Nonlinear waves in fluid flow through a viscoelastic tube

A system of nonlinear equations for describing the perturbations of the pressure and radius in fluid flow through a viscoelastic tube is derived. A differential relation between the pressure and the radius of a viscoelastic tube through which fluid flows is obtained. Nonlinear evolutionary equations for describing perturbations of the pressure and radius in fluid flow are derived. It is shown that the Burgers equation, the Korteweg-de Vries equation, and the nonlinear fourth-order evolutionary equation can be used for describing the pressure pulses on various scales. Exact solutions of the equations obtained are discussed. The numerical solutions described by the Burgers equation and the nonlinear fourth-order evolutionary equation are compared.     

非线性方程的求解—最优化样条函数康托诺维奇加权残值法

本文是得新提出的一种微分方程的新解法，最优化样条函数康托诺维奇加权残值法。来求解非线性微分方程。该法把优化理论引入微分方程的数值解法，揉最优化算法，加权残值法，样条函数法，康托诺维奇法于一体，具有精度高、收敛快、易于处理各种边界条件的优点，文中有基于原始微分方程的算例，对流体力学中Ｂｕｒｇｅｒｓ方程的成功求解，展示了该法的应用前景。     

ADAPTIVE INTERVAL WAVELET PRECISE INTEGRATION METHOD FOR PARTIAL DIFFERENTIAL EQUATIONS

IntroductionThepreciseintegrationmethod(PIM) [1],whichwasproposedforsolvingstructuraldynamicequations.Thismethodissimplerandpossesseshigherprecision .Forlinearsteadystructuraldynamicsystems,itsnumericalresultsattheintegrationpointsarealmostequaltothatoftheexactsolutioninmachineaccuracy .InthepreciseintegrationmethodforsolvingPDEs,theequationsshouldbediscretizedinthephysicalspaceforobtainingthesystemofODEsintime ,whichisoftenexecutedbythefinitedifferencemethodorthefiniteelementmethod .Inrec…     

Nonlinear Fluctuations of Weakly Asymmetric Interacting Particle Systems

We introduce what we call the second-order Boltzmann–Gibbs principle, which allows one to replace local functionals of a conservative, one-dimensional stochastic process by a possibly nonlinear function of the conserved quantity. This replacement opens the way to obtain nonlinear stochastic evolutions as the limit of the fluctuations of the conserved quantity around stationary states. As an application of this second-order Boltzmann–Gibbs principle, we introduce the notion of energy solutions of the KPZ and stochastic Burgers equations. Under minimal assumptions, we prove that the density fluctuations of one-dimensional, stationary, weakly asymmetric, conservative particle systems are sequentially compact and that any limit point is given by energy solutions of the stochastic Burgers equation. We also show that the fluctuations of the height function associated to these models are given by energy solutions of the KPZ equation in this sense. Unfortunately, we lack a uniqueness result for these energy solutions. We conjecture these solutions to be unique, and we show some regularity results for energy solutions of the KPZ/Burgers equation, supporting this conjecture.     

A Simple Fast Method in Finding Particular Solutions of Some Nonlinear PDE

1　ＡＴｒｉａｌＦｕｎｃｔｉｏｎａｎｄａＲｏｕｔｉｎｅｔｏＦｉｎｄＡｎａｌｙｔｉｃａｌＳｏｌｕｔｉｏｎｏｆＴｗｏＴｙｐｅｓｏｆＮｏｎｌｉｎｅａｒＰＤＥ　　Ｗｅｔｒｅａｔｔｈｅｎｏｎｌｉｎｅａｒｅｖｏｌｕｔｉｏｎｅｑｕａｔｉｏｎ ,ｗｈｉｃｈｉｓｆｏｒｍｅｄｂｙａｄｄｉｎｇｈｉｇｈｏｒｄｅｒｄｅｒｉｖａｔｉｖｅｔｅｒｍｓａｎｄｎｏｎｌｉｎｅａｒｔｅｒｍｓｔｏｔｈｅＢｕｒｇｅｒｓｅｑｕａｔｉｏｎ ｕ ｔ ｕ ｕ ｘ … ｕｐ ｕ ｘｑ α1 ｕ ｘ … αｎ ｎｕ ｘｎ =0 ,( 1)ｗｈｉｃｈｐ ,ｑ ,ｎａｎｄαｉ(ｉ =1,2…     

Sound wave propagation through incompressible flows

This paper derives a variable coefficient, multidimensional Burgers equation which models the propagation of a nonlinear sound wave through an incompressible background flow. The equation is derived from the compressible Euler equations by the combination of a weakly nonlinear acoustics expansion for the sound wave and an incompressible expansion for the background flow. The main effect of the incompressible flow on the sound wave is the advection of the sound wave by the transverse velocity component of the flow.     

Weakly nonlinear magnetosonic waves and structure of low-intensity discontinuities in magnetohydrodynamics

Mathematical techniques are proposed which make it possible to reduce the system of magnetohydrodynamic equations for a viscous heat-conducting gas with finite electric conductivity and a general equation of state to the model Burgers equation. On the basis of this equation the structure of weakly nonlinear magnetohydrodynamic shock waves is studied. In particular, the width of the shock wave is estimated.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.4, pp. 43–48, May–June, 1993.     

The Linear Stability of Shock Waves for the Nonlinear Schrödinger–Inviscid Burgers System

We investigate the coupling between the nonlinear Schrödinger equation and the inviscid Burgers equation, a system which models interactions between short and long waves, for instance in fluids. Well-posedness for the associated Cauchy problem remains a difficult open problem, and we tackle it here via a linearization technique. Namely, we establish a linearized stability theorem for the Schrödinger–Burgers system, when the reference solution is an entropy-satisfying shock wave to Burgers equation. Our proof is based on suitable energy estimates and on properties of hyperbolic equations with discontinuous coefficients. Numerical experiments support and expand our theoretical results.     

EXACT SOLUTION OF NAVIER-STOKES EQUATIONS-THE THEORY OF FUNCTIONS OF A COMPLEX VARIABLE UNDER DIRACPAULI REPRESENTATION AND ITS APPLICATION IN FLUID DYNAMICS (Ⅱ)

This work is the continuation of the discussion of ref. [1]. In ref. [1] we applied the theory of functions of a complex variable under Dirac-Pauli representation, introduced the Kaluza Ghost coordinate, and turned Navier-Stokes equations of viscofluid dynamics of homogeneous and incompressible fluid into nonlinear equation with only a pair of complex unknown functions. In this paper we again combine the complex independent variable except time, and cause it to decrease in a pair to the number of complex independent variables. Lastly, we turn Navier-Stokes equations into classical Burgers equation. The Cole-Hopf transformation join up with Burgers equation and the diffusion equation is Bäcklund transformation in fact and the diffusion equation has the general solution as everyone knows. Thus, we obtain the exact solution of Navier-Stokes equations by Bäcklund transformation.     

On the structure of characteristic surfaces related to partial differential equations of first and higher orders. II

The generalized method of characteristics is developed within the framework of the geometric Monge picture. Hopf-Lax-type extremality solutions are obtained for a broad class of Cauchy problems for nonlinear partial differential equations of the first and higher orders. A special Hamilton-Jacobi-type case is analyzed separately. An exact extremality Hopf-Lax-type solution of the Cauchy problem for the nonlinear Burgers equation is obtained, and its linearization to the Hopf-Cole expression and to the corresponding Airy-type linear partial differential equation is found and discussed. Published in Neliniini Kolyvannya, Vol. 8, No. 4, pp. 529–543, October–December, 2005.     

基于数据驱动的流场控制方程的稀疏识别

利用有限数据建立系统的非线性动力学模型是具有挑战性的重要课题. 数据驱动的稀疏识别方法是近年来发展的从数据识别动力系统控制方程的有效方法. 本文基于数据驱动稀疏识别方法对不同流场的控制方程进行了识别. 采用非线性动力学偏微分方程函数识别(partial differential equations functional identification of nonlinear dynamics, PDE-FIND)方法和最小绝对收缩和选择算子(least absolute shrinkage and selection operator, LASSO)方法对二维圆柱绕流、顶盖驱动方腔流、Rayleigh-Bénard (RB)对流和三维槽道湍流的控制方程进行了识别. 在稀疏识别过程中, 采用直接数值模拟得到的流场数据来计算过完备候选库中的每一项, 候选库中变量最高保留到二次, 变量导数最高保留到二阶, 非线性项最高保留到四阶. 结果发现PDE-FIND方法和LASSO方法对于不含有非线性项的控制方程, 如涡量输运方程、热输运方程和连续性方程, 都能准确识别. 对于含有强非线性项的控制方程, 如Navier-Stokes方程的识别, PDE-FIND方法正确地识别出了控制方程及流场的Rayleigh数和Reynolds数, 而LASSO方法识别结果不正确, 这是因为候选库中的项之间存在分组效应, LASSO方法通常只取分组中的一项. 本文还发现选择流动结构丰富的区域的数据进行控制方程的稀疏识别可以提高识别的准确性.      

Backlund transformation and exact solutions for whitham-broer-kaup equations in shallow water

I.IntroductionTodescribethepropagationofshallowwaterwave,manywell-knowncompletelyintegrablemodelsareintroduced,suchasKdVequatioll,Boussinesqequation,K-Pequation,WBKequation,etc.UnderBoussinesqapproximation,Whitham,BroerandKaupl"2'3]obtainednon-lilleurWBKequationwhereu=u(x,l)isthefieldofhorizontalvelocity;v=v(x,t)istheheightthatdeviatefromequilibriumpositionofliquid;a,gareconstantsthatrepresentdift'erentdispersivepower.TheEqs.(1.I),(l.2)areverygoodmodelstodescl.ibedispersivewave.Ifa=0,P…     

A Volterra series approach to the frequency domain analysis of non-linear viscous Burgers’ equation

In this paper, higher order frequency response functions, based on the Volterra series, are employed to characterise the input-output behaviour of the non-linear viscous Burgers?? equation subject to sinusoidal excitation. First, a formal Volterra series representation for each spatial location is derived for the solution of Burgers?? equation with a boundary condition as the input to the system. Then a systematic method is presented to obtain the higher order frequency kernels of the Volterra series at each spatial location by solving a series of ordinary differential equations. It is shown that the convergence region of the individual harmonics with respect to the magnitude of the input excitation can be estimated by using these higher order kernels. The frequency characteristics of Burgers?? equation is investigated and compared with numerical simulation.     

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.     

Weakly nonlinear propagation of small-wavelength,impulsive acoustic waves in a general atmosphere

Multiple-scale asymptotic analysis is applied to small-wavelength, weakly nonlinear propagation of an impulsive acoustic wave in a general (3D, in-motion and time dependent) atmosphere. In keeping with previous work on sonic booms and nonlinear acoustics in general, the result is a combination of ray tracing and a generalised Burgers equation describing evolution of the waveform carried by a ray. This is nonetheless, to our knowledge, the first derivation of such a model based on asymptotic analysis of the governing equations for a general atmosphere. Results are given, discussed and compared with measurements for the particular example of the test explosion known as Misty Picture.     

Nonlinear vibration of a two-mass system with nonlinear stiffnesses

An analytical approach is developed for nonlinear free vibration of a conservative, two-degree-of-freedom mass–spring system having linear and nonlinear stiffnesses. The main contribution of the proposed approach is twofold. First, it introduces the transformation of two nonlinear differential equations of a two-mass system using suitable intermediate variables into a single nonlinear differential equation and, more significantly, the treatment a nonlinear differential system by linearization coupled with Newton’s method and harmonic balance method. New and accurate higher-order analytical approximate solutions for the nonlinear system are established. After solving the nonlinear differential equation, the displacement of two-mass system can be obtained directly from the governing linear second-order differential equation. Unlike the common perturbation method, this higher-order Newton–harmonic balance (NHB) method is valid for weak as well as strong nonlinear oscillation systems. On the other hand, the new approach yields simple approximate analytical expressions valid for small as well as large amplitudes of oscillation unlike the classical harmonic balance method which results in complicated algebraic equations requiring further numerical analysis. In short, this new approach yields extended scope of applicability, simplicity, flexibility in application, and avoidance of complicated numerical integration as compared to the previous approaches such as the perturbation and the classical harmonic balance methods. Two examples of nonlinear two-degree-of-freedom mass–spring system are analyzed and verified with published result, exact solutions and numerical integration data.     

Method of relaxed streamline upwinding for hyperbolic conservation laws

In this work a new finite element based Method of Relaxed Streamline Upwinding is proposed to solve hyperbolic conservation laws. Formulation of the proposed scheme is based on relaxation system which replaces hyperbolic conservation laws by semi-linear system with stiff source term also called as relaxation term. The advantage of the semi-linear system is that the nonlinearity in the convection term is pushed towards the source term on right hand side which can be handled with ease. Six symmetric discrete velocity models are introduced in two dimensions which symmetrically spread foot of the characteristics in all four quadrants thereby taking information symmetrically from all directions. Proposed scheme gives exact diffusion vectors which are very simple. Moreover, the formulation is easily extendable from scalar to vector conservation laws. Various test cases are solved for Burgers equation (with convex and non-convex flux functions), Euler equations and shallow water equations in one and two dimensions which demonstrate the robustness and accuracy of the proposed scheme. New test cases are proposed for Burgers equation, Euler and shallow water equations. Exact solution is given for two-dimensional Burgers test case which involves normal discontinuity and series of oblique discontinuities. Error analysis of the proposed scheme shows optimal convergence rate. Moreover, spectral stability analysis gives implicit expression of critical time step.     

Nonlinear Boundary Control of the Generalized Burgers Equation

In this paper, the adaptive and non-adaptive stabilization of the generalized Burgers equation by nonlinear boundary control are analyzed. For the non-adaptive case, we show that the controlled system is exponentially stable in L². As for the adaptive case, we present a novel and elegant approach to show the L² regulation of the solution of the generalized Burgers system. Numerical results supporting and reinforcing the analytical ones of both the controlled and uncontrolled system for the non-adaptive and adaptive cases are presented using the Chebychev collocation method with backward Euler method as a temporal scheme.     

A comparative study of the performance of high resolution advection schemes in the context of data assimilation

High resolution advection schemes have been developed and studied to model propagation of flows involving sharp fronts and shocks. So far the impact of these schemes in the framework of inverse problem solution has been studied only in the context of linear models. A detailed study of the impact of various slope limiters and the piecewise parabolic method (PPM) on data assimilation is the subject of this work, using the nonlinear viscous Burgers equation in 1?D. Also provided are results obtained in 2?D using a global shallow water equations model. The results obtained in this work may point out to suitability of these advection schemes for data assimilation in more complex higher dimensional models. Copyright © 2005 John Wiley & Sons, Ltd.     

Improvement of the weighted essentially nonoscillatory scheme based on the interaction of smoothness indicators

The weighted essentially nonoscillatory scheme is improved by introducing new smoothness indicators that evaluate the interactions among the classical smoothness indicators suggested by Jiang and Shu. The effect of the key parameters in the new smoothness indicators on the scheme is systematically investigated. The improved scheme has smaller dissipation with larger weight assignment to the discontinuous stencils and higher numerical accuracy with weights closer to the ideal weights. To verify the theory, benchmark problems governed by the linear transport equation, the 1‐dimensional nonlinear Burgers equation, and the Euler equations are conducted and analyzed, respectively. Better computational performances both on numerical resolution and accuracy are shown in the comparisons with other classical weighted essentially nonoscillatory schemes.