equations of the first and second kind illustrative examples will be examined to support the proposed analysis

random analytic solution of coupled differential models with uncertain initial condition and source term this paper deals with the construction of random power series solution of vector initial value problems containing uncertainty in both initial condition and source term under appropriate hypothesis on the data we prove that the random series solution constructed by a random frobenius method is convergent in the mean square sense also the main statistical functions of the approximating stochastic process solution generated by truncation of the exact series solution are given finally we apply the proposed technique to several illustrative examples

solvability and optimal controls of impulsive hilfer fractional delay evolution inclusions with clarke subdifferential abstract we study the solvability and optimal controls of an impulsive nonlinear hilfer fractional delay evolution inclusion in banach spaces for the main results we use fractional calculus fixed point technique semigroup theory and multivalued analysis we introduce the notion of clarke delay subdifferential to continue and extend previous works in the field we prove an existence result of optimal control pair for lagrange problem l p under appropriate set of sufficient conditions finally as application we give an impulsive control partial differential inclusion with hilfer fractional derivative

regression models for nonstationary categorical time series asymptotic estimation theory for the analysis of nonstationary categorical time series a parsimonious and flexible class of models is proposed these models are generalizations of regression models for stochastically independent categorical observations consistency asymptotic normality and efficiency of the maximum likelihood estimator are shown under weak and easily verifiable requirements some models for binary time series are discussed in detail to demonstrate asymptotic properties a theorem is given addressing maximum likelihood estimation for general stochastic processes then it is shown that the assumptions of this theorem are consequences of the requirements for categorical time series for this proof some lemmas are used which may be of interest in similar cases

tree structured logistic model for over dispersed binomial data with application to modeling developmental effects this article proposes tree structured logistic regression modeling for over dispersed binomial data recursive partitioning is performed using a combination of statistical tests and residual analysis the splitting criterion in cross validation is based on the deviance function a nested grid algorithm to estimate the bootstrap parameters is developed the regression tree procedure provides a new approach for exploring in detail the relationship between the binomial response and explanatory variables the proposed procedure is used to model the relationship between the incidence of malformation and dose and fetal weight using data from a developmental experiment conducted at the national center for toxicological research a conditional gaussian chain model is used to account for the effect of fetal weight by dose

time dependent generalized polynomial chaos generalized polynomial chaos gpc has non uniform convergence and tends to break down for long time integration the reason is that the probability density distribution pdf of the solution evolves as a function of time the set of orthogonal polynomials associated with the initial distribution will therefore not be optimal at later times thus causing the reduced efficiency of the method for long time integration adaptation of the set of orthogonal polynomials with respect to the changing pdf removes the error with respect to long time integration in this method new stochastic variables and orthogonal polynomials are constructed as time progresses in the new stochastic variable the solution can be represented exactly by linear functions this allows the method to use only low order polynomial approximations with high accuracy the method is illustrated with a simple decay model for which an analytic solution is available and subsequently applied to the three mode kraichnan orszag problem with favorable results

models for longitudinal data a generalized estimating equation approach this article discusses extensions of generalized linear models for the analysis of longitudinal data two approaches are considered subject specific ss models in which heterogeneity in regression parameters is explicitly modelled and population averaged pa models in which the aggregate response for the population is the focus we use a generalized estimating equation approach to fit both classes of models for discrete and continuous outcomes when the subject specific parameters are assumed to follow a gaussian distribution simple relationships between the pa and ss parameters are available the methods are illustrated with an analysis of data on mother s smoking and children s respiratory disease

numerical inversion of a laplace transform solution of a diffusion equation with a mixed derivative term we consider a diffusion equation with a mixed derivative term modelling the transient flow of a second grade fluid the same diffusion equation with a mixed derivative term also comes up in the modelling of heat conduction in which the history of the temperature gradient is included the equations differ in the sign of the coefficient of the mixed derivative term we show that the method of lines solution is stable for the transient flow of a second grade fluid and unstable for heat conduction with the temperature gradient history included a laplace transform solution for the heat conduction case is then investigated the laplace transform solution is inverted numerically using a fourier based method a stable solution is obtained the solutions are compared and discussed

random airy type differential equations mean square exact and numerical solutions this paper deals with the construction of power series solutions of random airy type differential equations containing uncertainty through the coefficients as well as the initial conditions under appropriate hypotheses on the data we establish that the constructed random power series solution is mean square convergent over the whole real line in addition the main statistical functions such as the mean and the variance of the approximate solution stochastic process generated by truncation of the exact power series solution are given finally we apply the proposed technique to several illustrative examples which show substantial speed up and improvement in accuracy compared to other approaches such as monte carlo simulations

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

the approximate solutions of some stochastic differential equations using transformations in this paper the transformation technique together with a numerical technique is used to obtain an approximate probability density function pdf for the solution process of some stochastic differential equations the approach is applied on some first order and second order differential equations with making comparisons with the exact solution if possible

solving random mixed heat problems this paper develops a random mean square fourier transform approach to solve random partial differential heat problems with nonhomogeneous boundary value conditions random mean square operational rules for the random fourier sine and cosine transforms are stated and illustrative examples are included

solution of the stochastic transport equation of neutral particles with anisotropic scattering using rvt technique much of the literatures are directed toward the development of a mathematical formalism for a rigorous estimation of the ensemble average of the solution process of a stochastic differential equation sde the random variable transformation technique rvt is a powerful technique to get the complete solution for the sde represented by the probability density function of the solution process in this paper the rvt technique together with a simple integral transformation to the input stochastic process are implemented to get the complete solution of the one speed transport equation for neutral particles in a semi infinite stochastic medium with linear anisotropic scattering the extinction function of the medium input stochastic process is assumed to be a continuous random function of position the probability density function and hence the higher order statistical moments of the solution process are presented numerical results are given for different distributions of the input stochastic process

a comparative study of the numerical approximation of the random airy differential equation the aim of this paper is twofold first we deal with the extension to the random framework of the piecewise frobenius method to solve airy differential equations this extension is based on mean square stochastic calculus second we want to explore the capability to provide not only reliable approximations for both the average and the standard deviation functions associated to the solution stochastic process but also to save computational time as it happens in dealing with the analogous problem in the deterministic scenario this includes a comparison of the numerical results with respect to those obtained by other commonly used operational methods such as polynomial chaos and monte carlo simulations to conduct this comparative study we have chosen the airy random differential equation because it has highly oscillatory solutions this feature allows us to emphasize differences between all the considered approaches

potential for bias in estimating human fecundability parameters a comparison of statistical models fecundability studies where couples attempting pregnancy subject to failure conception one time in each menstrual cycle present a natural discrete failure time scenario because the biologic capacity to conceive varies among couples in the population a complication arises in choosing a method of analysis related to the fact that the maximum follow up time can vary from study to study and follow up time could potentially have different effects on parameters based on different approaches to modelling traditional development in fertility studies has been based on an implicit assumption that binary outcomes for different menstrual cycles are mutually independent we contrast traditional models to a random effects model where cycle viability is modelled as subject specific we clarify the interpretations for different parameters from different models we show that the traditional approach yields some regression parameters that depend on follow up time limiting the generalizability of inferences based on this analytic approach by contrast the subject specific model consistently estimates parameters of interest if the underlying distribution is properly specified data from a fecundability study carried out in north carolina serves to illustrate these points

approximate solution of a mixed nonlinear stochastic oscillator in this paper nonlinear oscillators under mixed quadratic and cubic nonlinearities with stochastic inputs are considered different methods are used to obtain second order approximations namely the wiener hermite and perturbation whep technique and the homotopy perturbation method hpm some statistical moments are computed for the different methods using mathematica 5 comparisons are illustrated through figures for different case studies

solving the random cauchy one dimensional advection diffusion equation numerical analysis and computing abstract in this paper a random finite difference scheme to solve numerically the random cauchy one dimensional advection diffusion partial differential equation is proposed and studied throughout our analysis both the advection and diffusion coefficients are assumed to be random variables while the deterministic initial condition is assumed to possess a discrete fourier transform for the sake of generality in our study we consider that the advection and diffusion coefficients are statistical dependent random variables under mild conditions on the data it is demonstrated that the proposed random numerical scheme is mean square consistent and stable finally the theoretical results are illustrated by means of two numerical examples

random effects models for serial observations with binary response this paper presents a general mixed model for the analysis of serial dichotomous responses provided by a panel of study participants each subject s serial responses are assumed to arise from a logistic model but with regression coefficients that vary between subjects the logistic regression parameters are assumed to be normally distributed in the population inference is based upon maximum likelihood estimation of fixed effects and variance components and empirical bayes estimation of random effects exact solutions are analytically and computationally infeasible but an approximation based on the mode of the posterior distribution of the random parameters is proposed and is implemented by means of the em algorithm this approximate method is compared with a simpler two step method proposed by korn and whittemore using data from a panel study of asthmatics originally described in that paper one advantage of the estimation strategy described here is the ability to use all of the data including that from subjects with insufficient data to permit fitting of a separate logistic regression model as required by the korn and whittemore method however the new method is computationally intensive

statistical moments of the random linear transport equation this paper deals with a numerical scheme to approximate the mth moment of the solution of the one dimensional random linear transport equation the initial condition is assumed to be a random function and the transport velocity is a random variable the scheme is based on local riemann problem solutions and godunov s method we show that the scheme is stable and consistent with an advective diffusive equation numerical examples are added to illustrate our approach

random matrix difference models arising in long term medical drug strategies abstract this paper deals with the construction of random discrete solutions of coupled linear difference equations incorporating uncertainty into both the initial condition and the source term first sufficient conditions in order to guarantee mean square stability of the solution are provided then the main statistical functions such as mean and covariance of the discrete solution stochastic process are given finally illustrative examples of potential interest in long time medical drug strategies are shown

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

theory of needle crystal abstract we investigate the integro differential equation giving the shape of a steady dendrite growing from a weakly undercooled melt either in two or in three dimensions using a generalized wkb analysis due to kruskal and segur we explicitly show that isotropic surface tension prohibits the growth of needle crystal introducing transcendentally small terms at the tip in agreement with our numerical data

the decomposition method for cauchy advection diffusion problems in this paper the solution of cauchy problems for the advection diffusion equation is obtained using the decomposition method in the case when the flow velocity is constant an analytical solution can be derived whilst for variable flow velocity symbolic numerical computations need to be performed

stochastic galerkin techniques for random ordinary differential equations over the last decade the stochastic galerkin method has become an established method to solve differential equations involving uncertain parameters it is based on the generalized wiener expansion of square integrable random variables although there exist very sophisticated variants of the stochastic galerkin method wavelet basis multi element approach convergence for random ordinary differential equations has rarely been considered analytically in this work we develop an asymptotic upper boundary for the l 2 error of the stochastic galerkin method furthermore we prove convergence of a local application of the stochastic galerkin method and confirm convergence of the multi element approach within this context

random variable transformation for generalized stochastic radiative transfer in finite participating slab media abstract the stochastic radiative transfer problem is studied in a participating planar finite continuously fluctuating medium the problem is considered for specular and diffusly reflecting boundaries with linear anisotropic scattering random variable transformation rvt technique is used to get the complete average for the solution functions that are represented by the probability density function pdf of the solution process in the rvt algorithm a simple integral transformation to the input stochastic process the extinction function of the medium is applied this linear transformation enables us to rewrite the stochastic transport equations in terms of the optical random variable x and the optical random thickness l then the transport equation is solved deterministically to get a closed form for the solution as a function of x and l so the solution is used to obtain the pdf of the solution functions applying the rvt technique among the input random variable l and the output process the solution functions the obtained averages of the solution functions are used to get the complete analytical averages for some interesting physical quantities namely reflectivity and transmissivity at the medium boundaries in terms of the average reflectivity and transmissivity the average of the partial heat fluxes for the generalized problem with internal source of radiation are obtained and represented graphically

biased estimates of treatment effect in randomized experiments with nonlinear regressions and omitted covariates summary certain important nonlinear regression models lead to biased estimates of treatment effect even in randomized experiments if needed covariates are omitted the asymptotic bias is determined both for estimates based on the method of moments and for maximum likelihood estimates the asymptotic bias from omitting covariates is shown to be zero if the regression of the response variable on treatment and covariates is linear or exponential and in regular cases this is a necessary condition for zero bias many commonly used models do have such exponential regressions thus randomization ensures unbiased treatment estimates in a large number of important nonlinear models for moderately censored exponential survival data analysis with the exponential survival model yields less biased estimates of treatment effect than analysis with the proportional hazards model of cox if needed covariates are omitted simulations confirm that calculations of asymptotic bias are in excellent agreement with the bias observed in experiments of modest size

longitudinal data analysis using generalized linear models summary this paper proposes an extension of generalized linear models to the analysis of longitudinal data we introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence the estimating equations are derived without specifying the joint distribution of a subject s observations yet they reduce to the score equations for multivariate gaussian outcomes asymptotic theory is presented for the general class of estimators specific cases in which we assume independence m dependence and exchangeable correlation structures from each subject are discussed efficiency of the proposed estimators in two simple situations is considered the approach is closely related to quasi likelih ood some key ironh estimating equation generalized linear model longitudinal data quasi likelihood repeated measures