a priori l 2 error estimates for finite element methods for nonlinear diffusion equations with memory this paper studies finite element approximations to the solutions of the nonlinear diffusion equations with memory an elliptic projection with memory associated with our equations is defined and then used in the derivations of optimal error estimates for semidiscrete and crank nicolson finite element approximations

time discretization of an integro differential equation of parabolic type the stability and convergence properties of two time discretizations of an integro differential equation of parabolic type are studied the methods reduce to the backward euler and crank nicolson methods if the integral term is absent the integral term is approximated in each case by a quadrature rule with relatively high order truncation error so that a relatively large time step can be used for the quadrature in order to reduce the memory and computational requirements of the method

maximum norm stability and error estimates in galerkin methods for parabolic equations in one space variable maximum norm stability and error estimates of best approximation and nonsmooth data types are derived for the approximate solution of a parabolic equation in one space variable using the continuous in time galerkin method based on piecewise polynomial approximating functions on a quasi uniform mesh

multiple imputation and posterior simulation for multivariate missing data in longitudinal studies summary this paper outlines a multiple imputation method for handling missing data in designed longitudinal studies a random coefficients model is developed to accommodate incomplete multivariate continuous longitudinal data multivariate repeated measures are jointly modeled specifically an i i d normal model is assumed for time independent variables and a hierarchical random coefficients model is assumed for time dependent variables in a regression model conditional on the time independent variables and time with heterogeneous error variances across variables and time points gibbs sampling is used to draw model parameters and for imputations of missing observations an application to data from a study of startle reactions illustrates the model a simulation study compares the multiple imputation procedure to the weighting approach of robins rotnitzky and zhao 1995 journal of the american statistical association90 106 121 that can be used to address similar data structures

sharp l 2 error estimates and superconvergence of mixed finite element methods for non fickian flows in porous media a sharper l2 error estimate is obtained for the non fickian flow of fluid in porous media by means of a mixed ritz volterra projection instead of the mixed ritz projection used in r e ewing y lin and j wang acta math univ comenian n s 70 2001 pp 75 84 moreover local l2 superconvergence for the velocity along the gauss lines and for the pressure at the gauss points is derived for the mixed finite element method via the ritz volterra projection and global l2 superconvergence for the velocity and the pressure is also investigated by virtue of an interpolation postprocessing technique on the basis of the superconvergence estimates some useful a posteriori error estimators are presented for this mixed finite element method

two splitting positive definite mixed finite element methods for parabolic integro differential equations abstract two novel mixed finite element procedures are established for parabolic integro differential equations which can be split into two independent symmetric positive definite sub schemes and do not need to solve a coupled system of equations without requiring the lbb consistency condition the convergence analysis shows that both methods lead to the optimal order l 2 norm error estimate for u and optimal h div norm error estimate for a numerical example is presented to illustrate the theoretical analysis

finite element methods of the two nonlinear integro differential equations abstract transformation of dependent variables such as the kirchhoff transformation is a classical tool for solving nonlinear partial differential equations in larsson et al this approach was used in connection with the finite element method zlamal used it in galerkin approximation in space the euler method in time and applied it to the solution of nonlinear heat equations here we give justification of the methods for the two nonlinear integro heat equations in the case that the discretization is carried through by piecewise linear polynomials in space and by the implicit euler method in time with quadrature rules which require the storage of relatively few values of the solution the estimate of the discretization error in the maximum norm is introduced the results depend on superconvergence in the gradient for a related elliptic problem which is shown to hold only for very special meshes

disorder induced phase transition in a one dimensional model of rice pile abstract we propose a one dimensional rice pile model which connects the 1d btw sandpile model phys rev a 38 1988 364 and the oslo rice pile model phys rev lett 77 1997 107 in a continuous manner we found that for a sufficiently large system there is a sharp transition between the trivial critical behaviour of the 1d btw model and the self organized critical soc behaviour when there is soc the model belongs to a known universality class with the avalanche exponent 1 53

unconditionally stable numerical method for a nonlinear partial integro differential equation the paper presents an unconditionally stable numerical scheme to solve a nonlinear integro differential equation which arises in mathematical modeling of the penetration of a magnetic field into a substance if the temperature is kept constant throughout the material numerical scheme comprises of the galerkin finite element method jangveladze and kiguradze 2011 for the spatial discretization followed by an implicit finite difference scheme for the time stepping we extended the results for stability estimates to a non homogeneous problem and derived optimal order error estimates for the semidiscretized and fully discretized equations using h 0 1 projection further to show the efficiency the proposed numerical method is demonstrated via numerical example

discontinuous galerkin method for an integro differential equation modeling dynamic fractional order viscoelasticity an integro differential equation modeling dynamic fractional order viscoelasticity with a mittag leffler type convolution kernel is considered a discontinuous galerkin method based on piecewise constant polynomials is formulated for temporal semidiscretization of the problem stability estimates of the discrete problem are proved that are used to prove optimal order a priori error estimates the theory is illustrated by a numerical example

time stepping schemes for systems of volterra integro differential equations linear volterra type systems of integro differential equations sides with kernels in the form of sums of complex exponentials are considered such systems arise for example in the semi discrete finite element model of dynamic viscoelasticity and of some dispersive wave equations and in the representation of dynamic subsystems on interfaces several new families of time stepping schemes are proposed here for the solution of this class of sides the accuracy stability and efficiency properties of these schemes are demonstrated via both theoretical analysis using a scalar model problem and numerical experiments applied to first and second order sides

finite element approximation of an evolution problem modeling shear band formation abstract an error analysis is provided for the spatial approximation by the finite element method of a nonlinear time dependent problem which models the antiplane shear deformations of a thermoplastic material this problem consists of a system of two coupled partial differential equations with velocity and temperature unknowns optimal order error estimates in the l 2 norm are given for each of these

analysis of fully discrete fem for miscible displacement in porous media with bear scheidegger diffusion tensor fully discrete galerkin finite element methods are studied for the equations of miscible displacement in porous media with the commonly used bear scheidegger diffusion dispersion tensor begin aligned d mathbf u gamma d m i mathbf u bigg alpha t i alpha l alpha t frac mathbf u otimes mathbf u mathbf u 2 bigg end aligned previous works on optimal order l infty 0 t l 2 norm error estimate required the regularity assumption nabla x partial td mathbf u x t in l infty 0 t l infty omega while the bear scheidegger diffusion dispersion tensor is only lipschitz continuous even for a smooth velocity field mathbf u in terms of the maximal l p regularity of fully discrete finite element solutions of parabolic equations optimal error estimate in l p 0 t l q norm and almost optimal error estimate in l infty 0 t l q norm are established under the assumption of d mathbf u being lipschitz continuous with respect to mathbf u

finite element methods with numerical quadrature for parabolic integrodifferential equations we study the effect of numerical quadrature in space on semidiscrete and fully discrete piecewise linear finite element methods for parabolic integrodifferential equations we give error estimates for both smooth and nonsmooth initial data our results generalize those obtained for the lumped mass method

boltzmann power laws abstract in this paper we propose a model based on the boltzmann distribution as a mechanism for generating power laws boltzmann power laws bpl some of these power laws are studied and compared to popular power laws such as 1 f noise and self organized criticality soc we will show how in some cases these bpls reproduce behaviors similar to the finite size scaling fss scenario which is typical of soc

finite element approximation of diffusion equations with convolution terms approximation of solutions to diffusion equations with memory represented by convolution integral terms is considered such problems arise from modeling of flows in fissured media convergence of the method is proved and results of numerical experiments confirming the theoretical results are presented the advantages of implementation of the algorithm in a multiprocessing environment are discussed

finite element methods for partial volterra integro differential equations on two dimensional unbounded spatial domains abstract the precise and effective way for solving a partial volterra integro differential equation on unbounded spatial domain is to derive artificial boundary conditions for some feasible and prescribed computational domain thus the unsolvable problem on unbounded domain is converted to the computational problem on bounded domain which is often called reduced problem a feasible numerical method will then be applied to solve the reduced problem with the artificial boundary conditions which often exhibit nonlocal properties in this paper the finite element method for the spatial space and the recently developed discontinuous galerkin time stepping method for the temporal space are considered to solve the reduced problem the convergence rate explicitly with those interesting parameters spatial grid size h temporal mesh size k the location of the artificial boundary d and the truncation number m is obtained this paper is the first theoretical analysis of the numerical methods for volterra integro differential equations on unbounded spatial domains

numerical solution of parabolic integro differential equations by the discontinuous galerkin method the numerical solution of a parabolic equation with memory is considered the equation is first discretized in time by means of the discontinuous galerkin method with piecewise constant or piecewise linear approximating functions the analysis presented allows variable time steps which as will be shown call then efficiently be selected to match singularities in the solution induced by singularities in the kernel of the memory term or by nonsmooth initial data the combination with finite element discretizat in space is also studied

imputation of missing categorical data by maximizing internal consistency this paper suggests a method to supplant missing categorical data by reasonable replacements these replacements will maximize the consistency of the completed data as measured by guttman s squared correlation ratio the text outlines a solution of the optimization problem describes relationships with the relevant psychometric theory and studies some properties of the method in detail the main result is that the average correlation should be at least 0 50 before the method becomes practical at that point the technique gives reasonable results up to 10 15 missing data

universality under conditions of self tuning we study systems with a continuous phase transition that tune their parameters to maximize a quantity that diverges solely at a unique critical point varying the size of these systems with dynamically adjusting parameters the same finite size scaling is observed as in systems where all relevant parameters are fixed at their critical values this scheme is studied using a self tuning variant of the ising model it is contrasted with a scheme where systems approach criticality through a target value for the order parameter that vanishes with increasing system size in the former scheme the universal exponents are observed in naive finite size scaling studies whereas in the latter they are not

time discretization via laplace transformation of an integro differential equation of parabolic type we consider the discretization in time of an inhomogeneous parabolic integro differential equation with a memory term of convolution type in a banach space setting the method is based on representing the solution as an integral along a smooth curve in the complex plane which is evaluated to high accuracy by quadrature using the approach in recent work of lopez fernandez and palencia this reduces the problem to a finite set of elliptic equations with complex coefficients which may be solved in parallel the method is combined with finite element discretization in the spatial variables to yield a fully discrete method the paper is a further development of earlier work by the authors which on the one hand treated purely parabolic equations and on the other an evolution equation with a positive type memory term

fem ids for a second order strongly damped wave equation with memory abstract a linear triangular finite element method implicit difference scheme fem ids for the solution of second order strongly damped wave equation sdwe with memory on domain with interfaces is proposed sufficient conditions that guarantee the existence of a unique solution are given the finite element discretization is such that the arbitrary but smooth interface is first approximated by a polygon with a boundary whose vertices all lie on the interface with the interface being at distance from the approximate interface linear interpolation operator incorporating the influence of isproved this together with ritz volterra operator and some auxiliary error estimates in the neighborhood of the interface are used to obtain the convergence estimate of the proposed scheme the scheme is applied to some test examples and the numerical results confirm the computational efficiency of the method

self organized criticality and absorbing states lessons from the ising model we investigate a suggested path to self organized criticality originally this path was devised to generate criticality in systems displaying an absorbing state phase transition but nothing in its definition forbids the mechanism to be used in any other continuous phase transition we used the ising model as well as the manna model to demonstrate how the finite size scaling exponents depend on the tuning of driving and dissipation rates with system size our findings limit the explanatory power of the mechanism as it is to nonuniversal critical behavior suggesting that the explanation of self organized criticality in terms of absorbing state phase transitions is incomplete

the calculation of posterior distributions by data augmentation abstract the idea of data augmentation arises naturally in missing value problems as exemplified by the standard ways of filling in missing cells in balanced two way tables thus data augmentation refers to a scheme of augmenting the observed data so as to make it more easy to analyze this device is used to great advantage by the em algorithm dempster laird and rubin 1977 in solving maximum likelihood problems in situations when the likelihood cannot be approximated closely by the normal likelihood maximum likelihood estimates and the associated standard errors cannot be relied upon to make valid inferential statements from the bayesian point of view one must now calculate the posterior distribution of parameters of interest if data augmentation can be used in the calculation of the maximum likelihood estimate then in the same cases one ought to be able to use it in the computation of the posterior distribution it is the purpose of this article to explain how this can be done the basic idea

on the discretization in time for a parabolic integrodifferential equation with a weakly singular kernel ii nonsmooth initial data abstract backward euler methods are considered for a parabolic integrodifferential equation with a memory term containing a weakly singular kernel the integral term is treated by means of a convolution quadrature suggested by lubich some results from lubich s discretized fractional calculus play a crucial role in error analyses the stability and convergence properties of the time discretizations are derived and are applied to the semidiscrete equations built by the galerkin finite element methods in the space variables

some new error estimates of a semidiscrete finite volume element method for a parabolic integro differential equation with nonsmooth initial data a semidiscrete finite volume element fve approximation to a parabolic integro differential equation pide is analyzed in a two dimensional convex polygonal domain an optimal order l 2 error estimate for smooth initial data and nearly the same optimal order l 2 error estimate for nonsmooth initial data are obtained more precisely for homogeneous equations an elementary energy technique and a duality argument are used to derive an error estimate of order o left t 1 h 2 ln h right in the l 2 norm for positive time when the given initial function is only in l 2

comment on self organized criticality and absorbing states lessons from the ising model according to pruessner and peters g pruessner and o peters phys rev e 73 025106 r 2006 the finite size scaling exponents of the order parameter in sandpile models depend on the tuning of driving and dissipation rates with system size we point out that the same is not true for avalanches in the slow driving limit

analysis of fully discrete approximations to parabolic problems with rough or distribution valued initial data we consider fully discrete approximations to a parabolic initial boundary value problem with rough or distribution valued initial data in two space dimensions for discretization in time and space we apply single step methods and the standard galerkin method with piecewise linear test functions respectively for spatial discretization of the initial condition we are however forced to use more involved constructions our main result is stability and error estimates of the discrete solutions

absolutely continuous invariant measures for one parameter families of one dimensional maps given a one parameter familyf x of maps of the interval 0 1 we consider the set of parameter values for whichf has an invariant measure absolutely continuous with respect to lebesgue measure we show that this set has positive measure for two classes of maps i f x f x where 0 4 andf x is a functionc3 near the quadratic mapx 1 x and ii f x f x mod 1 wheref isc3 f 0 f 1 0 andf has a unique nondegenerate critical point in 0 1

long time numerical solution of a parabolic equation with memory long time stability and convergence properties of two time discreti zation methods for an integro differential equation of parabolic type are studied the methods are based on the standard backward euler and second order back ward differencing methods the memory term is approximated by a quadrature rule with emphasis on such rules with reduced computational memory require ments discretization of the spatial partial differential operators by the finite element method is also considered

discrete schemes for treating hereditary problems of viscoelasticity and applications abstract mathematical models for treating problems of linear viscoelasticity involving hereditary constitutive relations for compressible solids are presented and their discretisation using finite element methods in space together with quadrature rules in time to treat the hereditary integrals is described theoretical error estimates in appropriate sobolev space settings are given both as they arise as a result of using a gronwall inequality and also from employing a more sensitive comparison theorem which for the quasistatic problem and under physically reasonable assumptions on the relaxation function yields much sharper constants in the estimates the range of applicability of the mathematical models and hence the numerical schemes and error estimates are discussed in the context of various materials primarily polymeric materials and extensions of the techniques to the modelling of manufacturing processes such as thermoforming are presented

critical behavior of a one dimensional fixed energy stochastic sandpile we study a one dimensional fixed energy version that is with no input or loss of particles of manna s stochastic sandpile model the system has a continuous transition to an absorbing state at a critical value of the particle density and exhibits the hallmarks of an absorbing state phase transition including finite size scaling critical exponents are obtained from extensive simulations which treat stationary and transient properties and an associated interface representation these exponents characterize the universality class of an absorbing state phase transition with a static conserved density in one dimension they differ from those expected at a linear interface depinning transition in a medium with point disorder and from those of directed percolation

non fickian convection diffusion models in porous media in this paper we propose a numerical scheme to approximate the solution of a non fickian coupled model that describes e g miscible transport in porous media the model is defined by a system of a quasilinear elliptic equation which governs the fluid pressure and a quasilinear integro differential equation which models the convection diffusion transport process the numerical scheme is based on a conforming piecewise linear finite element method for the discretization in space the fully discrete approximations is obtained with an implicit explicit method estimates for the continuous in time and the fully discrete methods are derived showing that the numerical approximation for the concentrations and the pressure are second order convergent in a discrete l 2 norm and in a discrete h 1 norm respectively

a parallel method for the numerical solution of integro differential equation with positive memory abstract an efficient parallel numerical method is proposed for an integro differential equation with positive memory instead of solving the equation in classical time marching methods which require massive storage of solutions of previous time steps in order to advance to a next time step the fourier laplace transformation in time is applied to obtain a set of complex valued elliptic problems parameterized by points on a contour in the complex plane using the independence of an elliptic problem corresponding to one contour point is independent of those elliptic problems corresponding to other contour points all elliptic problems can be solved in parallel essentially without data communications then the time domain solution can be obtained by the fourier laplace inversion formula an error analysis and the numerical implementation of this parallel method is presented

simulation of massive public health data by power polynomials situations in which multiple outcomes and predictors of different distributional types are collected are becoming increasingly common in public health practice and joint modeling of mixed types has been gaining popularity in recent years evaluation of various statistical techniques that have been developed for mixed data in simulated environments necessarily requires joint generation of multiple variables most massive public health data sets include different types of variables for instance in clustered or longitudinal designs often multiple variables are measured or observed for each individual or at each occasion this work is motivated by a need to jointly generate binary and possibly non normal continuous variables we illustrate the use of power polynomials to simulate multivariate mixed data on the basis of a real adolescent smoking study we believe that our proposed technique for simulating such intensive data has potential to be a handy methodological addition to public health researchers toolkit

space time spectral method for a weakly singular parabolic partial integro differential equation on irregular domains abstract the spectral method is proposed for the partial integro differential equations with a weakly singular kernel on irregular domains the space discretization is based on the nodal spectral element method using the lagrange polynomials basis associated with the gauss lobatto legendre quadrature nodes also the model is discretized in time with the legendre spectral galerkin method the discretization leads to conversion of the problem to a sylvester matrix equation which can be solved efficiently by the qz algorithm gardiner et al 1992 the convergence of the method is proven by providing a priori l 2 error estimate numerical results illustrate the efficiency and spectral accuracy of the proposed method

finite element methods for parabolic and hyperbolic partial integro differential equations on formule plusieurs methodes de galerkin a temps discret pour des equations integrodifferentielles paraboliques

multiple imputation after 18 years abstract multiple imputation was designed to handle the problem of missing data in public use data bases where the data base constructor and the ultimate user are distinct entities the objective is valid frequency inference for ultimate users who in general have access only to complete data software and possess limited knowledge of specific reasons and models for nonresponse for this situation and objective i believe that multiple imputation by the data base constructor is the method of choice this article first provides a description of the assumed context and objectives and second reviews the multiple imputation framework and its standard results these preliminary discussions are especially important because some recent commentaries on multiple imputation have reflected either misunderstandings of the practical objectives of multiple imputation or misunderstandings of fundamental theoretical results then criticisms of multiple imputation are considered and finally comparisons are made to alt

finite element and dg analysis for neutral type volterra integro differential equations with weakly singular kernels abstract in this paper we analyze the numerical solution of volterra integro differential equations of neutral type with weakly singular kernels we establish a priori error estimations for the finite element method semi discretization of the given problem by defining a suitable ritz volterra projection operator here the key point in the proof is the fact that the definition of the ritz volterra projection operator that is not related to the neutral term we then discuss the discontinuous galerkin time stepping method for the semi discretized equation together with a fully discretized form

gaussianization based quasi imputation and expansion strategies for incomplete correlated binary responses new quasi imputation and expansion strategies for correlated binary responses are proposed by borrowing ideas from random number generation the core idea is to convert correlated binary outcomes to multivariate normal outcomes in a sensible way so that re conversion to the binary scale after performing multiple imputation yields the original specified marginal expectations and correlations this conversion process ensures that the correlations are transformed reasonably which in turn allows us to take advantage of well developed imputation techniques for gaussian outcomes we use the phrase quasi because the original observations are not guaranteed to be preserved we argue that if the inferential goals are well defined it is not necessary to strictly adhere to the established definition of multiple imputation our expansion scheme employs a similar strategy where imputation is used as an intermediate step it leads to proportionally inflated observed patterns forcing the data set to a complete rectangular format the plausibility of the proposed methodology is examined by applying it to a wide range of simulated data sets that reflect alternative assumptions on complete data populations and missing data mechanisms we also present an application using a data set from obesity research we conclude that the proposed method is a promising tool for handling incomplete longitudinal or clustered binary outcomes under ignorable non response mechanisms

oslo rice pile model is a quenched edwards wilkinson equation the oslo rice pile model is a sandpile like paradigmatic model of self organized criticality soc in this paper it is shown that the oslo model is in fact exactly a discrete realization of the much studied quenched edwards wilkinson equation qew nattermann et al j phys ii france 2 1483 1992 this is possible by choosing the correct dynamical variable and identifying its equation of motion it establishes for the first time an exact link between soc models and the field of interface growth with quenched disorder this connection is obviously very encouraging as it suggests that established theoretical techniques can be brought to bear with full strength on some of the hitherto elusive problems of soc

mixed finite element approximations of parabolic integro differential equations with nonsmooth initial data we analyze the semidiscrete mixed finite element methods for parabolic integro differential equations that arise in the modeling of nonlocal reactive flows in porous media a priori l 2 error estimates for pressure and velocity are obtained with both smooth and nonsmooth initial data more precisely a mixed ritz volterra projection introduced earlier by ewing et al in siam j numer anal 40 2002 pp 1538 1560 is used to derive optimal l 2 error estimates for problems with initial data in h 2 cap h 0 1 in addition for homogeneous equations we derive optimal l 2 error estimates for initial data just in l 2 here we use an elementary energy technique and duality argument

multiple imputation under bayesianly smoothed pattern mixture models for non ignorable drop out conventional pattern mixture models can be highly sensitive to model misspecification in many longitudinal studies where the nature of the drop out and the form of the population model are unknown interval estimates from any single pattern mixture model may suffer from undercoverage because uncertainty about model misspecification is not taken into account in this article a new class of bayesian random coefficient pattern mixture models is developed to address potentially non ignorable drop out instead of imposing hard equality constraints to overcome inherent inestimability problems in pattern mixture models we propose to smooth the polynomial coefficient estimates across patterns using a hierarchical bayesian model that allows random variation across groups using real and simulated data we show that multiple imputation under a three level linear mixed effects model which accommodates a random level due to drop out groups can be an effective method to deal with non ignorable drop out by allowing model uncertainty to be incorporated into the imputation process

a new expanded mixed method for parabolic integro differential equations a new expanded mixed scheme is studied and analyzed for linear parabolic integro differential equations the proposed method s gradient belongs to the simple square integrable space replacing the classical h div space the new expanded mixed projection is introduced the existence and uniqueness of solution for semi discrete scheme are proved and the fully discrete error estimates based on both backward euler scheme are obtained moreover the optimal a priori error estimates in l 2 and h 1 norm for the scalar unknown u and the error results in l 2 norm for its gradient and its flux the coefficients times the negative gradient are derived finally some numerical results are calculated to verify our theoretical analysis

reply to comment on self organized criticality and absorbing states lessons from the ising model in r dickman m a mu noz a vespignani and s zapperi braz j phys 30 27 2000 dickman et al suggested that self organized criticality can be produced by coupling the activity of an absorbing state model to a dissipation mechanism and adding an external drive we analyzed the proposed mechanism in g pruessner and o peters phys rev e 73 025106 r 2006 and found that if this mechanism is at work the finite size scaling found in self organized criticality will depend on the details of the implementation of dissipation and driving in the preceding paper m j alava l laurson a vespignani and s zapperi phys rev e 77 048101 2008 alava et al show that one avalanche exponent in the absorbing state approach becomes independent of dissipation and driving in our reply we clarify their findings and put them in the context of the original paper

sticky grains do not change the universality class of isotropic sandpiles we revisit the sandpile model with sticky grains introduced by mohanty and dhar phys rev lett 89 104303 2002 whose scaling properties were claimed to be generically in the universality class of directed percolation for both isotropic and directed models while for directed models this conclusion is unquestionable for isotropic models we present strong evidence that the asymptotic scaling in the self organized regime in which a stationary critical state exists in the limit of slow driving and vanishing dissipation rate is like other stochastic sandpiles generically in the manna universality class this conclusion is drawn from extensive monte carlo simulations and is strengthened by the analysis of the langevin equations proposed by the same authors to account for this problem argued to converge upon coarse graining to the well established set of langevin equations for the manna class

error estimates for semidiscrete finite element methods for parabolic integro differential equations application des methodes d elements finis aux equations integro differentielles de type parabolique avec un noyau integral forme par un operateur aux derivees partielles d ordre 2

applications and numerical analysis of partial differential volterra equations a brief survey this article contains a concise survey of the numerical analysis of volterra equations and leads up to some recent results on a posteriori error estimation for finite element approximations

critical density of the abelian manna model via a multitype branching process a multitype branching process is introduced to mimic the evolution of the avalanche activity and determine the critical density of the abelian manna model this branching process incorporates partially the spatiotemporal correlations of the activity which are essential for the dynamics in particular in low dimensions an analytical expression for the critical density in arbitrary dimensions is derived which significantly improves the results over mean field theories as confirmed by comparison to the literature on numerical estimates from simulations the method can easily be extended to lattices and dynamics other than those studied in the present work

superconvergence of the gradient in piecewise linear finite element approximation to a parabolic problem some recent results concerning maximum norm superconvergence of the gradient in piecewise linear finite element approximations of an elliptic problem are carried over to a parabolic problem both the standard semidiscrete in space galerkin method and the lumped mass modification are analyzed for both smooth and nonsmooth data situations and the crank nicolson discretizations in time of these procedures are considered as examples of completely discrete schemes

laplace transform finite element method for non fickian flows abstract in this paper we consider numerical methods for integro differential problems based on time discretization via laplace transformation we focus our attention in models arising in the context of non fickian solute transport phenomena in porous media the mathematical models which describe the evolution of the solute concentrations are characterized by volterra equations we present and analyze an hybrid method which combines the laplace transformation with respect to the time variable with the finite element discretization in the spatial variables numerical results illustrate the performance of the method

a random effects ordinal regression model for multilevel analysis a random effects ordinal regression model is proposed for analysis of clustered or longitudinal ordinal response data this model is developed for both the probit and logistic response functions the threshold concept is used in which it is assumed that the observed ordered category is determined by the value of a latent unobservable continuous response that follows a linear regression model incorporating random effects a maximum marginal likelihood mml solution is described using gauss hermite quadrature to numerically integrate over the distribution of random effects an analysis of a dataset where students are clustered or nested within classrooms is used to illustrate features of random effects analysis of clustered ordinal data while an analysis of a longitudinal dataset where psychiatric patients are repeatedly rated as to their severity is used to illustrate features of the random effects approach for longitudinal ordinal data

the finite element numerical analysis for a class nonlinear evolution equations a complete stability and convergence analysis is given for two level scheme for the approximate solution of a class generalized nerve conduction equation defining a kind of truncation error and using a kind of technique optimal order estimates in l 2 is constructively obtained

an imputation strategy for incomplete longitudinal ordinal data a new quasi imputation strategy for correlated ordinal responses is proposed by borrowing ideas from random number generation the essential idea is collapsing ordinal levels to binary ones and converting correlated binary outcomes to multivariate normal outcomes in a sensible way so that re conversion to the binary and then ordinal scale after conducting multiple imputation yields the original marginal distributions and correlations this conversion process ensures that the correlations are transformed reasonably which in turn allows us to take advantage of well developed imputation techniques for gaussian outcomes we use the phrase quasi because the original observations are not guaranteed to be preserved we present an application using a data set from psychiatric research we conclude that the proposed method may be a promising tool for handling incomplete longitudinal or clustered ordinal outcomes

approach to criticality in sandpiles a popular theory of self organized criticality predicts that the stationary density of the abelian sandpile model equals the threshold density of the corresponding fixed energy sandpile we recently announced that this density conjecture is false when the underlying graph is any of z2 the complete graph kn the cayley tree the ladder graph the bracelet graph or the flower graph in this paper we substantiate this claim by rigorous proof and extensive simulations we show that driven dissipative sandpiles continue to evolve even after a constant fraction of the sand has been lost at the sink nevertheless we do find and prove a relationship between the two models the threshold density of the fixed energy sandpile is the point at which the driven dissipative sandpile begins to lose a macroscopic amount of sand to the sink

optimal long time l_p 0 t stability and semidiscrete error estimates for the volterra formulation of the linear quasistatic viscoelasticity problem the purpose of this article is to show how the solution of the linear quasistatic compressible viscoelasticity problem written in volterra form with fading memory may be sharply bounded in terms of the data if certain physically reasonable assumptions are satisfied the bounds are derived by making precise assumptions on the memory term which then make it possible to avoid the gronwall inequality and use instead a comparison theorem which is more sensitive to the physics of the problem once the data stability estimates are established we apply the technique also to deriving a priori error bounds for semidiscrete finite element approximations our bounds are derived for viscoelastic solids and fluids under the small strain assumption in terms of the eigenvalues of a certain matrix derived from the stress relaxation tensor for isotropic materials we can be explicit about the form of these bounds while for the general case we give a formula for their computation

maximum norm stability and maximal l p lp regularity of fems for parabolic equations with lipschitz continuous coefficients in this paper we study the semi discrete galerkin finite element method for parabolic equations with lipschitz continuous coefficients we prove the maximum norm stability of the semigroup generated by the corresponding elliptic finite element operator and prove the space time stability of the parabolic projection onto the finite element space in l infty q t l qt and l p 0 t l q omega lp 0 t lq 1

ritz volterra projections to finite element spaces and applications to integrodifferential and related equations the object of this paper is to investigate the convergence of finite element approximations to solutions of parabolic and hyperbolic integrodifferential equations and also of equations of sobolev and viscoelasticity type the concept of ritz volterra projection will be seen to unify much of the analysis for the different types of problems optimal order error estimates are obtained in l p for 2 leqq p infty and almost optimal order pointwise results given

numerical solution of semilinear integrodifferential equations of parabolic type with nonsmooth data error estimates are derived for the discretization in space by finite elements and in time by the backward euler method of a parabolic equation with a semilinear memory term special attention is paid to the case of nonsmooth data and the computation of the memory term by quadrature

finite element approximation of a parabolic integro differential equation with a weakly singular kernel we give error estimates for the numerical solution by means of the galerkin finite element method of an integro differential equation of parabolic type with a memory term containing a weakly singular kernel optimal order estimates are shown for spatially semidiscrete and completely discrete methods special attention is paid to the regularity of the exact solution

long time behavior of the two grid finite element method for fully discrete semilinear evolution equations with positive memory abstract based on two grid discretizations two fully discrete finite element algorithms for semilinear parabolic integro differential equations with positive memory are proposed with the backward euler scheme for the temporal discretization the basic idea of the space two grid finite element algorithms is to approximate the semilinear equations on a coarse space grid and to solve the linearized equations on a finer space grid at each time step to further decreases the amount of computational work a space time two grid algorithm based on a coarse space grid with large time stepsize t and a finer space grid with small time stepsize t for the evolutional equations is proposed in this paper the sharp long time stability and error estimates for the standard finite element method the space two grid finite element method and the space time two grid finite element method are derived it is showed that the two grid algorithms long time stability and error estimates are similar to those of the direct resolution of the semilinear problem on a fine grid

on the numerical approximation of an evolution problem in nonlinear viscoelasticity abstract an error analysis is furnished for the spatial approximation via the finite element method of certain nonlinear time dependent problems emanating in the theory of viscoelastic solids numerical examples are given first for a model equation and then for numerical equilibrium solutions in cases of a nonconvex double well potential and a convex potential the latter derived from the nonconvex one via relaxation by an equal area construction

a remark on split least squares mixed element procedures for pseudo parabolic equations abstract in this paper we introduce two novel split least squares mixed element procedures for pseudo parabolic equations by selecting the least squares functional properly each procedure can be split into two independent symmetric positive definite sub procedures one of sub procedures is for the primitive unknown variable u which is the same as the standard galerkin finite element procedure and the other is for the introduced flux variable optimal order error estimates are developed a numerical example is given to show the efficiency of the introduced schemes

a two grid block centered finite difference method for nonlinear non fickian flow model in this paper a two grid block centered finite difference scheme is introduced and analyzed to solve the nonlinear parabolic integro differential equation arising in modeling non fickian flow in porous media this method is considered where the nonlinear problem is solved only on a coarse grid of size h and a linear problem is solved on a fine grid of size h error estimates are established on non uniform rectangular grid which show that the discrete l l2 and l2 h1 errors are o t h 2 h 3 finally some numerical experiments are presented to show the efficiency of the two grid method and verify that the convergence rates are in agreement with the theoretical analysis

error estimates with sharp constants for a fading memory volterra problem in linear solid viscoelasticity the problem characterizing nonageing linear isothermal quasi static isotropic compressible solid viscoelasticity in the time interval 0 t is described this is essentially a volterra equation of the second kind arrived at by adding smooth fading memory to the elliptic linear elasticity equations we analyze the errors resulting from replacing the relaxation functions with practical approximations in a semidiscrete finite element approximation and in a fully discrete scheme derived by replacing the hereditary integral with the trapezoidal rule for numerical integration the error estimates are sharp in the sense that if certain bounds on the data are independent of t then so also are the constants involved in them this is a consequence of bypassing the usual gronwall lemmas with arguments that are more sensitive to the fading memory of the physical problem

a meshless method on non fickian flows with mixing length growth in porous media based on radial basis functions a comparative study the present study aims to introduce a solution for parabolic integro differential equations arising in heat conduction in materials with memory which naturally occur in many applications two radial basis functions rbfs collocation schemes are employed for solving this equation the first method tested is an unsymmetric method and the second one which appears to be more efficient is a symmetric one the convergence of these two schemes is accelerated as we use the cartesian nodes as the center nodes

how self organized criticality works a unified mean field picture we present a unified dynamical mean field theory based on the single site approximation to the master equation for stochastic self organized critical models in particular we analyze in detail the properties of sandpile and forest fire ff models in analogy with other nonequilibrium critical phenomena we identify an order parameter with the density of active sites and control parameters with the driving rates depending on the values of the control parameters the system is shown to reach a subcritical absorbing or supercritical active stationary state criticality is analyzed in terms of the singularities of the zero field susceptibility in the limit of vanishing control parameters the stationary state displays scaling characteristics of self organized criticality soc we show that this limit corresponds to the breakdown of space time locality in the dynamical rules of the models we define a complete set of critical exponents describing the scaling of order parameter response functions susceptibility and correlation length in the subcritical and supercritical states in the subcritical state the response of the system to small perturbations takes place in avalanches we analyze their scaling behavior in relation with branching processes in sandpile models because of conservation laws a critical exponents subset displays mean field values ensuremath nu frac 1 2 and ensuremath gamma 1 in any dimensions we treat bulk and boundary dissipation and introduce a critical exponent relating dissipation and finite size effects we present numerical simulations that confirm our results in the case of the forest fire model our approach can distinguish between different regimes soc ff and deterministic ff studied in the literature and determine the full spectrum of critical exponents