Exact Linearization and Invariant Solutions of the Generalized Burgers Equation with Linear Damping and Variable Viscosity

The generalized Burgers equation with linear damping and variable viscosity is subjected to Lie's classical method. Five distinct expressions for the variable viscosity are identified. Both the reduced ordinary differential equations and their corresponding Euler-Painlevé transcendents admit first integrals in the form of Bernoulli's equation and are linearized to obtain solutions in closed form.     

Numerical study of modified Adomian's method applied to Burgers equation

The application of Adomian's decomposition method to partial differential equations, when the exact solution is not reached, demands the use of truncated series. But the solution's series may have small convergence radius and the truncated series may be inaccurate in many regions. In order to enlarge the convergence domain of the truncated series, Padé approximants (PAs) to the Adomian's series solution have been tested and applied to partial and ordinary differential equations, with good results. In this paper, PAs, both in x$x$ and t$t$ directions, applied to the truncated series solution given by Adomian's decomposition technique for Burgers equation, are tested. Numerical and graphical illustrations show that this technique can improve the accuracy and enlarge the domain of convergence of the solution. It is also shown in this paper, that the application of Adomian's method to the ordinary differential equations set arising from the discretization of the spatial derivatives by finite differences, the so-called method of lines, may reduce the convergence domain of the solution's series.     

Existence and decay rates of solutions to the generalized Burgers equation

In this paper we study the generalized Burgers equation u_t+(u²/2)_x=f(t)u_xx, where f(t)>0 for t>0. We show the existence and uniqueness of classical solutions to the initial value problem of the generalized Burgers equation with rough initial data belonging to , as well it is obtained the decay rates of u in L^p norm are algebra order for p∈[1,∞[.     

Solutions of a Nonhomogeneous Burgers Equation

In this article, we construct solutions of a nonhomogeneous Burgers equation subject to certain unbounded initial profiles. In an interesting study, Kloosterziel [ 1 ] represented the solution of an initial value problem (IVP) for the heat equation, with initial data in , as a series of the self‐similar solutions of the heat equation. This approach quickly revealed the large time behavior for the solution of the IVP. Inspired by Kloosterziel [ 1 ]'s approach, we express the solution of the nonhomogeneous Burgers equation in terms of the self‐similar solutions of a linear partial differential equation with variable coefficients. Finally, we also obtain the large time behavior of the solution of the nonhomogeneous Burgers equation.     

A new transformation of Burger’s equation for an exact solution in a bounded region necessary for certain boundary conditions

In this work, the transient analytic solution is found for the initial-boundary-value Burgers equation in 0?x?L. The boundary conditions are a homogeneous Dirichlet condition at x=0 and a constant total flux at x=L. The technique used consists of applying the transformation that reduces Burgers equation to a linear diffusion-advection equation. Previous work on this equation in a bounded region has only applied the Cole-Hopf transformation , which transforms Burgers equation to the linear diffusion equation. The Cole-Hopf transformation can only solve Burgers equation with constant Dirichlet boundary conditions, or time-dependent Dirichlet boundary conditions of the form u(0,t)=F₁(t) and u(L,t)=F₂(t),0?x?L. In this work, it is shown that the Cole-Hopf transformation will not solve Burgers equation in a bounded region with the boundary conditions dealt with in this work.     

A Support Theorem for a Generalized Burgers SPDE

When the initial condition u ₀ to a parabolic Burgers SPDE (containing a quadratic term) belongs to L ^q [0,1],2q, the trajectories of the solution u(t,x) a.s. belong to the space C([0,T],L ^q [0,1]). We characterize the support of the law of u in this space; the proof is based on an approximation of u by a sequence of stochastic processes obtained by replacing the Brownian sheet by linear adapted interpolations.     

A Dimensional Splitting Method for Quasilinear Hyperbolic Equations with Variable Coefficients

A numerical method is presented for the variable coefficient, nonlinear hyperbolic equation u _t + _i=1 ^d V _i(x, t)f _i(u) _{x i} = 0 in arbitrary space dimension for bounded velocities that are Lipschitz continuous in the x variable. The method is based on dimensional splitting and uses a recent front tracking method to solve the resulting one-dimensional non-conservative equations. The method is unconditionally stable, and it produces a subsequence that converges to the entropy solution as the discretization of time and space tends to zero. Four numerical examples are presented; numerical error mechanisms are illustrated for two linear equations, the efficiency of the method compared with a high-resolution TVD method is discussed for a nonlinear problem, and finally, applications to reservoir simulation are presented.     

Global well-posedness of the critical Burgers equation in critical Besov spaces

We make use of the method of modulus of continuity [A. Kiselev, F. Nazarov, R. Shterenberg, Blow up and regularity for fractal Burgers equation, Dyn. Partial Differ. Equ. 5 (2008) 211-240] and Fourier localization technique [H. Abidi, T. Hmidi, On the global well-posedness of the critical quasi-geostrophic equation, SIAM J. Math. Anal. 40 (1) (2008) 167-185] [H. Abidi, T. Hmidi, On the global well-posedness of the critical quasi-geostrophic equation, SIAM J. Math. Anal. 40 (1) (2008) 167-185] to prove the global well-posedness of the critical Burgers equation t_∂u+ux_∂u+Λu=0 in critical Besov spaces with p∈[1,∞), where .     

Relatively Undistorted Waves. New Examples

New solutions of the wave equation with three space variables of the form u = g(x,y,z,t)f(), where the functions g and = (x,y,z,t) are some specified functions and f is an arbitrary function of one variable, are presented. Bibliography: 4 titles.     

Local controllability to stationary trajectories of a Burgers equation with nonlocal viscosity

This article studies the local controllability to trajectories of a Burgers equation with nonlocal viscosity. By linearization we are led to an equation with a non local term whose controllability properties are analyzed by using Fourier decomposition and biorthogonal techniques. Once the existence of controls is proved and the dependence of their norms with respect to the time is established for the linearized model, a fixed point method allows us to deduce the result for the nonlinear initial problem.     

Distributional products and global solutions for nonconservative inviscid Burgers equation

Burgers equation for inviscid fluids is a simplified case of Navier-Stokes equation which corresponds to Euler equation for ideal fluids. Thus, from a variational viewpoint, Burgers equation appears naturally in its nonconservative form. In this form, a consistent concept of a weak solution cannot be formulated because the classical distribution theory has no products which account for the term u(∂u/∂x). This leads several authors to substitute Burgers equation by the so-called conservative form, where one has in distributional sense. In this paper we will treat nonconservative inviscid Burgers equation and study it with the help of our theory of products; also, the relationship with the conservative Burgers equation is considered. In particular, we will be able to exhibit a Dirac-δ travelling soliton solution in the sense of global α-solution. Applying our concepts, solutions which are functions with jump discontinuities can also be obtained and a jump condition is derived. When we replace the concept of global α-solution by the concept of global strong solution, this jump condition coincides with the well-known Rankine-Hugoniot jump condition for the conservative Burgers equation. For travelling waves functions these concepts are all equivalent.     

Existence and uniform decay for a non‐linear viscoelastic equation with strong damping

This paper is concerned with the non‐linear viscoelastic equation We prove global existence of weak solutions. Furthermore, uniform decay rates of the energy are obtained assuming a strong damping Δu_t acting in the domain and provided the relaxation function decays exponentially. Copyright © 2001 John Wiley & Sons, Ltd.     

Absolute value equation solution via concave minimization

The NP-hard absolute value equation (AVE) Ax − |x| = b where and is solved by a succession of linear programs. The linear programs arise from a reformulation of the AVE as the minimization of a piecewise-linear concave function on a polyhedral set and solving the latter by successive linearization. A simple MATLAB implementation of the successive linearization algorithm solved 100 consecutively generated 1,000-dimensional random instances of the AVE with only five violated equations out of a total of 100,000 equations.     

Asymptotic agreement of moments and higher order contraction in the Burgers equation

The purpose of this paper is to investigate the relation between the moments and the asymptotic behavior of solutions to the Burgers equation. The Burgers equation is a special nonlinear problem that turns into a linear one after the Cole-Hopf transformation. Our asymptotic analysis depends on this transformation. In this paper an asymptotic approximate solution is constructed, which is given by the inverse Cole-Hopf transformation of a summation of n heat kernels. The k-th order moments of the exact and the approximate solution are contracting with order in Lp-norm as t→∞. This asymptotics indicates that the convergence order is increased by a similarity scale whenever the order of controlled moments is increased by one. The theoretical asymptotic convergence orders are tested numerically.     

Periodic solutions of non-smooth friction oscillators

We study the differential equation x"+g(x￠)+m(x) sgn x￠+f(x)=j(t)x''+g(x')+\mu(x)\,{\rm sgn}\, x'+f(x)=\varphi(t) with T-periodic right-hand side, which models e.g. a mechanical system with one degree of freedom subjected to dry friction and periodic external force. If, in particular, the damping term g is present and acts, up to a bounded difference, like a linear damping, we get existence of a T-periodic solution.¶In the more difficult case g = 0, we concentrate on the model equation x"+m(x) sgn x￠+x=j(t)x''+\mu(x)\,{\rm sgn}\,x'+x=\varphi(t) and obtain sufficient conditions for the existence of a T-periodic solution by application of Brouwer's fixed point theorem. For this purpose we show that a certain associated autonomous differential equation admits a periodic orbit such that the surrounded set (minus some neighborhood of the equilibria) is forward invariant for the equation above. Under additional assumptions on 7 we prove boundedness of all solutions.¶Finally, we provide a principle of linearized stability for periodic solutions without deadzones, where the "linearized" differential equation is an impulsive Hill equation.     

Asymptotics of Solutions to Pseudodifferential Equations of MELLIN Type

We consider pseudodifferential operators on the half-axis of the form where \documentclass{article}\pagestyle{empty}\begin{document}$ u(z)\; = \;\int\limits_0^\infty {{\rm t}^{{\rm z - 1}} u(t)} $\end{document} is the MELLIN transform of u and a(t, z) satisfies suitable smoothness properties in t and holomorphy and growth properties in z in some strip around the line Re z = 1/2. (1) is called pseudodifferential operator of MELLIN type or shortly MELLIN operator with the symbol a(t, z). For example, FUCHS ian differential operators, singular integral operators and integral operators with fixed singularities can be written in this form. In the paper we give a new composition theorem for MELLIN operators which has a natural extension to operators with symbols meromorphic in a left half-plane. The theorem can be used in the construction of left parametrices modulo compact operators in weighted SOBOLEV spaces. This approach yields rather precise results on the complete asymptotics of solutions at the point t = 0 for an equation a(t, δ) u = f when the right-hand side f has a prescribed asymptotical behaviour at t = 0. The results are extended to pseudodifferential equations of MELLIN type on a finite interval as well as to systems of such equations.     

Über eine Verallgemeinerung des Begriffs „Charakteristik”︁ bei partiellen Differentialgleichungen erster Ordnung

As, in general, the projections of characteristics into the x-space intersect for finite values of t, the global solution of a conservation law cannot be determined from the characteristic system of the equation, is considered. Only in the linear case, this equation coincides with the equation of the projections of characteristics. For convex h and all x₀ this equation has a solution almost everywhere, and the properties of this solution permit to construct a global solution of the conservation law using strips, in the same way as this is done for linear problems by the method of characteristics.     

Ill-posedness and local ill-posedness concepts in hilbert spaces *

In this paper, we study ill-posedness concepts of nonlinear and linear operator equations in a Hilbert space setting. Such ill-posedness information may help to select appropriate optimization approaches for the stable approximate solution of inverse problems, which are formulated by the operator equations. We define local ill-posedness of a nonlinear operator equation F(x) = y ₀ in a solution point x ₀:and consider the interplay between the nonlinear problem and its linearization using the Fréchet derivative F′(x ₀). To find a corresponding ill-posedness concept for the linearized equation we define intrinsic ill-posedness for linear operator equations A x = y and compare this approach with the ill-posedness definitions due to Hadamard and Nashed     

Exact N-Wave Solutions of Generalized Burgers Equations

Exact N-Wave solutions for the generalized Burgers equation where j, α, β, and γ are nonnegative constants and n is a positive integer, are obtained. These solutions are asymptotic to the (linear) old-age solution for large time and extend the validity of the latter so as to cover the entire time regime starting where the originally sharp shock has become sufficiently thick and the viscous effects are felt in the entire N wave.