SOLVING STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS BASED ON THE EXPERIMENTAL DATA

2003 ◽  
Vol 13 (03) ◽  
pp. 415-444 ◽  
Author(s):  
IVO BABUŠKA ◽  
KANG-MAN LIU ◽  
RAÚL TEMPONE
Keyword(s):  
Experimental Data ◽  
A Priori Estimates ◽  
Elliptic Boundary ◽  
A Priori ◽  
Independent Random Variables ◽  
Practical Reasons ◽  
Elliptic Boundary Value ◽  
Priori Estimates ◽  
Stochastic Coefficient ◽  
Mutually Independent

We consider a stochastic linear elliptic boundary value problem whose stochastic coefficient a(x, ω) is expressed by a finite number NKL of mutually independent random variables, and transform this problem into a deterministic one. We show how to choose a suitable NKL which should be as low as possible for practical reasons, and we give the a priori estimates for modeling error when a(x, ω) is completely known. When a random function a(x, ω) is selected to fit the experimental data, we address the estimation of the error in this selection due to insufficient experimental data. We present a simple model problem, simulate the experiments, and give the numerical results and error estimates.

scholarly journals A short proof of the existence of the solution to elliptic boundary problem

2015 ◽  
Vol 3 (3) ◽  
pp. 105
Author(s):  
Alexander G. Ramm
Keyword(s):  
Boundary Problem ◽  
A Priori Estimates ◽  
Elliptic Boundary ◽  
A Priori ◽  
Upper And Lower Solutions ◽  
Parameter Method ◽  
Elliptic Boundary Problem ◽  
Integral Equation Methods ◽  
Priori Estimates ◽  
Uniqueness Of The Solution

<p>There are several methods for proving the existence of the solution to the elliptic boundary problem \(Lu=f \text{in} D,\quad u|_S=0,\quad    (*)\). Here <em>L</em> is an elliptic operator of second order, f is a given function, and uniqueness of the solution to problem (*) is assumed. The known methods for proving the existence of the solution to (*) include variational methods, integral equation methods, method of upper and lower solutions. In this paper a method based on functional analysis is proposed. This method is conceptually simple. It requires some a priori estimates and a continuation in a parameter method, which is well-known.</p>

A priori estimates for solutions to elliptic equations on non-smooth domains

2002 ◽  
Vol 132 (4) ◽  
pp. 793-813 ◽  
Author(s):  
Daniel Daners
Keyword(s):  
A Priori Estimates ◽  
Elliptic Boundary ◽  
A Priori ◽  
Type Inequality ◽  
Good Control ◽  
Sobolev Type ◽  
Elliptic Boundary Value ◽  
Underlying Space ◽  
Wide Range ◽  
Domain Dependence

It is proved that elliptic boundary-value problems have a global smoothing property in Lebesgue spaces, provided the underlying space of weak solutions admits a Sobolev-type inequality. The results apply to all standard boundary conditions, and a wide range of non-smooth domains, even if the classical estimates fail. The dependence on the data is explicit. In particular, this provides good control over the domain dependence, which is important for applications involving varying domains.

The Dirichlet problem for the uniformly elliptic equation in generalized weighted Morrey spaces

2020 ◽  
Vol 57 (1) ◽  
pp. 68-90 ◽  
Author(s):  
Tahir S. Gadjiev ◽  
Vagif S. Guliyev ◽  
Konul G. Suleymanova
Keyword(s):  
Elliptic Equations ◽  
A Priori Estimates ◽  
A Priori ◽  
Morrey Spaces ◽  
Smooth Bounded Domain ◽  
Weighted Morrey Spaces ◽  
Priori Estimates ◽  
Muckenhoupt Class ◽  
Morrey Estimates ◽  
Dirichlet Boundary Problem

Abstract In this paper, we obtain generalized weighted Sobolev-Morrey estimates with weights from the Muckenhoupt class Ap by establishing boundedness of several important operators in harmonic analysis such as Hardy-Littlewood operators and Calderon-Zygmund singular integral operators in generalized weighted Morrey spaces. As a consequence, a priori estimates for the weak solutions Dirichlet boundary problem uniformly elliptic equations of higher order in generalized weighted Sobolev-Morrey spaces in a smooth bounded domain Ω ⊂ ℝn are obtained.

Convergence of the Rosenau-Korteweg-de Vries Equation to the Korteweg-de Vries One

2020 ◽  
Author(s):  
Giuseppe Maria Coclite ◽  
Lorenzo di Ruvo
Keyword(s):  
A Priori Estimates ◽  
Vries Equation ◽  
A Priori ◽  
Diffusion Parameter ◽  
Priori Estimates ◽  
De Vries ◽  
Wall Interactions

The Rosenau-Korteweg-de Vries equation describes the wave-wave and wave-wall interactions. In this paper, we prove that, as the diffusion parameter is near zero, it coincides with the Korteweg-de Vries equation. The proof relies on deriving suitable a priori estimates together with an application of the Aubin-Lions Lemma.

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