Burgers equation

... a Department of Basic Science, Benha Higher Institute of Technology, Benha University 13512, Egypt. b Department of Engineering Mathematics and Physics, Faculty of Engineering, CairoUniversity, Giza-Egypt. Received 1 March 2006; revised 3 May 2006. ...

Usually fluid mechanics problems are nonlinear. Thus the need of establishing a model for solving inviscid problem. In this paper we have analyzed such an equation which is popularly known as Burger’s Equation. This equation may be viewed as a simple analog of the Euler equation for the flow of an inviscid fluid. We will be using the LAX method to solve the problem. The error is of First Order; error is of dissipative nature. For LAX method as the ratio of Δt/Δx reaches unity the numerical solution is able to capture more of the shock waves resolution and vice versa. Again while using FTCS method it is observed that the method is unconditionally unstable. But by keeping Δt/Δx as small as possible a stable solution may be attained but there remain some source term errors. Again for the third case Adam’s Bashforth Method is used. This method need an initial run using any other Explicit scheme like FTCS. Adam’s 3rd order equation is used for thus analyzing purpose. Both from error analyses and Richardson’s extrapolation method it’s observed that LAX method has the highest accuracy among the compared schemes. As we are working with varying wave velocities in this problem it is to be noted that changing wave speed from constant to a functional velocity forms discontinuous solution (Weak Solution).

We study asymptotic behavior of solutions to multifractal Burgers- type equation ut + f(u)x = Au, where the operator A is a linear combination of fractional powers of the second derivative −∂2/∂x2 and f is a polynomial nonlinearity. Such equations appear in contin- uum mechanics as models with fractal diffusion. The results include decay rates of the Lp-norms, 1 ≤

This paper presents two methods for finding the soliton solutions to the nonlinear dispersive and dissipative KdV–Burgers equation. The first method is a numerical one, namely the finite differences with variable mesh. The stability of the numerical scheme is discussed. The second method is the semi-analytic Adomian decomposition method. Test example is given. A comparison between the two methods is carried out to illustrate the pertinent feature of the proposed algorithm.

In the present paper numerical solutions of the one-dimensional Burgers' equation are obtained by a method based on collocation of cubic B-splines over finite elements. The accuracy of the proposed method is demonstrated by three test problems. The numerical results are found in good agreement with exact solutions. Time-space integration of the Burgers' equation yields a system of difference equation which is shown to be unconditionally stable.

We develop a new method to uniquely solve a large class of heat equations, so called Kolmogorov equations in infinitely many variables. The equations are analyzed in spaces of sequentially weakly continuous functions weighted by proper (Lyapunov type) functions. This way for the first time the solutions are constructed everywhere without exceptional sets for equations with possibly non-locally Lipschitz drifts. Apart from general analytic interest, the main motivation is to apply this to uniquely solve martingale problems in the sense of Stroock-Varadhan given by stochastic partial differential equations from hydrodynamics, such as the stochastic Navier-Stokes equations. In this paper this is done in the case of the stochastic generalized Burgers equation. Uniqueness is shown in the sense of Markov flows.

The Korteweg-de Vries-Burgers (KdV-Burgers) equation and modified Korteweg-de Vries-Burgers equation are derived in strongly coupled dusty plasmas containing nonthermal ions and Boltzmann distributed electrons. It is found that solitary waves and shock waves can be produced in this medium. The effects of important parameters such as ion nonthermal parameter, temperature, density and velocity on the properties of shock waves and solitary waves are discussed.

Calculus has widespread applications in science and engineering. Optimization is one of its major subjects, where a problem can be mathematically formulated and its optimal solution is determined by using derivatives. However, this calculus-based derivative technique can only be applied to real-valued or continuous-valued functions rather than discrete-valued functions while there are many situations where design variables contain not continuous values but discrete values by nature. In order to consider these realistic design situations, this study proposes a novel derivative for discrete design variables based on a harmony search algorithm. Detailed analysis shows how this new stochastic derivative works in the bench-mark function and fluid-transport network design. Hopefully this new derivative, as a fundamental technology, will be utilized in various science and engineering problems.

This paper establishes a general theoretical framework for variance reduction based on arbitrary order derivatives of the solution with respect to the random parameters, known as sensitivity derivatives. The theoretical results are validated by two examples—the solution of the Burgers equation with viscosity as a single random parameter, and a test case involving five random variables. These examples illustrate that the first-order sensitivity derivative variance reduction method achieves an order of magnitude improvement in accuracy for both Monte Carlo and stratified sampling schemes. The second-order sensitivity derivative method improves the accuracy by another order of magnitude relative to the first-order method. Coupling it with stratified sampling yields yet another order of magnitude improvement in accuracy.

Quark-gluon plasmas formed in heavy ion collisions at high energies are well described by ideal classical fluid equations with nearly zero viscosity. It is believed that a similar fluid permeated the entire universe at about three microseconds after the big bang. The estimated Reynolds number for this quark-gluon plasma at 3 microseconds is approximately 10^19. The possibility that quantum mechanics may be an emergent property of a turbulent proto-fluid is tentatively explored. A simple relativistic fluid equation which is consistent with general relativity and is based on a cosmic dust model is studied. A proper time transformation transforms it into an inviscid Burgers equation. This is analyzed numerically using a spectral method. Soliton-like solutions are demonstrated for this system, and these interact with the known ergodic behavior of the fluid to yield a stochastic and chaotic system which is time reversible. Various similarities to quantum mechanics are explored.

In this paper we study the following Burgers equation du/dt + d/dx (u^2/2) = epsilon d^2u/dx^2 + f(x,t) where f(x,t)=dF/dx(x,t) is a random forcing function, which is periodic in x and white noise in t. We prove the existence and uniqueness of an invariant measure by establishing a ``one force, one solution'' principle, namely that for almost every realization of the force, there is a unique distinguished solution that exists for the time interval (-infty, +infty) and this solution attracts all other solutions with the same forcing. This is done by studying the so-called one-sided minimizers. We also give a detailed description of the structure and regularity properties for the stationary solutions. In particular, we prove, under some non-degeneracy conditions on the forcing, that almost surely there is a unique main shock and a unique global minimizer for the stationary solutions. Furthermore the global minimizer is a hyperbolic trajectory of the underlying system of characteristics.