Global existence and asymptotic behaviour of the solution of a generalized burger's equation with viscosity

We obtain the global existence and uniqueness for a generalized Burger's equation with viscosity and the initial value being in L^∞ by successive method. Moreover, under certain condition on the initial value the solution tends to the solution of a linear heat equation in H¹.     

On the propagation of 1D solitary waves in Mindlin-type microstructured solids

The Mindlin model and hierarchical approach by Engelbrecht and Pastrone are used for modelling 1D wave propagation in microstructured solids. After introducing the free energy function, one gets from Euler–Lagrange equations a system of equations of motion. Making use of the slaving principle, a nonlinear hierarchical wave equation can be derived. Equations are solved numerically under localized initial conditions. For numerical integration, the pseudospectral method based on the Fourier transform is used. The influence of free energy parameters on the character of dispersion and wave propagation is studied. Numerical results of hierarchical approximation and the full equation system will be compared and the quality of the approximation will be discussed.     

Solutions of nonlinear delay and advanced partial difference equations in the space lN×N1

A functional analytic method is used to prove a general theorem which establishes the existence and the uniqueness of a solution of a class of nonlinear delay and advanced partial difference equations in the Banach space l_N×N¹. The proof of the theorem has a constructive character, which enables us to obtain a bound of the solution and a region, depending on the initial conditions and the parameters of the equation under consideration, where this solution holds. Some known nonlinear partial difference equations, which appear in applications, are studied as particular cases of the theorem.     

Thermodynamic assessment of the KCl–K2CO3–NaCl–Na2CO3 system

Phase equilibria and thermodynamic properties of the KCl–K₂CO₃–NaCl–Na₂CO₃ system were analyzed on the basis of the thermodynamic evaluation of the KCl–NaCl,KCl–K₂CO₃,NaCl–Na₂CO₃,K₂CO₃–Na₂CO₃ and KCl–K₂CO₃–NaCl–Na₂CO₃ systems. The Gibbs energies of individual phases was approximated by two-sublattice models for ionic liquids and crystals. Most of the experimental information was well described by the present set of thermodynamic parameters. The lowest monovariant eutectic temperature in the KCl–NaCl–Na₂CO₃ system is located at 573 ^°C, with a composition of X_Na2CO3=0.31,X_KCl=0.35 and X_NaCl=0.34.     

Motion design with Euler–Rodrigues frames of quintic Pythagorean-hodograph curves

The paper presents an interpolation scheme for G¹ Hermite motion data, i.e., interpolation of data points and rotations at the points, with spatial quintic Pythagorean-hodograph curves so that the Euler–Rodrigues frame of the curve coincides with the rotations at the points. The interpolant is expressed in a closed form with three free parameters, which are computed based on minimizing the rotations of the normal plane vectors around the tangent and on controlling the length of the curve. The proposed choice of parameters is supported with the asymptotic analysis. The approximation error is of order four and the Euler–Rodrigues frame differs from the ideal rotation minimizing frame with the order three. The scheme is used for rigid body motions and swept surface construction.     

The two-electron atomic systems. S-states

A simple Mathematica program for computing the S-state energies and wave functions of two-electron (helium-like) atoms (ions) is presented. The well-known method of projecting the Schrödinger equation onto the finite subspace of basis functions was applied. The basis functions are composed of the exponentials combined with integer powers of the simplest perimetric coordinates. No special subroutines were used, only built-in objects supported by Mathematica. The accuracy of results and computation time depend on the basis size. The precise energy values of 7-8 significant figures along with the corresponding wave functions can be computed on a single processor within a few minutes. The resultant wave functions have a simple analytical form consisting of elementary functions, that enables one to calculate the expectation values of arbitrary physical operators without any difficulties.

Program summary

Program title: TwoElAtom-SCatalogue identifier: AEFK_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEFK_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 10 185No. of bytes in distributed program, including test data, etc.: 495 164Distribution format: tar.gzProgramming language: Mathematica 6.0; 7.0Computer: Any PCOperating system: Any which supports Mathematica; tested under Microsoft Windows XP and Linux SUSE 11.0RAM:?¹⁰⁹ bytesClassification: 2.1, 2.2, 2.7, 2.9Nature of problem: The Schrödinger equation for atoms (ions) with more than one electron has not been solved analytically. Approximate methods must be applied in order to obtain the wave functions or other physical attributes from quantum mechanical calculations.Solution method: The S-wave function is expanded into a triple basis set in three perimetric coordinates. Method of projecting the two-electron Schrödinger equation (for atoms/ions) onto a subspace of the basis functions enables one to obtain the set of homogeneous linear equations F.C=0 for the coefficients C of the above expansion. The roots of equation det(F)=0 yield the bound energies.Restrictions: First, the too large length of expansion (basis size) takes the too large computation time giving no perceptible improvement in accuracy. Second, the order of polynomial Ω (input parameter) in the wave function expansion enables one to calculate the excited nS-states up to n=Ω+1 inclusive.Additional comments: The CPC Program Library includes “A program to calculate the eigenfunctions of the random phase approximation for two electron systems” (AAJD). It should be emphasized that this fortran code realizes a very rough approximation describing only the averaged electron density of the two electron systems. It does not characterize the properties of the individual electrons and has a number of input parameters including the Roothaan orbitals.Running time: ∼10 minutes (depends on basis size and computer speed)     

Online shopping delivery delay: Finding a psychological recovery strategy by online consumer experiences

We propose a new type of psychological recovery strategy (i.e., online consumer experiences, OCEs) for online shopping delivery delay. We developed 4 types of OCEs based on the interactivity concept (low machine interactivity with low person interactivity, MI^L–PI^L; high machine interactivity with low person interactivity, MI^H–PI^L; low machine interactivity with high person interactivity, MI^L–PI^H, and high machine interactivity with high person interactivity, MI^H–PI^H). We conducted 2 studies, and 1078 online shoppers participated in this study. The results indicate that when consumers meet a delivery delay, proving them with OCEs could increase satisfaction and reduce complaint intention. Cognitive dissonance (CD) from inconsistency between OCEs and direct experience also moderates OCEs effects on satisfaction, repurchase intention, and complaint intention. Finally, desire for control (DC) and consumer susceptibility to interpersonal influence (CSII) moderate the OCEs effects on satisfaction, repurchase intention, and complaint intention: consumers with low DC and low CSII prefer MI^L–PI^L; consumers with high DC and low CSII prefer MI^H–PI^L, consumers with low DC and high CSII prefer MI^L–PI^H, and consumers with high DC and high CSII prefer MI^H–PI^H.     

A numerical investigation of errors arising in applying the galerkin method of the solution of nonlinear partial differential equations

The Galerkin method is applied to the solution of Burgers' equation and a nonlinear wave equation, using expansions of B-splines of increasing number of terms and order of spline. The accuracy of the solutions obtained numerically is compared with analytical solutions, and the effect upon accuracy of increasing the order of spline and number of terms in the expansion is considered for a variety of initial conditions corresponding to waves having a range of wavelengths.Burgers' equation is used as a model for the hydrodynamic shallow water equations, and results illustrate the importance of using a sufficient number of functions in the expansion to accurately model the distortion of a wave progressing into a shallow water region where shorter waves contribute appreciably to the total wave profile.     

Thermodynamic reassessment of the Y 2O3–Al2O3–SiO2 system and its subsystems

Phase equilibria and thermodynamic properties at 1 bar in the Y ₂O₃–Al₂O₃–SiO₂ ternary system and its constituent binaries Y ₂O₃–Al₂O₃ and Y ₂O₃–SiO₂ have been reevaluated using the CALPHAD approach. The liquid phase is described by the ionic two-sublattice model with the formula (Al⁺³,Y ⁺³)_P(AlO₂⁻¹,O⁻²,SiO₄⁻⁴,SiO₂⁰)_Q. The SiO₂ solubility in the YAM phase was described using a compound energy model. Two datasets of self-consistent model parameters are presented. However, the rather meagre and scattered experimental data imply that the present assessments should be regarded as provisional. Some critical experiments are suggested for this system.     

Controllability of a Fokker-Planck equation,the Schrödinger system,and a related stochastic optimal control (revised version)

A controllability problem for a Fokker-Planck equation is termedProblem A. Under proper assumptions, a solution (v^*, Ф^*) to that problem is constructed by a Theorem of Jamison. Theorem 2 gives a sufficiency condition concerning the given initial and terminal data for that solution to exist. Theorem 3 states that v^* is an optimal feedback control for a stochastic optimal control problem with constraint on the end-state, termedProblem B. Further, v^* corresponds to the minimum of an entropy distance. Finally, Problem A is transformed into a controllability problem for a stochastic differential equation, termedProblem C: the solution to Problem C corresponding to the one constructed in Problem A is the Markovian process satisfying the given end conditions in a set of reciprocal processes of Jamison.     

A note on stabilization of locally damped wave equations with time delay

In this paper, we consider the wave equation with internal distributed time delay and local damping in a bounded and smooth domain Ω⊂Rⁿ. When the local damping acts on a neighborhood of a suitable part of the boundary of Ω, we show that an exponential stability result holds if the coefficient of the delay term is sufficiently small.     

Sufficient conditions for the preservation of the boundedness in a numerical method for a physical model with transport memory and nonlinear damping

Departing from a finite-difference scheme to approximate solutions of a nonlinear, hyperbolic partial differential equation which generalizes the Burgers–Huxley equation from fluid dynamics, we investigate conditions on the model coefficients and the computational parameters under which positive and bounded initial data evolve into positive and bounded new approximations. The model under investigation includes nonlinear coefficients of damping and advection, and the reaction term extends the reaction law of the classical Fisher–Kolmogorov–Petrovsky–Piscounov equation. The method can be expressed in vector form in terms of a multiplicative matrix which, under certain parametric conditions, becomes an M-matrix. Using the fact that every M-matrix is non-singular and that the entries of its inverse are positive, real numbers, we establish sufficient conditions under which the method provides new, positive and bounded approximations from previous, positive and bounded data and boundary conditions. The numerical results confirm the fact that the conditions derived here are sufficient for the positivity and the boundedness of the approximations; moreover, computational experiments evidence the fact that the method still preserves these properties for values of the model and the numerical parameters outside of the analytic regions of positivity and boundedness. We point out that our simulations show a good agreement between the numerical approximations computed through our method and the corresponding, analytical solutions.     

Quantum billiards in multidimensional models with fields of forms

A Bianchi type I cosmological model in (n + 1)-dimensional gravity with several forms is considered. When the electric non-composite brane ansatz is adopted, the Wheeler-DeWitt (WDW) equation for the model, written in a conformally covariant form, is analyzed. Under certain restrictions, asymptotic solutions to the WDW equation near the singularity are found, which reduce the problem to the so-called quantum billiard on the (n ? 1)-dimensional Lobachevsky space ? ^n?1. Two examples of quantum billiards are considered: a 2-dimensional quantum billiard for a 4D model with three 2-forms and a 9D quantum billiard for an 11D model with 120 4-forms, whichmimics the SM2-brane sector of D = 11 supergravity. For certain solutions, vanishing of the wave function at the singularity is proved.     

Critical thermodynamic evaluation and optimization of the MnO–SiO2–“ TiO2 ”–“ Ti2O3 ” system

A complete review, critical evaluation, and thermodynamic optimization of phase equilibrium and thermodynamic properties of the MnO–SiO₂–“ TiO₂”–“ Ti₂O₃” systems at 1 bar pressure are presented. The molten oxide phase was described by the Modified Quasichemical Model. The Gibbs energies of the manganosite, spinel, pyrophanite and pseudobrookite and rutile solid solutions were taken from the previous study. A set of optimized model parameters for the molten oxide phase was obtained which reproduces all available reliable thermodynamic and phase equilibrium data within experimental error limits from 25 ^°C to above the liquidus temperatures over the entire range of compositions and oxygen partial pressure in the range of pO₂ from 10⁻²⁰ bar to 10⁻⁷ bar. Complex phase relationships in these systems have been elucidated, and discrepancies among the data have been resolved. The database of model parameters can be used along with software for Gibbs energy minimization in order to calculate any phase diagram section or thermodynamic properties.     

Applications of complete discrimination system for polynomial for classifications of traveling wave solutions to nonlinear differential equations

If a partial differential equation is reduced to an ordinary differential equation in the form u^′(ξ)=G(u,θ₁,…,θm) under the traveling wave transformation, where θ₁,…,θm are parameters, its solutions can be written as an integral form . Therefore, the key steps are to determine the parameters' scopes and to solve the corresponding integral. When G is related to a polynomial, a mathematical tool named complete discrimination system for polynomial is applied to this problem so that the parameter's scopes can be determined easily. The complete discrimination system for polynomial is a natural generalization of the discrimination △=b²−4ac of the second degree polynomial ax²+bx+c. For example, the complete discrimination system for the third degree polynomial F(w)=w³+d₂w²+d₁w+d₀ is given by and . In the paper, we give some new applications of the complete discrimination system for polynomial, that is, we give the classifications of traveling wave solutions to some nonlinear differential equations through solving the corresponding integrals. In finally, as a result, we give a partial answer to a problem on Fan's expansion method.     

Joint maximum likelihood estimation of unit root testing equations and GARCH processes: Some finite-sample issues

In recent research [B. Seo, Distribution theory for unit root tests with conditional heteroskedasticity, J. Econometrics 91 (1999) 113–144] has suggested that the examination of the unit root hypothesis in series exhibiting GARCH behaviour should proceed via joint maximum likelihood (ML) estimation of the unit root testing equation and GARCH process. The results presented show the asymptotic distribution of the resulting ML t-test to be a mixture of the Dickey–Fuller and standard normal distributions. In this paper, the relevance of these asymptotic arguments is considered for the finite samples encountered in empirical research. In particular, the influences of sample size, alternative values of the parameters of the GARCH process and the use of the Bollerslev–Wooldridge covariance matrix estimator upon the finite-sample distribution of the ML t-statistic are explored. It is shown that the resulting critical values for the ML t-statistic are similar to those of the Dickey–Fuller distribution rather than the standard normal, unless a large sample size and empirically unrealistic values of the volatility parameter of the GARCH process are considered. Use of the Bollerslev–Wooldridge standard covariance matrix estimator exaggerates this finding, causing a leftward shift in the finite-sample distribution of the ML t-statistic. The results of the simulation analysis are illustrated via an application to U.S. short term interest rates.     

An innovative method for dynamic update of initial water table in XXT model based on neural network technique

The initial subsurface flow of whole basin plays a quite important role in daily rainfall–runoff simulation. However, general physically based rainfall–runoff model, such as the XXT model (a hybrid model of TOPographic MODEL and the Xinanjiang model), is difficult to catch the non-linear factors and take full advantages of previous information of rainfall and runoff that is essential to the initial watershed average saturation deficit of each time step. In order to address the issue, this study selected the initial subsurface flow for the whole time series of the XXT model as the breakthrough point, and used the observed runoff and rainfall data of two days before the present day as the inputs of artificial neural network (ANN) and initial subsurface flow of the present day as the output, then integrated ANN into runoff generation module of XXT model and finally tested the integrated model for daily runoff simulation in large-scale and semi-arid Linyi watershed, eastern China. In addition, this work employ particle swarm optimization (PSO) algorithm to seek the best combination of 6 physical parameters in XXT and a great number of weights in ANN to avoid the local optimization. The results show that the integrated model performs much better than XXT in terms of Nash–Sutcliffe efficiency coefficient (NE) and root mean square error (RMSE). Hence, the new integrating approach proposed here is promising for daily rainfall–runoff modeling and can be easily extended to other process-based models.     

Tension spline approach for the numerical solution of nonlinear Klein-Gordon equation

The nonlinear Klein-Gordon equation describes a variety of physical phenomena such as dislocations, ferroelectric and ferromagnetic domain walls, DNA dynamics, and Josephson junctions. We derive approximate expressions for the dispersion relation of the nonlinear Klein-Gordon equation in the case of strong nonlinearities using a method based on the tension spline function and finite difference approximations. The resulting spline difference schemes are analyzed for local truncation error, stability and convergence. It has been shown that by suitably choosing the parameters, we can obtain two schemes of O(k²+k²h²+h²) and O(k²+k²h²+h⁴). In the end, some numerical examples are provided to demonstrate the effectiveness of the proposed schemes.     

Energy-efficient train control: From local convexity to global optimization and uniqueness

The optimal driving strategy for a train is essentially a power–speedhold–coast–brake strategy unless the track contains steep grades in which case the speedhold mode must be interrupted by phases of power for steep uphill sections and coast for steep downhill sections. The Energymiser^®  device is used on freight and passenger trains in Australia and the United Kingdom to provide on-board advice for drivers about energy-efficient driving strategies. Energymiser^®  uses a specialized numerical algorithm to find optimal switching points for each steep section of track. Although the algorithm finds a feasible strategy that satisfies the necessary optimality conditions there has been no direct proof that the corresponding switching points are uniquely defined. We use a comprehensive perturbation analysis to show that a key local energy functional is convex with a unique minimum and in so doing prove that the optimal switching points are uniquely defined for each steep section of track. Hence we also deduce that the global optimal strategy is unique. We present two examples using realistic parameter values.     

Critical thermodynamic evaluation and optimization of the MnO–“ TiO2 ”–“ Ti2O3 ” system

A complete review, critical evaluation, and thermodynamic optimization of the phase equilibrium and thermodynamic properties of the MnO–“ TiO₂”–“ Ti₂O₃” systems at 1 bar pressure are presented. The molten oxide phase was described by the Modified Quasichemical Model. The Gibbs energy of spinel, pyrophanite and pseudobrookite solid solutions were modeled using the Compound Energy Formalism, and rutile solid solution was treated as a simple Henrian solution. Manganosite solid solution was assumed to dissolve both Ti⁴⁺ and Ti³⁺. A set of optimized model parameters for all phases was obtained which reproduces all available reliable thermodynamic and phase equilibrium data within experimental error limits from 25 ^°C to above the liquidus temperatures over the entire composition ranges and in the range of pO₂ from 10⁻²⁰ to 10⁻⁷ bar. Complex phase relationships in these systems have been elucidated, and discrepancies among the data have been resolved. The database of model parameters can be used along with software for Gibbs energy minimization in order to calculate any phase diagram section or thermodynamic properties.