scholarly journals SENSITIVE VERSUS CLASSICAL SINGULAR PERTURBATION PROBLEM VIA FOURIER TRANSFORM

2006 ◽  
Vol 16 (11) ◽  
pp. 1783-1816 ◽  
Author(s):  
N. MEUNIER ◽  
E. SANCHEZ-PALENCIA
Keyword(s):  
Fourier Transform ◽  
Singular Perturbation ◽  
Elliptic Boundary ◽  
Characteristic Parameter ◽  
Elliptic Boundary Value ◽  
Perturbation Problem ◽  
Formal Asymptotics ◽  
Ill Posed ◽  
The Subject ◽  
Thin Shell Theory

We consider a class of singular perturbation elliptic boundary value problems depending on a parameter ε which are classical for ε > 0 but highly ill-posed for ε = 0 as the boundary condition does not satisfy the Shapiro–Lopatinskii condition. This kind of problems is motivated by certain situations in thin shell theory, but we only deal here with model problems and geometries allowing a Fourier transform treatment. We consider more general loadings and more singular perturbation terms than in previous works on the subject. The asymptotic process exhibits a complexification phenomenon: in some sense, the solution becomes more and more complicated as ε decreases, and the limit does not exist in classical distribution theory (it may only be described in spaces of analytical functionals not enjoying localization properties). This phenomenon is associated with the emergence of the new characteristic parameter |log ε|. Numerical experiments based on a formal asymptotics are presented, exhibiting features which are unusual in classical elliptic equations theory. We also give a Fourier transform treatment of classical singular perturbations in order to exhibit the drastic differences with the present situation.

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.

On Positive Solutions for Some Nonlinear Semipositone Elliptic Boundary Value

2006 ◽  
Vol 11 (4) ◽  
pp. 323-329 ◽  
Author(s):  
G. A. Afrouzi ◽  
S. H. Rasouli
Keyword(s):  
Boundary Value Problems ◽  
Positive Solution ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Laplacian Operator ◽  
Smooth Bounded Domain ◽  
Elliptic Boundary Value ◽  
Existence Of Positive Solutions

This study concerns the existence of positive solutions to classes of boundary value problems of the form−∆u = g(x,u), x ∈ Ω,u(x) = 0, x ∈ ∂Ω,where ∆ denote the Laplacian operator, Ω is a smooth bounded domain in RN (N ≥ 2) with ∂Ω of class C2, and connected, and g(x, 0) < 0 for some x ∈ Ω (semipositone problems). By using the method of sub-super solutions we prove the existence of positive solution to special types of g(x,u).

Exponentially Convergent Approximation to the Elliptic Solution Operator

2006 ◽  
Vol 6 (4) ◽  
pp. 386-404 ◽  
Author(s):  
Ivan. P. Gavrilyuk ◽  
V.L. Makarov ◽  
V.B. Vasylyk
Keyword(s):  
Green Function ◽  
Boundary Value Problem ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Solution Operator ◽  
Hyperbolic Operator ◽  
Elliptic Solution ◽  
Elliptic Boundary Value ◽  
Sine Family ◽  
The Green Function

AbstractWe develop an accurate approximation of the normalized hyperbolic operator sine family generated by a strongly positive operator A in a Banach space X which represents the solution operator for the elliptic boundary value problem. The solution of the corresponding inhomogeneous boundary value problem is found through the solution operator and the Green function. Starting with the Dunford — Cauchy representation for the normalized hyperbolic operator sine family and for the Green function, we then discretize the integrals involved by the exponentially convergent Sinc quadratures involving a short sum of resolvents of A. Our algorithm inherits a two-level parallelism with respect to both the computation of resolvents and the treatment of different values of the spatial variable x ∈ [0, 1].

On Fourier Series in Eigenfunctions of Elliptic Boundary Value Problems

2003 ◽  
Vol 10 (3) ◽  
pp. 401-410
Author(s):  
M. S. Agranovich ◽  
B. A. Amosov
Keyword(s):  
Fourier Series ◽  
Boundary Conditions ◽  
Fourier Coefficients ◽  
Elliptic Boundary ◽  
A Priori ◽  
Adjoint Problem ◽  
Elliptic Boundary Value ◽  
Right Hand ◽  
The Difference ◽  
The Right

Abstract We consider a general elliptic formally self-adjoint problem in a bounded domain with homogeneous boundary conditions under the assumption that the boundary and coefficients are infinitely smooth. The operator in 𝐿2(Ω) corresponding to this problem has an orthonormal basis {𝑢𝑙} of eigenfunctions, which are infinitely smooth in . However, the system {𝑢𝑙} is not a basis in Sobolev spaces 𝐻𝑡 (Ω) of high order. We note and discuss the following possibility: for an arbitrarily large 𝑡, for each function 𝑢 ∈ 𝐻𝑡 (Ω) one can explicitly construct a function 𝑢0 ∈ 𝐻𝑡 (Ω) such that the Fourier series of the difference 𝑢 – 𝑢0 in the functions 𝑢𝑙 converges to this difference in 𝐻𝑡 (Ω). Moreover, the function 𝑢(𝑥) is viewed as a solution of the corresponding nonhomogeneous elliptic problem and is not assumed to be known a priori; only the right-hand sides of the elliptic equation and the boundary conditions for 𝑢 are assumed to be given. These data are also sufficient for the computation of the Fourier coefficients of 𝑢 – 𝑢0. The function 𝑢0 is obtained by applying some linear operator to these right-hand sides.

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