A non-homogeneous boundary value problem for the Kuramoto-Sivashinsky equation posed in a finite interval

2020 ◽  
Vol 26 ◽  
pp. 43 ◽  
Author(s):  
Jing Li ◽  
Bing-Yu Zhang ◽  
Zhixiong Zhang
Keyword(s):  
Boundary Value Problem ◽  
Finite Interval ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Solution Map ◽  
Ill Posed ◽  
Initial Boundary Value ◽  
Well Posed ◽  
Sivashinsky Equation

This paper studies the initial boundary value problem (IBVP) for the dispersive Kuramoto-Sivashinsky equation posed in a finite interval (0, L) with non-homogeneous boundary conditions. It is shown that the IBVP is globally well-posed in the space Hs(0, L) for any s > −2 with the initial data in Hs(0, L) and the boundary value data belonging to some appropriate spaces. In addition, the IBVP is demonstrated to be ill-posed in the space Hs(0, L) for any s < −2 in the sense that the corresponding solution map fails to be in C2.

scholarly journals Regularization method for parabolic equation with variable operator

2005 ◽  
Vol 2005 (4) ◽  
pp. 383-392
Author(s):  
Valentina Burmistrova
Keyword(s):  
Boundary Value Problem ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Approximation Solution ◽  
Elliptic Boundary Value ◽  
Dirichlet Data ◽  
Ill Posed ◽  
Initial Boundary Value

Consider the initial boundary value problem for the equationut=−L(t)u,u(1)=won an interval[0,1]fort>0, wherew(x)is a given function inL2(Ω)andΩis a bounded domain inℝnwith a smooth boundary∂Ω.Lis the unbounded, nonnegative operator inL2(Ω)corresponding to a selfadjoint, elliptic boundary value problem inΩwith zero Dirichlet data on∂Ω. The coefficients ofLare assumed to be smooth and dependent of time. It is well known that this problem is ill-posed in the sense that the solution does not depend continuously on the data. We impose a bound on the solution att=0and at the same time allow for some imprecision in the data. Thus we are led to the constrained problem. There is built an approximation solution, found error estimate for the applied method, given preliminary error estimates for the approximate method.

scholarly journals Comparative Analysis of Methods for Regularizing an Initial Boundary Value Problem for the Helmholtz Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Sergey Igorevich Kabanikhin ◽  
M. A. Shishlenin ◽  
D. B. Nurseitov ◽  
A. T. Nurseitova ◽  
S. E. Kasenov
Keyword(s):  
Comparative Analysis ◽  
Boundary Value Problem ◽  
Helmholtz Equation ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Stability Estimate ◽  
Ill Posed ◽  
Continuation Problem ◽  
Initial Boundary Value

We consider an ill-posed initial boundary value problem for the Helmholtz equation. This problem is reduced to the inverse continuation problem for the Helmholtz equation. We prove the well-posedness of the direct problem and obtain a stability estimate of its solution. We solve numerically the inverse problem using the Tikhonov regularization, Godunov approach, and the Landweber iteration. Comparative analysis of these methods is presented.

scholarly journals WELL-POSED INITIAL-BOUNDARY VALUE PROBLEM FOR A CONSTRAINED EVOLUTION SYSTEM AND RADIATION-CONTROLLING CONSTRAINT-PRESERVING BOUNDARY CONDITIONS

2007 ◽  
Vol 04 (04) ◽  
pp. 587-612 ◽  
Author(s):  
ALEXANDER M. ALEKSEENKO
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Boundary Value ◽  
Time Derivative ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Evolution System ◽  
Constrained Evolution ◽  
Initial Boundary Value ◽  
Well Posed

A well-posed initial-boundary value problem is formulated for the model problem of the vector wave equation subject to the divergence-free constraint. Existence, uniqueness and stability of the solution is proved by reduction to a system evolving the constraint quantity statically, i.e. the second time derivative of the constraint quantity is zero. A new set of radiation-controlling constraint-preserving boundary conditions is constructed for the free evolution problem. Comparison between the new conditions and the standard constraint-preserving boundary conditions is made using the Fourier–Laplace analysis and the power series decomposition in time. The new boundary conditions satisfy the Kreiss condition and are free from the ill-posed modes growing polynomially in time.

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