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1
Content available Radial solutions for nonlinear elliptic equation with nonlinear nonlocal boundary conditions
Kossowski Igor
Opuscula Mathematica
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2023
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Vol. 43, no. 5
675--687
EN
In this article, we prove existence of radial solutions for a nonlinear elliptic equation with nonlinear nonlocal boundary conditions. Our method is based on some fixed point theorem in a cone.
2
Content available remote Convergence of random oscillatory integrals in the presence of long-range dependence and application to homogenization
Lechiheb A., Nourdin I., Zheng G., Haouala E.
Probability and Mathematical Statistics
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2018
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Vol. 38, Fasc. 2
271--286
EN
This paper deals with the asymptotic behavior of random oscillatory integrals in the presence of long-range dependence. As a byproduct, we solve the corrector problem in random homogenization of onedimensional elliptic equations with highly oscillatory random coefficients displaying long-range dependence, by proving convergence to stochastic integrals with respect to Hermite processes.
3
Content available Numerical Implementation of the Fictitious Domain Method for Elliptic Equations
Temirbekov A. N., Wójcik W.
International Journal of Electronics and Telecommunications
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2014
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Vol. 60, No. 3
219-223
EN
In this paper, we consider an elliptic equation with strongly varying coefficients. Interest in the study of these equations is connected with the fact that this type of equation is obtained when using the fictitious domain method. In this paper, we propose a special method for the numerical solution of elliptic equations with strongly varying coefficients. A theorem is proved for the rate of convergence of the iterative process developed. A computational algorithm and numerical calculations are developed to illustrate the effectiveness of the proposed method.
4
Content available remote Boundary element method for elliptic equation : determination of fundamental solution
Klekot J.
Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej
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2010
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Vol. 9, nr 1
61-70
EN
Elliptic equation with source term dependent on the first derivative of unknown function is considered. To solve this equation by means of the boundary element method the fundamental solution should be known. In the paper the fundamental solutions for 1D,2D and 3D problems are derived.
5
Content available remote Optimality conditions for state-constrained PDE control problems with time-dependent controls
Los Reyes J. C. de, Merino P., Rehberg F., Troltzsch F.
Control and Cybernetics
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2008
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Vol. 37, no 1
5-38
EN
The paper deals with optimal control problems for semilinear elliptic and parabolic PDEs subject to pointwise state constraints. The main issue is that the controls are taken from a restricted control space. In the parabolic case, they are Rm -vector-valued functions of time, while they are vectors of Rm in elliptic problems. Under natural assumptions, first- and second-order sufficient optimality conditions are derived. The main result is the extension of second-order sufficient conditions to semilinear parabolic equations in domains of arbitrary dimension. In the elliptic case, the problems can be handled by known results of semi-infinite optimization. Here, different examples are discussed that exhibit different forms of active sets and where second-order sufficient conditions are satisfied at the optimal solution.
6
Content available remote On the regularization error of state constrained Neumann control problems
Krumbiegel K., Rosch A.
Control and Cybernetics
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2008
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Vol. 37, no 2
369-392
EN
A linear elliptic optimal control problem with point-wise state constraints in the interior of the domain is considered. Furthermore, the control is given on the boundary with associated constraints. An artificial distributed control is introduced in the cost functional, in the state equation and in the state constraints. Since there are no control constraints for the artificial control, efficient numerical methods can be easily established. Based on a possible violation of the pure pointwise state constraints, an error estimate for the regularization error is derived. The theoretical results are illustrated by numerical tests.
7
Content available remote Boundary value problem for the equation [formula] in some unbounded domain in En
Filar M., Szałajko K.
Opuscula Mathematica
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2002
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Vol. 22
11-19
EN
The aim of this paper is to find a classical solution u of the non-linear equation. The function u is the solution of suitable system of integral equations. The existence and uniqueness of the solution of this system follows from the Banach fixed point theorem.
PL
Celem pracy jest znalezienie klasycznego rozwiązania u równania nieliniowego [...]. Funkcja u jest rozwiązaniem odpowiedniego układu równań całkowych. Istnienie i jednoznaczność rozwiązania tego układu równań wynika z twierdzenia Banacha o punkcie stałym.
8
Content available remote Some nonlinear problem to the equation [Delta]u-cu=f for a sphere
Barański F., Tal-Figiel B.
Fasciculi Mathematici
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2002
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Nr 33
21-26
EN
The subject of the paper is the construction of a solution of the differential equation delta u (x) - c(x)u(x) = f(x, u(x)), in the spherical domain D = {x = (xt, x2, x,): | x | < R}, satisfying the Dirichlet boundary-value condition u(x) = h(x) for x e B(D). Let (x, y) -ť G(x, y) denote the Green function to the Laplace equation deltaG(x, y) = 0 in the sphere D and to the homogeneous boundary-value condition G(x, y) = 0 for x belong B(D), y belong D. Applying the change of the unknown function x -ť u(x) = u (x) + w(x), where u is a solution to the equation delta u (x, y) = 0 0, x belong D, with the boundary-value condition u(x) = h(x), x belong B(D), and w is a a new unknown function, we obtain the equation deltaw(x) = u + w(x)) + f(x,u(x) + w(x)) or the equation deltaw(x) = F(1)(x,w(x) = c(x)(u(x) + F(x,w(x)) with F(x, w(x)) = c(x)~w(x) + f(x, u(x) + w(x)), where x e D. Inverting the last problem by the Green function G, we obtain the integral equation w(x) = f, (x) + +fffF(y(w(y))G(x,y)dy, xbelongD, with f, (x) = fff c(y)u (y)G(x, y)dy, x belong D, and the homogeneous boundary-value condition w(x) = 0 for x e B(D). Solving by the Banach fixed point method the last equation we obtain w and u = u + w.
9
Content available remote Trefftz spectral method for elliptic equations of general type
Reutskiy S., Tirozzi B.
Computer Assisted Mechanics and Engineering Sciences
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2001
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Vol. 8, No. 4
629-644
EN
A new numerical method for 2D linear elliptic partial differential equations in an arbitrary geometry is presented. The special feature of the method presented is that the trial functions, which are used to approximate a solution, satisfy the PDE only approximately. This reduction of the requirement to the trial functions extends the field of application of the Trefftz method. The method is tested on several one- and two-dimensional problems.
10
Content available Optimal control of semilinear elliptic equation with state constraint : maximum principle for minimizing sequence, regularity, normality, sensitivity
Sumin M.
Control and Cybernetics
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2000
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Vol. 29, no 2
449-472
EN
This article deals with state constrained optimal control problem for semilinear elliptic equation in a domain Omega. The state constraint is lumped on the compactum X contained in/implied by Omega n and contains a functional parameter q in C(X ). It is shown that any minimizing approximate solution (m.a.s.) in the sense of J. Warga satisfies the pointwise maximum principle (the maximum principle for m.a.s.) if the problem is meaningful, i.e., the value of the problem is finite. It is also shown that a condition of Slater's type is sufficient for the normality in the so-called "linear-convex" problem, and the normality of the problem for some fixed value of the parameter q in C(X ) implies the Lipschitz continuity of its value function in a neighborhood of q. The paper contains illustrative examples.
11
Content available Lipschitz stability of solutions to parametric optimal control for elliptic equations
Malanowski K., Troeltzsch F.
Control and Cybernetics
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2000
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Vol. 29, no 1
237-256
EN
: A family of parameter dependent elliptic optimal control problems with nonlinear boundary control is considered. The control function is subject to amplitude constraints. A characterization of conditions is given under which solutions to the problems exist, are locally unique and Lipschitz continuous in a neighborhood of the reference value of the parameter.
12
Content available remote Nonexistence criteria for positive solutions of a discrete elliptic equation
Cheng S. S., Zhang B.-G.
Fasciculi Mathematici
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1998
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Nr 28
19-30
EN
A nonlinear elliptic type partial difference equation with a forcing terrn is studied in this paper. By means of an averaging technique, the problem of non-existence of positive solutions is reduced to that of forced recurrence relations. Several sample non-existence criteria are given for these recurrence relations which in turn yield non-existence criteria for the discrete elliptic equation.
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