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1
A note on the validity of the Schrödinger approximation for the Helmholtz equation
Klumpp Maximilian, Schneider Guido
Journal of Applied Analysis
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2022
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Vol. 28, nr 1
67--72
EN
Time-harmonic electromagnetic waves in vacuum are described by the Helmholtz equation Δu + ω2u = 0 for (x, y, z) ∈ ℝ3. For the evolution of such waves along the z-axis, a Schrödinger equation can be derived through a multiple scaling ansatz. It is the purpose of this paper to justify this formal approximation by proving bounds between this formal approximation and true solutions of the original system. The challenge of the presented validity analysis is the fact that the Helmholtz equation is ill-posed as an evolutionary system along the z-axis.
2
Content available remote Weighted difference schemes for systems of quasilinear first order partial functional differential equations
Szafrańska A.
Mathematica Applicanda
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2015
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Vol. 43, No. 2
225--251
EN
The paper deals with initial boundary value problems of the Dirichlet type for system of quasilinear functional differential equations. We investigate weighted difference methods for these problems. A complete convergence analysis of the considered difference methods is given. Nonlinear estimates of the Perron type with respect to functional variables for given functions are assumed. The proof of the stability of difference problems is based on a comparison technique. The results obtained here can be applied to differential integral problems and differential equations with deviated variables. Numerical examples are presented.
PL
Praca dotyczy zagadnień początkowo brzegowych typu Dirichlet’a dla układów quasiliniowych równań różniczkowo-funkcyjnych. Zamieszczona jest konstrukcja ważonych metod różnicowych dla wyjściowych zagadnień różniczkowych oraz przeprowadzona jest pełna analiza zbieżności. Niezbędne założenia obejmują oszacowania typu Perrona dla funkcji danych względem argumentów funkcyjnych. Dowód stabilności metody różnicowej opiera się na technice porównawczej. Teoretyczne rezultaty zobrazowane są na przykładzie całkowego równania różniczkowego oraz równań różniczkowych z odchylonym argumentem.
3
Content available Difference functional inequalities and applications
Szafrańska A.
Opuscula Mathematica
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2014
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Vol. 34, no. 2
405--423
EN
The paper deals with the difference inequalities generated by initial boundary value problems for hyperbolic nonlinear differential functional systems. We apply this result to investigate the stability of constructed difference schemes. The proof of the convergence of the difference method is based on the comparison technique, and the result for difference functional inequalities is used. Numerical examples are presented.
4
Content available remote Convergence estimates for the acoustic scattering problem approximated by NURBS
Karafiat A.
Computer Assisted Methods in Engineering and Science
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2013
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Vol. 20, no. 4
289--307
EN
The paper contains some estimates of an approximation to the solution of the problem of acoustic waves’s scattering by an elastic obstacle in two dimensions. The problem is approximated by the isogeometric adaptive method based on the known NURBS functions. The estimates show how the error of an approximation depends on the size of intervals and the degree of functions.
5
Content available Pseudospectral method for semilinear partial functional differential equations
Czernous W.
Opuscula Mathematica
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2010
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Vol. 30, no. 2
133-145
EN
We present a convergence result for two spectral methods applied to an initial boundary value problem with functional dependence of Volterra type. Explicit condition of Courant-Friedrichs-Levy type is assumed on time step τ and the number N of collocation points. Stability statements and error estimates are written using continuous norms in weighted Jacobi spaces.
6
Content available remote A note on variational discretization of elliptic Neumann boundary control
Hinze M., Matthes U.
Control and Cybernetics
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2009
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Vol. 38, no 3
577-591
EN
We consider variational discretization of Neumann-type elliptic optimal control problems with constraints on the control. In this approach the cost functional is approximated by a sequence of functionals, which are obtained by discretizing the state equation with the help of linear finite elements. The control variable is not discretized. Error bounds for control and state are obtained both in two and three space dimensions. Finally, we discuss some implementation issues of a generalized Newton method applied to the numerical solution of the problem class under consideration.
7
Content available remote A new stopping criterion for iterative solvers for control optimal problems
Krumbiegel K., Rosch A.
Archives of Control Sciences
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2008
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Vol. 18, no. 1
17-42
EN
Linear quadratic optimal control problems governed by PDEs with pointwise control constraints are considered. We derive error estimates for feasible and infeasible controls of the problem. Based on this theory an error estimator is constructed for different discretization schemes. Moreo ver, we establish the estimator as a stopping criterion for several optimization methods. Furthermore, additional errors caused by solving the linear systems are discussed. The theory is illustrated by numerical examples.
8
Content available remote Error analysis of discrete approximations to bang-bang optimal control problems: the linear case
Veliov V. M.
Control and Cybernetics
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2005
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Vol. 34, no 3
967-982
EN
The paper presents an error estimate for Runge-Kutta direct discretizations of terminal optimal control problems for linear systems. The optimal control for such problems is typically discontinuous, and Lipschitz stability of the solution with respect to perturbations does not necessarily hold. The estimate (in terms of the optimal controls) is of first order if certain recently obtained sufficient conditions for structural stability hold, and of fractional order, otherwise. The main tool in the proof is the established relation between the local convexity index of the reachable set and the multiplicity of zeros of appropriate switching functions associated with the problem.
9
Content available remote Numerical approximation of first order partial differential equations with deviated variables
Baranowska A., Kamont Z.
Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae
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2003
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[Z] 43(1)
1-31
EN
Classical solutions of the local Cauchy problem on the Haar pyramid are approximated in the paper by solutions of suitable quasilinear systems of difference equations. The proof of the stability of the difference problem is based on a comparison technique with nonlinear estimates of the Perron type. This new approach to the numerical solving of nonlinear equations with deviated variables is generated by a quasilinearization method for initial problems. Numerical examples are given.
10
Content available Error estimates for the finite-element approximation of a semilinear elliptic control problem
Casas E., Troeltzsch F.
Control and Cybernetics
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2002
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Vol. 31, no 3
695-712
EN
We consider the finite-element approximation of a distributed optimal control problem governed by a semilinear elliptic partial differential equation, where pointwise constraints on the control are given. We prove the existence of local approximate solutions converging to a given local reference solution. Moreover, we derive error estimates for local solutions in the maximum norm.
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