A modification of the exponential time-differencing fourth-order Runge–Kutta method for solving stiff nonlinear PDEs is presented that solves the problem of numerical instability in the scheme as proposed by Cox and Matthews and generalizes the method to nondiagonal operators. A comparison is made of the performance of this modified exponential time-differencing (ETD) scheme against the competi...

In this paper, we study irreducibility of Kuramoto-Sivashinsky equation which is driven by an additive noise acting only on a finite number Fourier modes. order to obtain the irreducibility, first investigate approximate controllability finite-dimensional force, proof based Agrachev-Sarychev type geometric control approach. Next, continuity solving operator for deterministic equation. Finally, ...

We prove that the Kuramoto-Sivashinsky equation is locally controllable in 1D and in 2D with one boundary control. Our method consists in combining several general results in order to reduce the nullcontrollability of this nonlinear parabolic equation to the exact controllability of a linear beam or plate system. This improves known results on the controllability of Kuramoto-Sivashinsky equatio...

The Kuramoto-Sivashinsky equation arises in a variety of applications, among which are modeling reaction-diffusion systems, flame-propagation and viscous flow problems. It is considered here, as a prototype to the larger class of generalized Burgers equations: those consist of quadratic nonlinearity and arbitrary linear parabolic part. We show that such equations are well-posed, thus admitting ...

In this paper, we study some controllability and observability properties for a coupled system of time-discrete fourth- second-order parabolic equations. This can be regarded as simplification the well-known stabilized Kumamoto-Sivashinsky equation. Unlike continuous case, prove only relaxed inequality which yields $ \phi(\triangle t) $-controllability result. result tells that cannot reach exa...

We develop an algorithm based on the nudging data assimilation scheme for concurrent (on-the-fly) estimation of scalar parameters a system evolutionary dissipative partial differential equations in which state is partially observed. The takes advantage error that results from with incorrect true system. intuitive nature makes its extension to several different systems immediate, and it allows r...

This paper deals with the null-controllability of a system mixed parabolic-elliptic pdes at any given time T>0. More precisely, we consider Kuramoto-Sivashinsky–Korteweg-de Vries equation coupled second order elliptic posed in interval (0,1). We first show that linearized is globally null-controllable by means localized interior control acting on either KS-KdV or equation. Using Carleman approa...

Motivated by the work of Foias and Temam [C. Foias, R. Temam, Gevrey class regularity for the solutions of the Navier–Stokes equations, J. Funct. Anal. 87 (1989) 359–369], we prove the existence and Gevrey regularity of local solutions to the Kuramoto–Sivashinsky equation in Rn with initial data in the space of distributions. The control on the Gevrey norm provides an explicit estimate of the a...

A method is developed for model predictive control of nonlinear stochastic partial differential equations (PDEs) to regulate the state variance, which physically represents the roughness of a surface in a thin film growth process, to a desired level. Initially a nonlinear stochastic PDE is formulated into a system of infinite nonlinear stochastic ordinary differential equations by using Galerki...