Steklov eigenvalues of reflection-symmetric nearly circular planar domains

We consider Steklov eigenvalues of reflection-symmetric, nearly circular, planar domains. Treating such domains as perturbations of the disc, we obtain a second-order formal asymptotic estimate in the domain perturbation parameter. We conclude with a discussion of implications for isoperimetric inequalities. Namely, our results corroborate the results of Weinstock and Brock that state, respectively, that the disc is the maximizer for the area and perimeter constrained problems. They also support the result of Hersch, Payne and Schiffer that the product of the first two eigenvalues is maximal among all open planar sets of equal perimeter. In addition, our results imply that the disc is not the maximizer of the area constrained problems for higher even numbered Steklov eigenvalues, as suggested by previous numerical results.

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An inequality for Steklov eigenvalues for planar domains

Zeitschrift für angewandte Mathematik und Physik ◽

10.1007/bf00945933 ◽

1994 ◽

Vol 45 (3) ◽

pp. 493-496 ◽

Cited By ~ 8

Author(s):

Julian Edward

Keyword(s):

Planar Domains ◽

Steklov Eigenvalues

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Modified Taylor wavelets approach to the numerical results of second order differential equations

International Journal of Applied Nonlinear Science ◽

10.1504/ijans.2021.10043789 ◽

2021 ◽

Vol 3 (2) ◽

pp. 136

Author(s):

Ankit Kumar ◽

Sag Ram Verma

Keyword(s):

Differential Equations ◽

Second Order ◽

Numerical Results ◽

Second Order Differential Equations

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Numerical solution of a source identification problem: Almost coercivity

Journal of Inverse and Ill-Posed Problems ◽

10.1515/jiip-2017-0072 ◽

2019 ◽

Vol 27 (4) ◽

pp. 457-468 ◽

Cited By ~ 1

Author(s):

Allaberen Ashyralyev ◽

Abdullah Said Erdogan ◽

Ali Ugur Sazaklioglu

Keyword(s):

Numerical Solution ◽

Source Identification ◽

Identification Problem ◽

Second Order ◽

Numerical Results ◽

Difference Schemes ◽

Stability Estimates ◽

Order Of Accuracy ◽

Accuracy Difference ◽

Coercive Stability

Abstract The present paper is devoted to the investigation of a source identification problem that describes the flow in capillaries in the case when an unknown pressure acts on the system. First and second order of accuracy difference schemes are presented for the numerical solution of this problem. Almost coercive stability estimates for these difference schemes are established. Additionally, some numerical results are provided by testing the proposed methods on an example.

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Control of Oscillations in Second-Order Differential Equation

Research Papers Faculty of Materials Science and Technology Slovak University of Technology ◽

10.2478/rput-2014-0035 ◽

2014 ◽

Vol 22 (35) ◽

pp. 57-62

Author(s):

Zuzana Šutová ◽

Róbert Vrábeľ ◽

Bohuslava Juhásová ◽

Martin Juhás

Keyword(s):

Oscillation Frequency ◽

Order Differential Equation ◽

Frequency Control ◽

Nonlinear Dynamical System ◽

Perturbation Parameter ◽

Second Order ◽

Frequency Change ◽

Nonlinear Dynamical ◽

Second Order Differential Equations ◽

Control Of Oscillations

Abstract The article deals with the control of oscillations in a specific type of second-order differential equations. The purpose of the research is to prove the possibility of oscillation frequency control based on a change in the value of a singular perturbation parameter placed into a mathematical model of a nonlinear dynamical system at the highest derivative. The oscillation frequency change caused by a different value of the parameter is verified by numerically modelling the system.

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Existence results for a nonlinear nonautonomous transmission problem via domain perturbation

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/prm.2021.60 ◽

2021 ◽

pp. 1-26

Author(s):

Matteo Dalla Riva ◽

Riccardo Molinarolo ◽

Paolo Musolino

Keyword(s):

Boundary Value Problem ◽

Laplace Equation ◽

Boundary Value ◽

Perturbation Parameter ◽

Existence Results ◽

Transmission Problem ◽

Perforated Domain ◽

Analytic Dependence ◽

Domain Perturbation ◽

Fixed Domain

In this paper we study the existence and the analytic dependence upon domain perturbation of the solutions of a nonlinear nonautonomous transmission problem for the Laplace equation. The problem is defined in a pair of sets consisting of a perforated domain and an inclusion whose shape is determined by a suitable diffeomorphism $\phi$ . First we analyse the case in which the inclusion is a fixed domain. Then we will perturb the inclusion and study the arising boundary value problem and the dependence of a specific family of solutions upon the perturbation parameter $\phi$ .

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A Second Order Weighted Numerical Scheme on Nonuniform Meshes for Convection Diffusion Parabolic Problems

European Journal of Computational Mechanics ◽

10.13052/ejcm2642-2085.2854 ◽

2020 ◽

Author(s):

Lolugu Govindarao ◽

Jugal Mohapatra

Keyword(s):

Time Derivative ◽

Perturbation Parameter ◽

Rectangular Domain ◽

Shishkin Mesh ◽

Second Order ◽

Parabolic Problems ◽

Uniform Mesh ◽

Central Difference ◽

Convection Diffusion ◽

Convection Diffusion Equation

In this article, a singularly perturbed parabolic convection-diffusion equation on a rectangular domain is considered. The solution of the problem possesses regular boundary layer which appears in the spatial variable. To discretize the time derivative, we use two type of schemes, first the implicit Euler scheme and second the implicit trapezoidal scheme on a uniform mesh. For approximating the spatial derivatives, we use the monotone hybrid scheme, which is a combination of midpoint upwind scheme and central difference scheme with variable weights on Shishkin-type meshes (standard Shishkin mesh, Bakhvalov-Shishkin mesh and modified Bakhvalov-Shishkin mesh). We prove that both numerical schemes converge uniformly with respect to the perturbation parameter and are of second order accurate. Thomas algorithm is used to solve the tri-diagonal system. Finally, to support the theoretical results, we present a numerical experiment by using the proposed methods.

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The Approximate Calculation of Eigenvalues for Certain Second-Order Linear Differential Equations

Journal of the Royal Aeronautical Society ◽

10.1017/s0368393100074472 ◽

1961 ◽

Vol 65 (605) ◽

pp. 360-360 ◽

Cited By ~ 1

Author(s):

W. J. Goodey

Keyword(s):

Approximate Solution ◽

Differential Equations ◽

Simple Formula ◽

Approximate Calculation ◽

Technical Note ◽

Second Order ◽

Numerical Results ◽

Linear Differential Equations ◽

Order Linear ◽

Linear Differential

In a recent technical note, Squire discussed the approximate solution of certain second-order linear differential equations by the method attributed variously to Riccati, Madelung, Wentzel, Kramers and Brillouin (the W.K.B. method), and others. The problem of eigenvalues, frequently met with in this type of equation, does not, however, appear to have received much attention by this method, and in this note a simple formula is developed which appears to give excellent numerical results in many cases.

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