Abel inversion of axially-symmetric shock wave flows

Finite-fringe interferograms produced for axisymmetric shock wave flows are analyzed by Fourier transform fringe analysis and an Abel inversion method to produce density field data for the validation of numerical models. For the Abel inversion process, we use basis functions to model phase data from axially-symmetric shock wave structure. Steady and unsteady flow problems are studied, and compared with numerical simulations. Good agreement between theoretical and experimental results is obtained when one set of basis functions is used during the inversion process, but the shock front is smeared when another is used. This is because each function in the second set of basis functions is infinitely differentiable, making them poorly-suited to the modelling of a step function as is required in the representation of a shock wave.Received: 12 November 2003, Accepted: 21 October 2004, Published online: 31 March 2005[/PUBLISHED]PACS: 47.40.-x, 42.40.Kw, 02.30.Zz     

Parametric excitation of beams moving over supports

Studied in this work are the formulation of equations of motion and the response to parametric excitation of a uniform cantilever beam moving longitudinally over a single bilateral support. The equations of motion are generated by using assumed modes to discretize the beam, by regarding the support as a kinematic constraint, and by employing an alternate form of Kane's method that is particularly well suited to systems subject to constraints. Instability information is developed using the results of perturbation analysis for harmonic longitudinal motions of small amplitude and with Floquet theory for general periodic motions of any amplitude. Results demonstrate that definitive instability information can be obtained for a beam moving longitudinally over supports based on the frequencies of free transverse vibration of a beam that is longitudinally fixed.     

Equivalent Boundary Integral Equations with indirect unknowns for thin elastic plate bending theory

Equivalent Boundary Integral Equations (EBIE) with indirect unknowns for thin elastic plate bending theory, which is equivalent to the original boundary value problem, is established rigorously by mathematical technique of non-analytic continuation and is fully proved by means of the variational principle. The previous three kinds of boundary integral equations with indirect unknowns are discussed thoroughly and it is shown that all previous results are not EBIE. Paper from SUN Huan-chun, Member of Editorial Commitee, AMM Biography: ZHANG Yao-ming (1962-)     

Dichotomy Spectrum for Nonautonomous Differential Equations

For nonautonomous linear differential equations x=A(t) x with locally integrable A: RR ^N×N the so-called dichotomy spectrum is investigated in this paper. As the closely related dichotomy spectrum for skew product flows with compact base (Sacker–Sell spectrum) our dichotomy spectrum for nonautonomous differential equations consists of at most N closed intervals, which in contrast to the Sacker–Sell spectrum may be unbounded. In the constant coefficients case these intervals reduce to the real parts of the eigenvalues of A. In any case the spectral intervals are associated with spectral manifolds comprising solutions with a common exponential growth rate. The main result of this paper is a spectral theorem which describes all possible forms of the dichotomy spectrum.     

An Accurate Finite Difference Scheme for Boussinesq Equations

In this paper, a stable and accurate finite difference scheme using a space-staggered grid is proposed for solving the extended Boussinesq-type equations as derived by Nwogu [Journal of Waterway, Port Coastal and Ocean Engineering, ASCE, 119, (1993) 618–638]. The alternate direction iterative method combined with an efficient predictor–corrector scheme is adopted for the numerical solution of the governing differential equations. The proposed method is verified by two test cases where experimental data are available for comparison. The first case is wave focusing by bottom topography as studied by Whalin [The limit of applicability of linear wave refraction theory in a convergence zone. Res. Rep. H-71-3, U.S.Army Corps of Engrs. Waterways Expt. Station, Vicksburg (1971)]. The second case is wave runup around a circular cylinder as investigated experimentally by Isaacson (Journal of the Waterway, Port, Coastal and Ocean Division, ASCE, 104, (1978), 69–79). Numerical results agree very well with the corresponding experimental data in both cases.     

矩阵黎卡提方程的精细积分法

选择恰当的参数，将２￣Ｎ类算法用于代数与微分黎卡提方程。证明了算得的解是如此精确，几乎是计算机上的精确解。数例验证了该结论。     

Perturbation theory for the double sine-Gordon equation

This paper presents a perturbation theory for the double sine-Gordon equation. We obtain a system of differential equations that shows the soliton parameters modification under the influence of the perturbation. For the particular case λ=0 the results transform into the well-known perturbation theory for the sine-Gordon equation.     

Constitutive Equations in Continuum Mechanics

One of the main research areas in continuum mechanics is constitutive equations. There is no generally accepted viewpoint on or method for this research. Several concepts have appeared, but further studies are still necessary. The paper elucidates some problems and viewpoints of such research     

Periodic Solutions for Some Systems of Delay Differential Equations

In this paper, we give sufficient conditions for the existence of periodic orbits of some systems of delay differential equations with a unique delay having 3, 4 or n equations. Moreover, we provide examples of delay systems satisfying the different sets of sufficient conditions.     

Wave Dynamics of Saturated Porous Media and Evolutionary Equations

Nonlinear wave dynamics of an elastically deformed saturated porous media is investigated following the Biot approach. Mathematical models under research are the Biot model and its generalization by consideration of viscous stresses inside liquids. Using two-scales and linear WKB methods, the classical Biot system is transformed to a first-order wave equation. To construct the solution of the other system, an asymptotic modified two-scales method is developed. Initial system of equations is transformed to a nonlinear generalized Korteweg–de Vries–Burgers equation for quick elastic wave. Distinctions of wave propagation in the context of the Biot model and its generalization are shown.     

Equations of motion for mechanical systems: A unified approach

This paper presents a unified framework from which emerge the Lagrange equations, the Gibbs-Appell Equations and the Generalized Inverse Equations for describing the motion of constrained mechanical systems. The unified approach extends the applicability of the first two approaches to systems where the constraints are non-linear functions of the generalized velocities and are not necessarily independent. Furthermore, the approach leads to the Explicit Gibbs-Appell Equations.     

Laplace Type Invariants for Parabolic Equations

The Laplace invariants pertain to linear hyperbolic differentialequations with two independent variables. They were discovered byLaplace in 1773 and used in his integration theory of hyperbolicequations. Cotton extended the Laplace invariants to ellipticequations in 1900. Cotton's invariants can be obtained from the Laplaceinvariants merely by the complex change of variables relating theelliptic and hyperbolic equations.To the best of my knowledge, the invariants for parabolic equations werenot found thus far. The purpose of this paper is to fill this gap byconsidering what will be called Laplace type invariants (or seminvariants), i.e. the quantities that remain unaltered under the linear transformation of the dependent variable. Laplace type invariants are calculated here for all hyperbolic, elliptic and parabolic equations using the unified infinitesimal method. A new invariant is found forparabolic equations.     

Boundary Integral Equations in Dynamic Problems for Elastic Plates

Initial-boundary value problems with Dirichlet and Neumann conditions arising in the theory of bending of plates with transverse shear deformation are reduced to time-dependent boundary integral equations by means of layer potentials. The solvability of these equations is then investigated in Sobolev-type spaces. This revised version was published online in June 2006 with corrections to the Cover Date.     

Properties and Averaged Equations for Flows of Bubbly Liquids

Averaged properties of bubbly liquids in the limit of large Reynolds and small Weber numbers are determined as functions of the volume fraction, mean relative velocity, and velocity variance of the bubbles using numerical simulations and a pair interaction theory. The results of simulations are combined with those obtained recently for sheared bubbly liquids [19] and the mixture momentum and continuity equations to propose a complete set of averaged equations and closure relations for the flows of bubbly liquids at large Reynolds and small Weber numbers.     

Convergence for Semilinear Degenerate Parabolic Equations in Several Space Dimensions

We show that any global nonnegative and bounded solution to the degenerate parabolic problemu_t-u^m+f(u)=0 qquad {\rm on} quad R^N,u|{}=0converges to a single stationary state as time goes to infinity. Here m>0, f is a restriction of a real analytic function defined on a sector containing the half-line [0, ), and f(u ^1/m ) is a continuously differentiable function of u.     

Recursive Conditional Moment Equations for Advective Transport in Randomly Heterogeneous Velocity Fields

Flow and transport parameters such as hydraulic conductivity, seepage velocity, and dispersivity have been traditionally viewed as well-defined local quantities that can be assigned unique values at each point in space-time. Yet in practice these parameters can be deduced from measurements only at selected locations where their values depend on the scale (support volume) and mode (instruments and procedure) of measurement. Quite often, the support of the measurements is uncertain and the data are corrupted by experimental and interpretive errors. Estimating the parameters at points where measurements are not available entails an additional random error. These errors and uncertainties render the parameters random and the corresponding flow and transport equations stochastic. The stochastic flow and transport equations can be solved numerically by conditional Monte Carlo simulation. However, this procedure is computationally demanding and lacks well-established convergence criteria. An alternative to such simulation is provided by conditional moment equations, which yield corresponding predictions of flow and transport deterministically. These equations are typically integro-differential and include nonlocal parameters that depend on more than one point in space-time. The traditional concept of a REV (representative elementary volume) is neither necessary nor relevant for their validity or application. The parameters are nonunique in that they depend not only on local medium properties but also on the information one has about these properties (scale, location, quantity, and quality of data). Darcy's law and Fick's analogy are generally not obeyed by the flow and transport predictors except in special cases or as localized approximations. Such approximations yield familiar-looking differential equations which, however, acquire a non-traditional meaning in that their parameters (hydraulic conductivity, seepage velocity, dispersivity) and state variables (hydraulic head, concentration) are information-dependent and therefore, inherently nonunique. Nonlocal equations contain information about predictive uncertainty, localized equations do not. We have shown previously (Guadagnini and Neuman, 1997, 1998, 1999a, b) how to solve conditional moment equations of steady-state flow numerically on the basis of recursive approximations similar to those developed for transient flow by Tartakovsky and Neuman (1998, 1999). Our solution yields conditional moments of velocity, which are required for the numerical computation of conditional moments associated with transport. In this paper, we lay the theoretical groundwork for such computations by developing exact integro-differential expressions for second conditional moments, and recursive approximations for all conditional moments, of advective transport in a manner that complements earlier work along these lines by Neuman (1993).     

Classification of Traveling Waves for a Class of Nonlinear Wave Equations

We classify the weak traveling wave solutions for a class of one-dimensional non-linear shallow water wave models. The equations are shown to admit smooth, peaked, and cusped solutions, as well as more exotic waves such as stumpons and composite waves. We also explain how some previously studied traveling wave solutions of the models fit into this classification.     

一种求解柔性多体系统动力学方程的新方法

柔性多体系统控制方程是具有stiff性质的刚柔耦合非线性代数一微分方程组,本文提出了一种求解该类刚性方程组的数值方法,在每一时间步,利用Newmark-β直接积分法计算迭代初值,基于控制方程及约束方程的泰勒展开,推导出Newton-Raphson迭代公式,对位移及拉格朗日乘子进行修正,最后,引用Blajer提出的违约修正方法对数值积分过程中约束方程的违约进行修正。就两个典型算例进行了数值仿真,结果证明了本文方法的有效性。     

Symmetries of Integro-Differential Equations: A Survey of Methods Illustrated by the Benny Equations

Classical Lie group theory provides a universal tool for calculatingsymmetry groups for systems of differential equations. However Lie'smethod is not as much effective in the case of integral orintegro-differential equations as well as in the case of infinitesystems of differential equations.This paper is aimed to survey the modern approaches to symmetriesof integro-differential equations. As an illustration, an infinitesymmetry Lie algebra is calculated for a system of integro-differentialequations, namely the well-known Benny equations. The crucial idea is tolook for symmetry generators in the form of canonical Lie–Bäcklundoperators.