Korteweg-de Vries-Burgers Equation on a Segment

Abstract

We study the following initial-boundary value problem for the Korteweg-de Vries-Burgers equation on the interval (0, 1)We prove that if the initial data u0 ∈ L 2 , then there exists a unique solution u ∈ C [0, ∞) ; L 2 ∪ C (0, ∞) ; H 1 of the initial-boundary value problem (0.1). We also obtain the large time asymptotic of solution uniformly with respect to x ∈ (0, 1) as t → ∞. E.I. Kaikina et. al. CUBO 12, 1 (2010) RESUMENEstudiamos el siguiente problema de valor inicial en la frontera para la ecuación de Kortewegde Vries-Burgers en el intervalo (0, 1)Provamos que si el dato inicial u0 ∈ L 2 , entonces existe unaúnica solución u ∈ C [0, ∞) ; L 2 ∪ C (0, ∞) ; H 1 del problema de valor inicial en la frontera (0.1). También obtenemos comportamiento asintótico de la solución con respecto a x ∈ (0, 1) cuando t → ∞.

Chebyshev and Legendre via Galerkin method for solving KdV Burgers' equation

“…The boundary conditions, of course, are imposed naturally in the weak formula (4) and the integrals above are meaningful. The motivation for our study is the KdVB equation presented in [18]. We started by considering the approximation:…”

“…Hence, the equation (18) can be re-written as the system of ordinary differential equations (ODEs) in time as follows:…”

“…Recently, the variety Galerkin algorithms have been applied in [6], [7], [8] and [21] on the numerical solution of the ordinary differential equations. Finally, the author in [18] proved that if the initial data . In this work, the KdV Burgers' equation is solved by variety Galerkin methods.…”