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Difference Equations (difference + equation)

Distribution by Scientific Domains

Engineering	60%
Mathematics and Statistics	25%
3 Other Domains	15%

Selected Abstracts

Difference Equations for the Higher Order Moments and Cumulants of the INAR(p) Model

JOURNAL OF TIME SERIES ANALYSIS, Issue 1 2005
Maria Eduarda Silva
Abstract., Here we obtain difference equations for the higher order moments and cumulants of a time series {Xt} satisfying an INAR(p) model. These equations are similar to the difference equations for the higher order moments and cumulants of the bilinear time series model. We obtain the spectral and bispectral density functions for the INAR(p) process in state,space form, thus characterizing it in the frequency domain. We consider a frequency domain method , the Whittle criterion , to estimate the parameters of the INAR(p) model and illustrate it with the series of the number of epilepsy seizures of a patient. [source]

The Kalman filter for the pedologist's tool kit

EUROPEAN JOURNAL OF SOIL SCIENCE, Issue 6 2006
R. Webster
Summary The Kalman filter is a tool designed primarily to estimate the values of the ,state' of a dynamic system in time. There are two main equations. These are the state equation, which describes the behaviour of the state over time, and the measurement equation, which describes at what times and in what manner the state is observed. For the discrete Kalman filter, discussed in this paper, the state equation is a stochastic difference equation that incorporates a random component for noise in the system and that may include external forcing. The measurement equation is defined such that it can handle indirect measurements, gaps in the sequence of measurements and measurement errors. The Kalman filter operates recursively to predict forwards one step at a time the state of the system from the previously predicted state and the next measurement. Its predictions are optimal in the sense that they have minimum variance among all unbiased predictors, and in this respect the filter behaves like kriging. The equations can also be applied in reverse order to estimate the state variable at all time points from a complete series of measurements, including past, present and future measurements. This process is known as smoothing. This paper describes the ,predictor,corrector' algorithm for the Kalman filter and smoother with all the equations in full, and it illustrates the method with examples on the dynamics of groundwater level in the soil. The height of the water table at any one time depends partly on the height at previous times and partly on the precipitation excess. Measurements of the height of water table and their errors are incorporated into the measurement equation to improve prediction. Results show how diminishing the measurement error increases the accuracy of the predictions, and estimates achieved with the Kalman smoother are even more accurate. Le filtre de Kalman comme outil pour le pédologue Résumé Le filtre de Kalman est un outil conçu essentiellement pour estimer les valeurs de l'état d'un système dynamique dans le temps. Il comprend deux équations principales. Celles-ci sont l'équation d'état, qui décrit l'évolution de l'état pendant le temps, et l'équation de mesure qui decrit à quel instants et de quelle façon on observe l'état. Pour le filtre discret de Kalman, décrit dans cet article, l'équation d'état est une équation stochastique différentielle qui comprend une composante aléatoire pour le bruit dans le système et qui peut inclure une force extérieure. On définit l'équation de mesure de façon à ce qu'elle puisse traiter des mesures indirectes, des vides dans des séquences de mesures et des erreurs de mesure. Le filtre de Kalman fonctionne récursivement pour prédire en avance une démarche à temps l'état du système de la démarche prédite antérieure plus l'observation prochaine. Ses prédictions sont optimales dans le sens qu'elles minimisent la variance parmi toutes les prédictions non-biasées, et à cet égard le filtre se comporte comme le krigeage. On peut appliquer, aussi, les équations dans l'ordre inverse pour estimer la variable d'état à toutes pointes à toutes les instants d'une série complète d'observations, y compris les observations du passé, du présent et du futur. Ce processus est connu comme ,smoothing'. Cet article décrit l'algorithme ,predictor,corrector' du filtre de Kalman et le ,smoother' avec toutes les équations entières. Il illustre cette méthode avec des exemples de la dynamique du niveau de la nappe phréatique dans le sol. Le niveau de la nappe à un instant particulier dépend en partie du niveau aux instants précédents et en partie de l'excès de la précipitation. L'équation d'état fournit la relation générale entre les deux variables et les prédictions. On incorpore les mesures du niveau de la nappe et leurs erreurs pour améliorer les prédictions. Les résultats mettent en évidence que lorsqu'on diminue l'erreur de mesure la précision des prédictions augmente, et aussi que les estimations avec le ,smoother' de Kalman sont encore plus précises. [source]

An efficient finite difference scheme for free-surface flows in narrow rivers and estuaries

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 3 2003
XinJian ChenArticle first published online: 13 MAY 200
Abstract This paper presents a free-surface correction (FSC) method for solving laterally averaged, 2-D momentum and continuity equations. The FSC method is a predictor,corrector scheme, in which an intermediate free surface elevation is first calculated from the vertically integrated continuity equation after an intermediate, longitudinal velocity distribution is determined from the momentum equation. In the finite difference equation for the intermediate velocity, the vertical eddy viscosity term and the bottom- and sidewall friction terms are discretized implicitly, while the pressure gradient term, convection terms, and the horizontal eddy viscosity term are discretized explicitly. The intermediate free surface elevation is then adjusted by solving a FSC equation before the intermediate velocity field is corrected. The finite difference scheme is simple and can be easily implemented in existing laterally averaged 2-D models. It is unconditionally stable with respect to gravitational waves, shear stresses on the bottom and side walls, and the vertical eddy viscosity term. It has been tested and validated with analytical solutions and field data measured in a narrow, riverine estuary in southwest Florida. Model simulations show that this numerical scheme is very efficient and normally can be run with a Courant number larger than 10. It can be used for rivers where the upstream bed elevation is higher than the downstream water surface elevation without any problem. Copyright © 2003 John Wiley & Sons, Ltd. [source]

The Liapunov's second method for continuous time difference equations

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL, Issue 15 2003
P. PepeArticle first published online: 10 OCT 200
Abstract Among many other cases such as economic and lossless propagation models, continuous time difference equations are encountered as the internal dynamics in a class of non-linear time delay systems, when controlled by a suitable state feedback which drives the output exponentially to zero. The Liapunov's second method for these infinite dimensional systems has not been extensively investigated in the literature. This paper has the aim of filling this gap. Liapunov's second method theorems for checking the stability and the asymptotic stability of this class of infinite dimensional systems are built up, in both a finite and an infinite dimensional setting. In the finite dimensional setting, the Liapunov function is defined on finite dimensional sets. The conditions for stability are given as inequalities on continuous time. No derivatives are involved, as in the dynamics of the studied systems. In the infinite dimensional setting, the continuous time difference equation is transformed into a discrete time system evolving on an infinite dimensional space, and then the classical Liapunov theorem for the system in the new form is written. In this paper the very general case is considered, that is non-linear continuous time difference equations with multiple non commensurate delays are considered, and moreover the functions involved in the dynamics are allowed to be discontinuous, as well as the initial state. In order to study the stability of the internal dynamics in non-linear time delay feedback systems, an exogenous disturbance is added, which goes to zero exponentially as the time goes to infinity. An example is considered, from non-linear time delay feedback theory. While the results available in the literature are inconclusive as far as the stability of that example is concerned, such stability is proved to hold by the theorems developed in this paper, and is validated by simulation results. Copyright © 2003 John Wiley & Sons, Ltd. [source]

Generating potentials via difference equations

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 16 2006
S. D. Maharaj
Abstract The condition for pressure isotropy, for spherically symmetric gravitational fields with charged and uncharged matter, is reduced to a recurrence equation with variable, rational coefficients. This difference equation is solved, in general, using mathematical induction leading to an exact solution to the Einstein field equations which extends the isotropic model of John and Maharaj. The metric functions, energy density and pressure are well behaved, which suggests that this model could be used to describe a relativistic sphere. The model admits a barotropic equation of state, which approximates a polytrope close to the stellar centre. Copyright © 2006 John Wiley & Sons, Ltd. [source]

Fast direct solver for Poisson equation in a 2D elliptical domain

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 1 2004
Ming-Chih Lai
Abstract In this article, we extend our previous work M.-C. Lai and W.-C. Wang, Numer Methods Partial Differential Eq 18:56,68, 2002 for developing some fast Poisson solvers on 2D polar and spherical geometries to an elliptical domain. Instead of solving the equation in an irregular Cartesian geometry, we formulate the equation in elliptical coordinates. The solver relies on representing the solution as a truncated Fourier series, then solving the differential equations of Fourier coefficients by finite difference discretizations. Using a grid by shifting half mesh away from the pole and incorporating the derived numerical boundary value, the difficulty of coordinate singularity can be elevated easily. Unlike the case of 2D disk domain, the present difference equation for each Fourier mode is coupled with its conjugate mode through the numerical boundary value near the pole; thus, those two modes are solved simultaneously. Both second- and fourth-order accurate schemes for Dirichlet and Neumann problems are presented. In particular, the fourth-order accuracy can be achieved by a three-point compact stencil which is in contrast to a five-point long stencil for the disk case. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 72,81, 2004 [source]

The evolution of mathematical immunology

IMMUNOLOGICAL REVIEWS, Issue 1 2007
Yoram Louzoun
Summary:, The types of mathematical models used in immunology and their scope have changed drastically in the past 10 years. Classical models were based on ordinary differential equations (ODEs), difference equations, and cellular automata. These models focused on the ,simple' dynamics obtained between a small number of reagent types (e.g. one type of receptor and one type of antigen or two T-cell populations). With the advent of high-throughput methods, genomic data, and unlimited computing power, immunological modeling shifted toward the informatics side. Many current applications of mathematical models in immunology are now focused around the concepts of high-throughput measurements and system immunology (immunomics), as well as the bioinformatics analysis of molecular immunology. The types of models have shifted from mainly ODEs of simple systems to the extensive use of Monte Carlo simulations. The transition to a more molecular and more computer-based attitude is similar to the one occurring over all the fields of complex systems analysis. An interesting additional aspect in theoretical immunology is the transition from an extreme focus on the adaptive immune system (that was considered more interesting from a theoretical point of view) to a more balanced focus taking into account the innate immune system also. We here review the origin and evolution of mathematical modeling in immunology and the contribution of such models to many important immunological concepts. [source]

A coupled simulation of an explosion inside a lined cavity surrounded by a plastic compressible medium

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 10 2007
V. R. Feldgun
Abstract The paper develops a coupled approach to simulate an axisymmetric explosion inside a buried lined cavity. The approach allows accounting for all the stages of the process: detonation of the internal charge; the shock wave propagation in the internal gas with further interaction with the lining, including multiple reflections; soil,structure dynamic interaction, including multiple gap openings and closures and wave propagation in the surrounding compressible plastic medium. The interaction problem is solved by a combination of the variational difference method and of the modified Godunov's method based on the fixed Eulerian mesh with the so-called mixed cell. The contact pressures acting on the lining due to both detonation products and soil,lining interaction are computed through the solution of the joint system of finite difference equations of gas, shell and soil dynamics using a simple iteration method. Copyright © 2007 John Wiley & Sons, Ltd. [source]

Stability of linear time-periodic delay-differential equations via Chebyshev polynomials

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 7 2004
Eric A. Butcher
Abstract This paper presents a new technique for studying the stability properties of dynamic systems modeled by delay-differential equations (DDEs) with time-periodic parameters. By employing a shifted Chebyshev polynomial approximation in each time interval with length equal to the delay and parametric excitation period, the dynamic system can be reduced to a set of linear difference equations for the Chebyshev expansion coefficients of the state vector in the previous and current intervals. This defines a linear map which is the ,infinite-dimensional Floquet transition matrix U'. Two different formulas for the computation of the approximate U, whose size is determined by the number of polynomials employed, are given. The first one uses the direct integral form of the original system in state space form while the second uses a convolution integral (variation of parameters) formulation. Additionally, a variation on the former method for direct application to second-order systems is also shown. An error analysis is presented which allows the number of polynomials employed in the approximation to be selected in advance for a desired tolerance. An extension of the method to the case where the delay and parametric periods are commensurate is also shown. Stability charts are produced for several examples of time-periodic DDEs, including the delayed Mathieu equation and a model for regenerative chatter in impedance-modulated turning. The results indicate that this method is an effective way to study the stability of time-periodic DDEs. Copyright © 2004 John Wiley & Sons, Ltd. [source]

Incorporating spatially variable bottom stress and Coriolis force into 2D, a posteriori, unstructured mesh generation for shallow water models

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 3 2009
D. Michael Parrish
Abstract An enhanced version of our localized truncation error analysis with complex derivatives (LTEA,CD ) a posteriori approach to computing target element sizes for tidal, shallow water flow, LTEA+CD , is applied to the Western North Atlantic Tidal model domain. The LTEA + CD method utilizes localized truncation error estimates of the shallow water momentum equations and builds upon LTEA and LTEA,CD-based techniques by including: (1) velocity fields from a nonlinear simulation with complete constituent forcing; (2) spatially variable bottom stress; and (3) Coriolis force. Use of complex derivatives in this case results in a simple truncation error expression, and the ability to compute localized truncation errors using difference equations that employ only seven to eight computational points. The compact difference molecules allow the computation of truncation error estimates and target element sizes throughout the domain, including along the boundary; this fact, along with inclusion of locally variable bottom stress and Coriolis force, constitute significant advancements beyond the capabilities of LTEA. The goal of LTEA + CD is to drive the truncation error to a more uniform, domain-wide value by adjusting element sizes (we apply LTEA + CD by re-meshing the entire domain, not by moving nodes). We find that LTEA + CD can produce a mesh that is comprised of fewer nodes and elements than an initial high-resolution mesh while performing as well as the initial mesh when considering the resynthesized tidal signals (elevations). Copyright © 2008 John Wiley & Sons, Ltd. [source]

Using a piecewise linear bottom to fit the bed variation in a laterally averaged, z -co-ordinate hydrodynamic model

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 11 2004
XinJian Chen
Abstract In developing a 3D or laterally averaged 2D model for free-surface flows using the finite difference method, the water depth is generally discretized either with the z -co-ordinate (z -levels) or a transformed co-ordinate (e.g. the so-called , -co-ordinate or , -levels). In a z -level model, the water depth is discretized without any transformation, while in a , -level model, the water depth is discretized after a so-called , -transformation that converts the water column to a unit, so that the free surface will be 0 (or 1) and the bottom will be -1 (or 0) in the stretched co-ordinate system. Both discretization methods have their own advantages and drawbacks. It is generally not conclusive that one discretization method always works better than the other. The biggest problem for the z -level model normally stems from the fact that it cannot fit the topography properly, while a , -level model does not have this kind of a topography-fitting problem. To solve the topography-fitting problem in a laterally averaged, 2D model using z -levels, a piecewise linear bottom is proposed in this paper. Since the resulting computational cells are not necessarily rectangular looking at the x,z plane, flux-based finite difference equations are used in the model to solve the governing equations. In addition to the piecewise linear bottom, the model can also be run with full cells or partial cells (both full cell and partial cell options yield a staircase bottom that does not fit the real bottom topography). Two frictionless wave cases were chosen to evaluate the responses of the model to different treatments of the topography. One wave case is a boundary value problem, while the other is an initial value problem. To verify that the piecewise linear bottom does not cause increased diffusions for areas with steep bottom slopes, a barotropic case in a symmetric triangular basin was tested. The model was also applied to a real estuary using various topography treatments. The model results demonstrate that fitting the topography is important for the initial value problem. For the boundary value problem, topography-fitting may not be very critical if the vertical spacing is appropriate. Copyright © 2004 John Wiley & Sons, Ltd. [source]

Positivity-preserving, flux-limited finite-difference and finite-element methods for reactive transport

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 2 2003
Robert J. MacKinnon
Abstract A new class of positivity-preserving, flux-limited finite-difference and Petrov,Galerkin (PG) finite-element methods are devised for reactive transport problems. The methods are similar to classical TVD flux-limited schemes with the main difference being that the flux-limiter constraint is designed to preserve positivity for problems involving diffusion and reaction. In the finite-element formulation, we also consider the effect of numerical quadrature in the lumped and consistent mass matrix forms on the positivity-preserving property. Analysis of the latter scheme shows that positivity-preserving solutions of the resulting difference equations can only be guaranteed if the flux-limited scheme is both implicit and satisfies an additional lower-bound condition on time-step size. We show that this condition also applies to standard Galerkin linear finite-element approximations to the linear diffusion equation. Numerical experiments are provided to demonstrate the behavior of the methods and confirm the theoretical conditions on time-step size, mesh spacing, and flux limiting for transport problems with and without nonlinear reaction. Copyright © 2003 John Wiley & Sons, Ltd. [source]

Stability and robust stability of positive linear Volterra difference equations

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL, Issue 5 2009
Pham Huu Anh Ngoc
Abstract We first introduce a class of positive linear Volterra difference equations. Then, we offer explicit criteria for uniform asymptotic stability of positive equations. Furthermore, we get a new Perron,Frobenius theorem for positive linear Volterra difference equations. Finally, we study robust stability of positive equations under structured perturbations and affine perturbations. Two explicit stability bounds with respect to these perturbations are given. Copyright © 2008 John Wiley & Sons, Ltd. [source]

Exponential estimates for neutral time delay systems with multiple delays

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL, Issue 2 2006
Vladimir Kharitonov
Abstract Exponential estimates and sufficient conditions for the exponential stability of linear neutral time delay for systems with multiple delays are given. The case of systems with uncertainties, including uncertainties in the difference operator, is considered. The proofs follows from new results on non-homogeneous difference equations evolving in continuous time combined with the Lyapunov,Krasovskii functionals approach. The conditions are expressed in terms of linear matrix inequalities. The particular case of neutral time delay systems with commensurate delays, which leads to less restrictive exponential estimates, is also addressed. Copyright © 2005 John Wiley & Sons, Ltd. [source]

The Liapunov's second method for continuous time difference equations

Difference Equations for the Higher Order Moments and Cumulants of the INAR(p) Model

Generating potentials via difference equations

A Macroeconomic Model with Hysteresis in Foreign Trade

METROECONOMICA, Issue 4 2001
Matthias Gocke
The continuous non-linear macro-hysteresis loop is approximated by a rhombus shaped path which therefore shows a closer affinity to the genuine concept of hysteresis than conventional techniques via difference equations. This linearized model is applied to implement foreign trade hysteresis in a standard macroeconomic simultaneous equation model demonstrating the persisting consequences of only temporary exogenous shocks on national income, interest rate and the determination of the exchange rate. Since hysteresis in foreign trade is analysed in a macroeconomic framework, the feedback of hysteresis caused by exchange rate variations on the exchange rate itself can be illustrated. [source]