Convergence of the spline function for functional-differential equation of neutral type

The existence, uniqueness and stability for the functional-differential equation of neutral type using spline of deficiency 3 with stepsize 3h spline function of degree four are presented in Ref. [1]. In this paper, we extend the study to the convergence of our proposed spline method. We prove that, if the local error is of order p, then the global error is of order p as well. Numerical examples are presented to illustrate the convergence of the method.     

A Taylor Collocation Method for the Solution of Linear Integro-Differential Equations

In this study, a matrix method called the Taylor collocation method is presented for numerically solving the linear integro-differential equations by a truncated Taylor series. Using the Taylor collocation points, this method transforms the integro-differential equation to a matrix equation which corresponds to a system of linear algebraic equations with unknown Taylor coefficients. Also the method can be used for linear differential and integral equations. To illustrate the method, it is applied to certain linear differential, integral, and integro-differential equations and the results are compared.     

Uniqueness and numerical scheme for the Robin coefficient identification of the time-fractional diffusion equation

We study an inverse problem of determining the Robin coefficient of fractional diffusion equation from a nonlocal boundary condition. Based on the property of Caputo fractional derivative, the uniqueness is proved. The numerical schemes for the direct problem and the inverse problem are developed. Three examples are given to show the effectiveness of the presented methods.     

有滞后的中立型控制系统与无滞后控制系统在镇定理论中的等价性

本文对有滞后的中立型控制系统与无滞后控制系统在镇定理论中的等价性进行了讨论,用不等式估值方法及分析技巧给出了其等价的时滞范围,为简化相应的中立型控制系统的分析和设计提供了理论依据.     

On the Convergence of a Spline Method for Singular Two Point Boundary Value Problems Arising in Physiology

The second order spline method developed by Iyengar et al. [11] for the problems $$eqalign{ x^{-alpha}(x^alpha y^prime)^prime & = f(x,y),quad 0     

An Effective High-Order Interval Method for Validating Existence and Uniqueness of the Solution of an IVP for an ODE

Validated methods for initial value problems for ordinary differential equations produce bounds that are guaranteed to contain the true solution of a problem. When computing such bounds, these methods verify that a unique solution to the problem exists in the interval of integration and compute a priori bounds for the solution in this interval. A major difficulty in this verification phase is how to take as large a stepsize as possible, subject to some tolerance requirement. We propose a high-order enclosure method for proving existence and uniqueness of the solution and computing a priori bounds.     

Stability and convergence of the two parameter cubic spline collocation method for delay differential equations

In this paper, we propose the cubic spline collocation method with two parameters for solving delay differential equations (DDEs). Some results of the local truncation error and the convergence of the spline collocation method are given. We also obtain some results of the linear stability and the nonlinear stability of the method for DDEs. In particular, we design an algorithm to obtain the ranges of the two parameters α,β which are necessary for the P-stability of the collocation method. Some illustrative examples successfully verify our theoretical results.     

On the Liapunov-Krasovskii methodology for the ISS of systems described by coupled delay differential and difference equations

The input-to-state stability of time-invariant systems described by coupled differential and difference equations with multiple noncommensurate and distributed time delays is investigated in this paper. Such equations include neutral functional differential equations in Hale’s form (which model, for instance, partial element equivalent circuits) and describe lossless propagation phenomena occurring in thermal, hydraulic and electrical engineering. A general methodology for systematically studying the input-to-state stability, by means of Liapunov-Krasovskii functionals, with respect to measurable and locally essentially bounded inputs, is provided. The technical problem concerning the absolute continuity of the functional evaluated at the solution has been studied and solved by introducing the hypothesis that the functional is locally Lipschitz. Computationally checkable LMI conditions are provided for the linear case. It is proved that a linear neutral system in Hale’s form with stable difference operator is input-to-state stable if and only if the trivial solution in the unforced case is asymptotically stable. A nonlinear example taken from the literature, concerning an electrical device, is reported, showing the effectiveness of the proposed methodology.     

A Correspondence between Martin-Löf Type Theory, the Ramified Theory of Types and Pure Type Systems

In Russell's Ramified Theory of Types RTT, two hierarchical concepts dominate:orders and types. The use of orders has as a consequencethat the logic part of RTT is predicative.The concept of order however, is almost deadsince Ramsey eliminated it from RTT. This is whywe find Church's simple theory of types (which uses the type concept without the order one) at the bottom of the Barendregt Cube rather than RTT.Despite the disappearance of orders which have a strong correlation with predicativity, predicative logic still plays an influential role in Computer Science.An important example is the proof checker Nuprl, which is basedon Martin-Löf's Type Theory which uses type universes. Those type universes,and also degrees of expressions in AUTOMATH, are closely related toorders. In this paper, we show that orders have not disappeared frommodern logic and computer science, rather, orders play a crucial role in understanding the hierarchy of modern systems. In order to achieve our goal, we concentrate on a subsystem of Nuprl.The novelty of our paper lies in: (1) a modest revival of Russell'sorders, (2) the placing of the historical system RTTunderlying the famous Principia Mathematica in a context with a modernsystem of computer mathematics (Nuprl) and modern type theories(Martin-Löf's type theory and PTSs), and (3) the presentation of acomplex type system (Nuprl) as a simple and compact PTS.     

Parallel Difference Schemes For Heat Conduction Equations * The work is supported by the Chinese National Key Program for Developing Basic Science,G1999032801, the National Natural Science Foundation of China (19932010) and the Foundation of CAEP (2X0107)

In this paper a parallel difference scheme based on Dufort-Frankel scheme and the classic implicit scheme for linear heat conduction equations is studied. In this procedure, the values at subdomain interfaces are calculated by using the Dufort-Frankel scheme, and then these values serve as Dirichlet boundary data for the implicit scheme in the subdomains. The weak necessary condition of the unconditional stability of the parallel difference scheme is proved. Numerical experiments indicates that the parallel difference scheme has good parallelism, and has better accuracy than the fully implicit scheme.     

The Use of the B Formal Method for the Design and the Validation of the Transaction Mechanism for Smart Card Applications

This document describes an industrial application of the B method in smart card applications. In smart card memory, data modification may be interrupted due to a card withdrawal or a power loss. The EEPROM memory may result in an unstable state and the values subsequently read may be erroneous. The transaction mechanism provides a secure means for modifying data located in the EEPROM. As the security in smart card applications is paramount, the use of the B formal method brings high confidence and provides mathematical proof that the design of the transaction mechanism fulfills the security requirements.     

A Galerkin Method for the Simulation of the Transient 2-D/2-D and 3-D/3-D Linear Boltzmann Equation

Many production steps used in the manufacturing of integrated circuits involve the deposition of material from the gas phase onto wafers. Models for these processes should account for gaseous transport in a range of flow regimes, from continuum flow to free molecular or Knudsen flow, and for chemical reactions at the wafer surface. We develop a kinetic transport and reaction model whose mathematical representation is a system of transient linear Boltzmann equations. In addition to time, a deterministic numerical solution of this system of kinetic equations requires the discretization of both position and velocity spaces, each two-dimensional for 2-D/2-D or each three-dimensional for 3-D/3-D simulations. Discretizing the velocity space by a spectral Galerkin method approximates each Boltzmann equation by a system of transient linear hyperbolic conservation laws. The classical choice of basis functions based on Hermite polynomials leads to dense coefficient matrices in this system. We use a collocation basis instead that directly yields diagonal coefficient matrices, allowing for more convenient simulations in higher dimensions. The systems of conservation laws are solved using the discontinuous Galerkin finite element method. First, we simulate chemical vapor deposition in both two and three dimensions in typical micron scale features as application example. Second, stability and convergence of the numerical method are demonstrated numerically in two and three dimensions. Third, we present parallel performance results which indicate that the implementation of the method possesses very good scalability on a distributed-memory cluster with a high-performance Myrinet interconnect.     

Gibbs2: A new version of the quasiharmonic model code. II. Models for solid-state thermodynamics, features and implementation

In the second article of the series, we present the Gibbs2 code, a Fortran90 reimplementation of the original Gibbs program [Comput. Phys. Commun. 158 (2004) 57] for the calculation of pressure–temperature dependent thermodynamic properties of solids under the quasiharmonic approximation. We have taken advantage of the detailed analysis carried out in the first paper to implement robust fitting techniques. In addition, new models to introduce temperature effects have been incorporated, from the simple Debye model contained in the original article to a full quasiharmonic model that requires the phonon density of states at each calculated volume. Other interesting novel features include the empirical energy corrections, that rectify systematic errors in the calculation of equilibrium volumes caused by the choice of the exchange-correlation functional, the electronic contributions to the free energy and the automatic computation of phase diagrams. Full documentation in the form of a user?s guide and a complete set of tests and sample data are provided along with the source code.

Program summary

Program title:Gibbs2Catalogue identifier: AEJI_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEJI_v1_0.htmlProgram obtainable from: CPC Program Library, Queen?s University, Belfast, N. IrelandLicensing provisions: GNU General Public License, v3No. of lines in distributed program, including test data, etc.: 936 087No. of bytes in distributed program, including test data, etc.: 8 596 671Distribution format: tar.gzProgramming language: Fortran90Computer: Any running Unix/LinuxOperating system: Unix, GNU/LinuxClassification: 7.8External routines: Part of the minpack, pppack and slatec libraries (downloaded from www.netlib.org) are distributed along with the program.Nature of problem: Given the static E(V) curve, and possibly vibrational information such as the phonon density of states, calculate the equilibrium volume and thermodynamic properties of a solid at arbitrary temperatures and pressures in the framework of the quasiharmonic approximation.Additional comments: A detailed analysis concerning the fitting of equations of state has been carried out in the first part of this article, and implemented in the code presented here.Running time: The tests provided only take a few seconds to run.     

(n+1)维空间Cn[a,b]上规范B基存在的充要条件

给出(n 1)维空间C^n[a,b]上规范B基存在的充要条件，并就摆线的规范B基的非退化表示进行讨论，得到了函数空间Pl＝span{l，t，cost,sint}中摆线的非退化规范B基表示．     

Method of calculating the collision integral and solution of the Boltzmann kinetic equation for simple gases,gas mixtures and gases with rotational degrees of freedom

The conservative method of calculating the Boltzmann collision integral for simple gases, gas mixtures and gases with rotational degrees of freedom of molecules is presented. In all cases the common approach based on the projection technique of summing up the contributions in the collision integral is used. The method is applied for solving the Boltzmann kinetic equation for two fundamental rarefied gas flow problems: the heat transfer problem and the problem of shock wave structure. A comparison with experimental and numerical data of other authors is reported. It is shown that the considered numerical method allows one to solve the Boltzmann equation for real gases with high accuracy.