Related papers: Approximate Solution of Kuramoto-Sivashinsky Equat…
This paper is presented to give numerical solutions of some cases of nonlinear wave-like equations with variable coefficients by using Reduced Differential Transform Method (RDTM). RDTM can be applied most of the physical, engineering,…
The generalized Kuramoto-Sivashinsky equation in the case of the power nonlinearity with arbitrary degree is considered. New exact solutions of this equation are presented.
We show that the only locally integrable stationary solutions to the integrated Kuramoto-Sivashinsky equation in $R$ and $R^2$ are the trivial constant solutions. We extend our technique and prove similar results to other nonlinear elliptic…
We present a novel control methodology to control the roughening processes of semilinear parabolic stochastic partial differential equations in one dimension, which we exemplify with the stochastic Kuramoto-Sivashinsky equation. The…
In this paper, we solve Laplace equation analytically by using differential transform method. For this purpose, we consider four models with two Dirichlet and two Neumann boundary conditions and obtain the corresponding exact solutions. The…
We consider data-driven reduced-order models of partial differential equations with shift equivariance. Shift-equivariant systems typically admit traveling solutions, and the main idea of our approach is to represent the solution in a…
In this work, we describe how to approximate solutions of some partial differential equations using the finite difference method defined on the Minkowski self-similar curve.
We report numerical simulations of one-dimensional cellular solutions of the stabilized Kuramoto-Sivashinsky equation. This equation offers a range of generic behavior in pattern-forming instabilities of moving interfaces, such as a host of…
The modification of simplest equation method to look for exact solutions of nonlinear partial differential equations is presented. Using this method we obtain exact solutions of generalized Korteweg-de Vries equation with cubic source and…
We present a new topological method for the study of the dynamics of dissipative PDE's. The method is based on the concept of the self-consistent apriori bounds, which allows to justify rigorously the Galerkin projection. As a result we…
We analyse the nonlinear Kuramoto-Sivashinsky equation to develop an accurate finite difference approximation to its dynamics. The analysis is based upon centre manifold theory so we are assured that the finite difference model accurately…
It is shown that an oblique projection based feedback control is able to stabilize the state of the Kuramoto-Sivashinsky equation, evolving in rectangular domains, to a given time-dependent trajectory. The number of actuators is finite and…
We propose an approximate model for the 2D Kuramoto-Sivashinsky equations (KSE) of flame fronts and crystal growth. We prove that this new ``calmed'' version of the KSE is globally well-posed, and moreover, its solutions converge to…
This article introduces and analyzes a new explicit, easily implementable, and full discrete accelerated exponential Euler-type approximation scheme for additive space-time white noise driven stochastic partial differential equations…
The concept of quasi-exact solutions of nonlinear differential equations is introduced. Quasi-exact solution expands the idea of exact solution for additional values of parameters of differential equation. These solutions are approximate…
Initial-boundary value problems for the $n$-dimensional ($n$ is a natural number from the interval [2,7]) Kuramoto-Sivashinsky equation posed on smooth bounded domains in $\mathbb{R}^n$ were considered. The existence and uniqueness of…
A numerical method is developed leading to algebraic systems based on generalized Lyapunov-Sylvester operators to approximate the solution of two-dimensional Boussinesq equation. It consists of an order reduction method and a finite…
The nonlocal Kuramoto-Sivashinsky equation arises in the modeling of the flow of a thin film of viscous liquid falling down an inclined plane, subject to an applied electric field. In this paper, the authors show that, as the coefficient of…
A recent type of B-spline functions, namely trigonometric cubic B-splines, are adapted to the collocation method for the numerical solutions of the Kuramoto-Sivashinsky equation. Having only first and second order derivatives of the…
In the present paper, we present some numerical methods for computing approximate solutions to some large differential linear matrix equations. In the first part of this work, we deal with differential generalized Sylvester matrix equations…