A new formulation of the general solution to Burgers'' equation

We present a new method to solve Burgers' equation in one space dimension for an arbitrary incident pulse of finite length. It is based on Cole-Hopf linearization and Fourier analysis of the pulse in a special set of orthogonal functions. Each of these functions corresponds to a stable, well behaved, solitary wave solution of Burgers' equation. After Cole-Hopf back-transformation the Fourier series of the pulse gives an exact solution of Burgers' equation, valid all the way from the boundary, through the shock-forming phase, the subsequent damping phase and all the way to infinity. The solution also provides a basis for qualitative discussions of the evolution of arbitrary pulses.     

A CLASS OF ALTERNATING GROUP METHOD OF BURGERS' EQUATION

IntroductionWeconsiderthefollowingnonlinearBurgers’equation : u t u u x=ε 2 u x2 ,　　 0 0 ,( 1 )withtheinitialandtheboundaryconditions　　　　　u(x,0 ) =f(x) ,　　 0 相似文献   

A control problem for Burgers' equation with bounded input/output

A stabilization problem for Burgers' equation is considered. Using linearization, various controllers are constructed which minimize certain weighted energy functionals. These controllers produce the desired degree of stability for the closed-loop nonlinear system. A numerical scheme for computing the feedback gain functional is developed and several numerical experiments are performed to illustrate the theoretical results.     

The study of quasi wavelets based numerical method applied to Burgers' equations

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Exact solutions to the KdV-Burgers'' equation

This paper presents two different methods for the construction of exact solutions to the KdVB equation. The first is a direct one based on a combination of solutions to the KdV equation and Burgers' equation. In this approach a number of unknown constants are involved, and it is shown that the equations leading to their determination are properly determined and are capable of solution.

The second method involves a series, and is essentially an extension of Hirota's method. This approach is capable of solving the KdVB equation exactly, and also of generalization to higher order equations with a KdVB-type nonlinearity.     

超音速粘性流动的SUPG有限元数值解法

本文构造了准简化 N-S 方程组的 SUPG(Streamline Upwind/Petrov-Galerk-in)加权剩余式,并利用该方法对 Burgers 方程、无粘性激波反射问题、以及超音速平板和压缩拐角的层流流动作了数值求解。计算结果表明,本文方法是精确、收敛和稳定的。     

小波插值方法自适应数值求解时间进化微分方程

应用小波自相关函数的插值性质,得到任意给定函数的插值小波表达式,然后对其直接求导,可以得到函数导数的表达式。导数运算不再应用差分算法,扩展了小波方法在数值求解微分方程中的应用。由于小波基函数的有限支撑特点,小波方法可以有效地处理微分方程中解的局部突变问题。通过设定小波系数阀值,实现了求解过程的自适应。本文给出了两个算例,结果表明了算法的自适应特点及其向二维空间问题推广的有效性。     

Interpolation of Scattered 3D PTV Data to a Regular Grid

The velocity data obtained by many 3D measurement methods such as particle tracking velocimetry (PTV) are not regularly distributed in3D space. We revised three numerical schemes to interpolate the scattered velocity vectors to a regularly spaced grid. Additionally, two techniques were examined to smooth the resulting flow field. The different algorithms were tested for a synthetic data set based on the analytical solution of Burgers' vortex. To study the impact of measurement errors a Gaussian noise was superimposed on the exact solution. It was found that an interpolation scheme of higher order does not necessarily perform better than one of lower order. The most‘robust’ algorithm was used to process 3D PTV data, which were obtained from measurements of a separating flow in a forward facing step configuration. Information on the 3D streamlines and vortex structures was obtained. This revised version was published online in July 2006 with corrections to the Cover Date.     

Assessment of spatial derivatives determined from scattered 3D PTV data

Based on a linear regression model, we propose a numerical method to determine the spatial derivatives from scattered 3D PTV (particle-tracking velocimetry) data. Various quantities allowing an assessment of the numerically calculated gradients are introduced and their reliability is investigated. The performance of the numerical scheme and of the “quality estimators” was examined for different synthetic data sets obtained from Burgers' vortex. The energy dissipation was computed from experimental PTV measurements performed for a forward-facing step configuration. Received: 11 March 1999/Accepted: 8 August 2000     

ON THE NUMERICAL SOLUTION OF QUASILINEAR WAVE EQUATION WITH STRONG DISSIPATIVE TERM

The numerical solution for a type of quasilinear wave equation is studied. The three-level difference scheme for quasi-linear waver equation with strong dissipative term is constructed and the convergence is proved. The error of the difference solution is estimated. The theoretical results are controlled on a numerical example.     

A numerical method for separation of stresses in photo-orthotropic elasticity

A numerical method is suggested for separation of stresses in photo-orthotropic elasticity using the numerical solution of compatibility equation for orthotropic case. The compatibility equation is written in terms of a stress parameter S analogous to the sum of principal stresses in two-dimensional isotropic case. The solution of this equation provides a relation between the normal stresses. The photoelastic data give the shear stress and another relation between the two normal stresses. The accuracy of the numerical method and its application to practical problems are illustrated with examples.     

Drying of solids with irregular geometry: numerical study and application using a three-dimensional model

This article proposes a numerical solution for the diffusion equation applied to solids with arbitrary geometry using non-orthogonal structured grids for the boundary condition of the first kind. A transient three-dimensional mathematical formulation written in boundary fitted coordinates and numerical formalism to discretize the diffusion equation by using the finite volume method, including numerical analysis of the computational solution are presented. To validate the proposed solution, the results obtained in this work were compared with well-known numerical solution available in literature and good agreement was observed. In order to verify the potential of the proposed numerical solution, it was applied to describe mass transfer inside ceramic roof tiles during drying. For that, it was used experimental data of the drying kinetics at the following temperatures: 55.6; 69.7; 82.7 and 98.6 °C. An optimization technique using experimental dataset has been presented to estimation of transport properties. The obtained statistical indicators enable to conclude that the numerical solution satisfactorily describes the drying processes.     

The solution of a convection–diffusion–reaction equation arising from cathodic reduction in a pulsating crack

This paper derives the convection–diffusion-reaction equation governing the reaction between the dissolved oxygen in sea-water and the steel walls of a pulsating crack. By the neglect of the diffusion term it is shown that an exact solution of the convection-reaction equation can be obtained. A numerical method for the solution of the complete convection–diffusion-reaction equation is derived by the use of finite differences. The numerical computation of the initial transient and the final periodic steady-state values is also discussed.     

弹性力学混合状态方程的小波解法

应用小波理论求解弹性力学混合状态方程，讨论了解的收敛性。从文中的数值算例不难看出，该方法不失为混合状态方程一种新的求解途径。     

双相介质P波波动方程小波域多尺度模拟

相比于单相介质理论而言,双相介质理论更接近实际地层的真实情况,因此在地球物理勘探、地震工程和岩土动力学等领域有着广泛的应用。传统的波动方程数值解法由于本身固有的不足不利于求解诸如双相介质波动方程等复杂的非线性和不规则性问题;而小波方法则由于自身良好的特性可以用来构建解决此类问题的自适应性算法。本文详细推导了双相介质P波波动方程的有限差分矩阵表示形式,利用小波变换将其转移到小波域,设置阈值形成更为稀疏的迭代矩阵以构建自适应算法,从而达到减少计算量,增加地震波场数值模拟灵活性和准确性的目的。地球物理勘探的数值模拟实例验证了方法的有效性。     

Solution of the laminar duct flow problem by a boundary integral equation method

Summary A boundary integral equation method is proposed for approximate numerical and exact analytical solutions to fully developed incompressible laminar flow in straight ducts of multiply or simply connected cross-section. It is based on a direct reduction of the problem to the solution of a singular integral equation for the vorticity field in the cross section of the duct. For the numerical solution of the singular integral equation, a simple discretization of it along the cross-section boundary is used. It leads to satisfactory rapid convergency and to accurate results. The concept of hydrodynamic moment of inertia is introduced in order to easily calculate the flow rate, the main velocity, and the fRe-factor. As an example, the exact analytical and, comparatively, the approximate numerical solution of the problem of a circular pipe with two circular rods are presented. In the literature, this is the first non-trivial exact analytical solution of the problem for triply connected cross section domains. The solution to the Saint-Venant torsion problem, as a special case of the laminar duct-flow problem, is herein entirely incorporated.     

Further numerical methods for the Falkner-Skan equations: Shooting and continuation techniques

We consider in this paper the numerical solution of the Falkner-Skan differential equation, modelling under some similarity assumptions the boundary layer equation. We look for the extremal solution of this third order differential equation. The methods we use are basically the Newton method with a shooting process, which is coupled with a continuation method: they allow us to follow the solution arcs which contain regular and turning point solutions.     

Numerical study of single-phase gas filtration in a porous medium

This paper describes a method for numerical solution of single-phase nonstationary gas filtration in a porous medium. Linearization of the classical Leibenzon equation is performed. To solve the resulting linear equation, an efficient numerical algorithm was constructed without saturation over spatial variables and time.