We analyze a Crank–Nicolson–type finite difference scheme for the Kuramoto– Sivashinsky equation in one space dimension with periodic boundary conditions. We discuss linearizations of the scheme and derive second–order error estimates.

In this article we study the solution of the Kuramoto–Sivashinsky equation on a bounded interval subject to a random forcing term. We show that a unique solution to the equation exists for all time and depends continuously on the initial data.

In this work we estimate rates of the linear transient growth of the perturbations of cellular flames governed by the Sivashinsky equation. The possibility and significance of such a growth was indicated earlier in both computational and analytical investigations. Numerical investigation of the norm of the resolvent of the linear operator associated with the Sivashinsky equation linearized in a...

We consider the Kuramoto-Sivashinsky (KS) equation in one dimension with periodic boundary conditions. We apply a Lyapunov function argument similar to the one first introduced by Nicolaenko, Scheurer, and Temam [18], and later improved by Collet, Eckmann, Epstein and Stubbe[1], and Goodman [10] to prove that lim sup t→∞ ||u||2 ≤ CL 3 2 . This result is slightly weaker than that recently announ...

where μ, ν are positive constants. This equation, in the case μ = 0, was derived independently by Sivashinsky [1] and Kuramoto [2] with the purpose to model amplitude and phase expansion of pattern formations in different physical situations, for example, in the theory of a flame propagation in turbulent flows of gaseous combustible mixtures, see Sivashinsky [1], and in the theory of turbulence...

In this article, a nonic B-spline collocation approach is applied to solve the approximate solution of Kuramoto-Sivashinsky equation (KSE). Here nonlinear term KSE linearizing using Taylo...

Nonlinear partial differential equations are widely used to describe complex phenomena in various fields of science, for example the Korteweg-de Vries-Kuramoto-Sivashinsky equation (KdV-KS equation) and the Ablowitz-Kaup-Newell-Segur shallow water wave equation (AKNS-SWW equation). To our knowledge the exact solutions for the first equation were still not obtained and the obtained exact solutio...