China 
Africa 
Explore chapters and articles related to this topic
Application of Numerical Methods to Selected Model Equations
Published in Dale A. Anderson, John C. Tannehill, Richard H. Pletcher, Munipalli Ramakanth, Vijaya Shankar, Computational Fluid Mechanics and Heat Transfer, 2020
Dale A. Anderson, John C. Tannehill, Richard H. Pletcher, Munipalli Ramakanth, Vijaya Shankar
This nondimensional parameter gives the ratio of convection to diffusion and plays an important role in determining the character of the solution for Burgers’ equation. The mesh (cell) Reynolds number (also called Peclet number) can be expressed in terms of ν and r in the following manner: ReΔx=cΔxμ=cΔtΔx(Δx)2μΔt=νr
High-Amplitude, Ultrashort Strain Solitons in Solids
Published in Kong-Thon Tsen, and Nanostructures, 2018
Otto L. Muskens, Jaap I. Dijkhuis
Here, ε denotes the viscosity constant of the medium, which can be obtained experimentally from the thermal scattering length of ultrasonic waves.34 The combination of nonlinearity, dispersion, and viscosity in the wave equation is called the Kortewegde Vries-Burgers (KdV-Burgers) equation. Unlike the KdV initial value problem, this is not an integrable equation, and a solution can only be obtained numerically. For special situations, however, we can neglect any one of the three constituent terms and recover the KdV-, Burgers-, or a linear wave equation. At this point we consider the problem of a small dispersive term relative to the viscous term. Neglecting the dispersion, one recovers the Burgers equation that can be solved in an analytical way. Starting from the Burgers equation in the form st+12(s2)y=vsyy, one can apply the Cole-Hopf transformation s = −2vuy/u to arrive at () ut=vuyy.
K
Published in Splinter Robert, Illustrated Encyclopedia of Applied and Engineering Physics, 2017
[computational, fluid dynamics, mechanics] The Korteweg–de Vries–Burgers equation describes a mathematical model for a nonperiodic shock wave. This is based on the work by Johannes Martinus (Jan) Burgers (1895–1981) in 1948 which describes weak acoustic burst waves (solitary wave) as a solution to his expression:[df(x,t)/dt]+c1′f(x,t)[df(x,t)/dx]=c2′[d2f(x,t)/dx2], which is combined with the Koreteweg–de Vries equation. The Korteweg–de Vries–Burgers equation is defined as: [df(x,t)/dt]+c1f(x,t)[df(x,t)/dx]−c2[d2f(x,t)/dx2]+c3[d2f(x,t)/dx3]=0, where the term c2[d2f(x,t)/dx2] represents viscous dissipation, and ci are constants. When the term c3 = 0 this represents Burgers’ equation.
On adjustable undular bore profiles based on the modified steady KdV–Burgers equation
Published in Journal of Hydraulic Research, 2023
The KdV–Burgers equation is the KdV-type equation with an additional damping term. The KdV equation in different versions with various constant coefficients arises in many physical applications, which can be proved equivalent after simple transformation (Cheng & Liu, 2023; Gardner et al., 1974). Korteweg and de Vries (1895) first derived it to describe theoretically the observed solitary wave in an open channel. The KdV equation is also derived in plasma physics (Berezin & Karpman, 1967) and in anharmonic lattices (Zabusky, 1969). By considering dissipation, the KdV–Burgers equation further extends the theory to other realistic phenomena with the solution of decaying oscillation. If inhomogeneous media and non-uniform boundaries are considered, the variable-coefficient KdV/KdV–Burgers type equations can be obtained, which have attracted much research interest, e.g. Karczewska et al. (2014) and Kumar et al. (2018, 2021).
Gaussian Process Assisted Active Learning of Physical Laws
Published in Technometrics, 2021
Jiuhai Chen, Lulu Kang, Guang Lin
Burgers’ equation is one of the most important PDEs applied in various areas of physics, such as fluid mechanics, nonlinear acoustics, gas dynamics, and traffic flow. It can be derived from the Navier–Stokes equation for the velocity by dropping the pressure gradient term. For the one-dimensional space, that is, , the Burgers’ equation iswhere the parameters are set to be . The initial condition is chosen as
Well-posedness of the generalized Burgers equation on a finite interval
Published in Applicable Analysis, 2019
Jing Li, Bing-Yu Zhang, Zhixiong Zhang
The classical Burgers equation is a nonlinear parabolic partial differential equation introduced by J. M. Burgers [1] in 1948. It is a model for studying the interaction between nonlinear and dissipative phenomena in fluid flow. The Burgers equation can be viewed as a toy model for the Navier-Stokes equation. E. Hopf [2] and J. D. Cole [3] showed that the Burgers equation can be turned to a linear heat equation through the well-known Hopf-Cole transformation, which is a nonlinear transformation named after them.