Abstract
In this work, the asymptotic stability result for Rosenau-Burgers equation is established, under appropriate assumptions on steady state eigenvalue problem and the forcing function. In addition, we propose and analyze a linearized numerical method for solving this nonlinear Rosenau-Burgers equation. We prove that the numerical scheme is unconditionally stable, and the error estimate shows that the numerical method is in the order of O(Δt2+N2-m), where Δt, N, and m are, respectively, step of time, polynomial degree, and regularity of u. Numerical examples are illustrated to verify the theoretical results.
Highlights
In the research of dynamic dense discrete system, it was shown that the KdV equation can not completely describe the interaction between waves and waves, and in order to overcome the shortcoming of this KdV equation, Rosenau [1, 2] proposed the following Rosenau equation: ut + uxxxxt + ux + uux = 0. (1)For further consideration of dissipation in dynamic system, the term −αuxx is included in the above equatoin
We prove that the numerical scheme is unconditionally stable, and the error estimate shows that the numerical method is in the order of O(Δt2 + N2−m), where Δt, N, and m are, respectively, step of time, polynomial degree, and regularity of u
This article studies the asymptotic stability of RosenauBurgers equation
Summary
Kinami [8] & Mei [9] considered the asymptotic behavior of solution for Benjamin-BonaMahony-Burgers equation They proved that the solution asymptotically converges to 0. Chung [10] & Sank [11] introduced finite element Galerkin method for solving a Rosenau equation They obtained the existence and uniqueness of solutions and the error estimates of the solutions are discussed. We consider the following Rosenau-Burgers equation: ut + uxxxxt − αuxx + ux + uux = f (x, t) , (3). We present numerical method to solve Rosenau-Burgers equation, and the proposed scheme is performed by combining Crank-Nicolson approach in time and Fourier-spectral in space.
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