The eigenvalues of ∇2u + λu=0 when the boundary conditions are given on semi-infinite domains

1953 ◽  
Vol 49 (4) ◽  
pp. 668-684 ◽  
Author(s):  
D. S. Jones
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Continuous Spectrum ◽  
Discrete Spectrum ◽  
Cross Section ◽  
Radiation Condition ◽  
Finite Width ◽  
Infinite Domains ◽  
Counter Example ◽  
Submerged Cylinder

ABSTRACTThe spectrum of −∇2 (and of −∇2 + b) is investigated when the boundary conditions are given on surfaces which extend to infinity. Simple criteria are obtained for determining whether point-eigenvalues are present in the lower part of the spectrum.Semi-infinite domains which are conical at infinity are found to possess purely continuous spectra when the boundary condition is u = 0 or ∂u/∂v = 0; the radiation condition ensures a unique solution. A counter-example shows that this is not true in general for the boundary condition ∂u/∂v + σu = 0.Semi-infinite domains which are cylindrical at infinity have a continuous spectrum with a discrete spectrum embedded in it. An example is given.The results are applied to the theory of surface waves. It is shown that Ursell's ‘trapping modes’ can occur in a canal of finite width when the bed has a protrusion over a finite longth but is otherwise of uniform depth. Trapping modes can also occur when the canal contains a submerged cylinder (not necessarily small in cross-section).

scholarly journals Ranges of Backwater Curves in Lower Odra

2018 ◽  
Vol 28 (4) ◽  
pp. 25-35
Author(s):  
Robert Mańko
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Cross Section ◽  
River Network ◽  
Hydrological Conditions ◽  
Complicated System ◽  
Odra River ◽  
The Cross

Abstract In the paper, the backwater curve ranges at the mouth of the Odra River with changing boundary conditions were analysed. The aim of the study is to determine which boundary condition, i.e. stage of lower cross-section or flow in upper cross-section, has a greater impact on the formation of the backwater curve at the mouth of the Odra River. Due to the complicated system of the Lower Odra River network (Międzyodrze and Dabie Lake), the analysis takes into consideration a section of the Odra River from a weir in Widuchowa upwards, thereby accepting as an axiom that the cross-section in Widuchowa is within the range of sea impact, regardless of other hydrological conditions.

scholarly journals A boundary value problem with a discontinuous coefficient and containing a spectral parameter in the boundary condition

1995 ◽  
Vol 18 (1) ◽  
pp. 133-140 ◽  
Author(s):  
A. A. Darwish
Keyword(s):  
Boundary Condition ◽  
Boundary Value Problem ◽  
Continuous Spectrum ◽  
Discrete Spectrum ◽  
Spectral Parameter ◽  
Boundary Value ◽  
Adjoint Problem ◽  
Discontinuous Coefficient ◽  
Eigenvalue Equation

A singular non-self-adjoint boundary value problem is considered. This problem has a discontinuous coefficient with a spectral parameter in the boundary condition. Some solutions of the eigenvalue equation are given. The discrete spectrum is studied and the resolvent is obtained. Formulation of the adjoint problem is deduced and hence the continuous spectrum of the considered problem is given. Furthermore, the spectrum of the adjoint problem is investigated.

scholarly journals An accurate and efficient series solution for the longitudinal vibration of elastically restrained rods with arbitrarily variable cross sections

2019 ◽  
Vol 38 (2) ◽  
pp. 403-414 ◽  
Author(s):  
Deshui Xu ◽  
Jingtao Du ◽  
Zhigang Liu
Keyword(s):  
Boundary Condition ◽  
Fourier Series ◽  
Boundary Conditions ◽  
Cross Section ◽  
Longitudinal Vibration ◽  
Variable Cross ◽  
Arbitrary Boundary ◽  
Cross Section Area ◽  
Section Area ◽  
Area Variation

Longitudinal vibration of non-uniform rod has been of great significance in various engineering occasions. The existing works are usually limited to the certain area variation and/or classical boundary condition. Motivated by this limitation, an efficient accurate solution is developed for the longitudinal vibration of a general variable cross-section rod with arbitrary boundary condition. Displacement function is invariantly expressed as the summation of standard Fourier series and supplementary polynomials, with an aim to make the calculation of all derivatives more straightforwardly in the whole solving region [0, L]. Energy principle is employed for system dynamics formulation, with the elastic boundaries considered as potential energy stored in the restraining spring. Arbitrary cross-section area variation is uniformly expanded into Fourier series. Numerical examples are presented for the natural frequency and mode shapes of non-uniform rod of free and clamped boundary conditions and compared with literature data. Results show good agreement with the previous analytical solutions. Influence of cross section area variation on vibration characteristics of non-uniform rods is then studied and discussed. One of the most advantages of the proposed model is that there is no need to reformulate the problem or rewrite the codes when the cross-section area distribution and/or boundary conditions change arbitrarily.

An asymptotic approximation of the maximum runup produced by a Tsunami wave train entering an inclined bay with parabolic cross section

2020 ◽  
Author(s):  
Nazmi Postacioglu ◽  
M. Sinan Özeren ◽  
Ebubekir Çelik
Keyword(s):  
Boundary Condition ◽  
Integral Equation ◽  
Boundary Conditions ◽  
Cross Section ◽  
Incident Wave ◽  
Dirichlet Boundary ◽  
Decay Rates ◽  
Dirichlet Boundary Conditions ◽  
Open Boundary ◽  
Open Sea

<p>Investigation of the behavior of various types of Tsunami wave trains entering bays is of practical importance for coastal hazard assessments. The linear shallow water equations admit two types of solutions inside an inclined bay with parabolic cross section: Energy transmitting modes and decaying modes. In low frequency limit there is only one mode susceptible of transmitting energy to the inland tip of the bay. The decay rates of decaying modes are controlled by the boundary conditions at the sides of the bay. Therefore a complicated eigenvalue problem needs to be solved in order to compute these decay rates. To determine the amplitude of the energy transmitting mode one should solve an integral equation, involving not just the energy transmitting mode but also decaying modes, the scattered field into the open sea, the incident wave and the reflected wave in the open sea. However, in the long wave limit, all these complications can be avoided if one applies the Dirichlet boundary conditions at the open boundary. That is to take the displacement of the free surface at the open boundary being equal to the twice of the disturbance associated with the incident wave in the open sea, just like a wall boundary condition. The runup produced by the solution obtained from this Dirichlet boundary condition, can be easily calculated using a series of images. In this model no energy is allowed to escape from the bay therefore the error arising from the simplification of the boundary conditions at the open boundary grows with time. Nevertheless the maximum runup occurs before this error becomes significant. If the characteristic wavelength of the incident wave train is equal to 5 times the width of the bay then this simple solution overestimates the first maximum of the runup only by %15 compared to the “exact” solution derived from the integral equation. This overestimation is partly due to the fact that Dirichlet boundary conditions violates the continuity of depth integrated velocities. The solution associated with Dirichlet boundary condition is perturbed in order to match fluxes inside and outside of the bay. This perturbation does not use the decaying modes inside the bay. The height of the first maximum of the runup coming from the perturbation theory is in excellent agreement with that obtained using the integral equation. This perturbation theory can also be applied to narrow bays with arbitrary cross section as long as their depth does not not change in the longitudinal direction.</p>

Sturm-Liouville Problem for Stationary Differential Operator with Nonlocal Two-Point Boundary Conditions

2006 ◽  
Vol 11 (1) ◽  
pp. 47-78 ◽  
Author(s):  
S. Pečiulytė ◽  
A. Štikonas
Keyword(s):  
Boundary Condition ◽  
Differential Operator ◽  
Boundary Conditions ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Complex Case ◽  
Qualitative Behavior ◽  
Point Boundary ◽  
Sturm Liouville

The Sturm-Liouville problem with various types of two-point boundary conditions is considered in this paper. In the first part of the paper, we investigate the Sturm-Liouville problem in three cases of nonlocal two-point boundary conditions. We prove general properties of the eigenfunctions and eigenvalues for such a problem in the complex case. In the second part, we investigate the case of real eigenvalues. It is analyzed how the spectrum of these problems depends on the boundary condition parameters. Qualitative behavior of all eigenvalues subject to the nonlocal boundary condition parameters is described.

scholarly journals Stiffness Estimation and Equivalence of Boundary Conditions in FEM Models

2021 ◽  
Vol 11 (4) ◽  
pp. 1482
Author(s):  
Róbert Huňady ◽  
Pavol Lengvarský ◽  
Peter Pavelka ◽  
Adam Kaľavský ◽  
Jakub Mlotek
Keyword(s):  
Boundary Condition ◽  
Finite Element ◽  
Boundary Conditions ◽  
Model Updating ◽  
Element Model ◽  
Computational Time ◽  
Stiffness Coefficients ◽  
First Case ◽  
Numerical Approaches

The paper deals with methods of equivalence of boundary conditions in finite element models that are based on finite element model updating technique. The proposed methods are based on the determination of the stiffness parameters in the section plate or region, where the boundary condition or the removed part of the model is replaced by the bushing connector. Two methods for determining its elastic properties are described. In the first case, the stiffness coefficients are determined by a series of static finite element analyses that are used to obtain the response of the removed part to the six basic types of loads. The second method is a combination of experimental and numerical approaches. The natural frequencies obtained by the measurement are used in finite element (FE) optimization, in which the response of the model is tuned by changing the stiffness coefficients of the bushing. Both methods provide a good estimate of the stiffness at the region where the model is replaced by an equivalent boundary condition. This increases the accuracy of the numerical model and also saves computational time and capacity due to element reduction.

scholarly journals Bootstrapping boundary-localized interactions

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Connor Behan ◽  
Lorenzo Di Pietro ◽  
Edoardo Lauria ◽  
Balt C. van Rees
Keyword(s):  
Boundary Condition ◽  
Scalar Field ◽  
Boundary Conditions ◽  
Parameter Space ◽  
Three Dimensional ◽  
Conformal Boundary ◽  
Dimensional Boundary ◽  
Dirichlet And Neumann Conditions ◽  
Boundary Fields ◽  
Boundary Dynamics

Abstract We study conformal boundary conditions for the theory of a single real scalar to investigate whether the known Dirichlet and Neumann conditions are the only possibilities. For this free bulk theory there are strong restrictions on the possible boundary dynamics. In particular, we find that the bulk-to-boundary operator expansion of the bulk field involves at most a ‘shadow pair’ of boundary fields, irrespective of the conformal boundary condition. We numerically analyze the four-point crossing equations for this shadow pair in the case of a three-dimensional boundary (so a four-dimensional scalar field) and find that large ranges of parameter space are excluded. However a ‘kink’ in the numerical bounds obeys all our consistency checks and might be an indication of a new conformal boundary condition.

scholarly journals An asymptotically compatible approach for Neumann-type boundary condition on nonlocal problems

2020 ◽  
Vol 54 (4) ◽  
pp. 1373-1413 ◽  
Author(s):  
Huaiqian You ◽  
XinYang Lu ◽  
Nathaniel Task ◽  
Yue Yu
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Nonlocal Boundary ◽  
Nonlocal Diffusion ◽  
Order Convergence ◽  
Flux Boundary Condition ◽  
Nonlocal Interactions ◽  
Local Case ◽  
Neumann Type ◽  
Type Boundary

In this paper we consider 2D nonlocal diffusion models with a finite nonlocal horizon parameter δ characterizing the range of nonlocal interactions, and consider the treatment of Neumann-like boundary conditions that have proven challenging for discretizations of nonlocal models. We propose a new generalization of classical local Neumann conditions by converting the local flux to a correction term in the nonlocal model, which provides an estimate for the nonlocal interactions of each point with points outside the domain. While existing 2D nonlocal flux boundary conditions have been shown to exhibit at most first order convergence to the local counter part as δ → 0, the proposed Neumann-type boundary formulation recovers the local case as O(δ2) in the L∞ (Ω) norm, which is optimal considering the O(δ2) convergence of the nonlocal equation to its local limit away from the boundary. We analyze the application of this new boundary treatment to the nonlocal diffusion problem, and present conditions under which the solution of the nonlocal boundary value problem converges to the solution of the corresponding local Neumann problem as the horizon is reduced. To demonstrate the applicability of this nonlocal flux boundary condition to more complicated scenarios, we extend the approach to less regular domains, numerically verifying that we preserve second-order convergence for non-convex domains with corners. Based on the new formulation for nonlocal boundary condition, we develop an asymptotically compatible meshfree discretization, obtaining a solution to the nonlocal diffusion equation with mixed boundary conditions that converges with O(δ2) convergence.

Chaotic Vibration of a Two-dimensional Non-strictly Hyperbolic Equation

2018 ◽  
Vol 61 (4) ◽  
pp. 768-786 ◽  
Author(s):  
Liangliang Li ◽  
Jing Tian ◽  
Goong Chen
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Hyperbolic Equation ◽  
Method Of Characteristics ◽  
Nonlinear Boundary ◽  
Chaotic Oscillation ◽  
Nonlinear Boundary Conditions ◽  
Rigorous Proof ◽  
Two Dimensional ◽  
Chaotic Vibration

AbstractThe study of chaotic vibration for multidimensional PDEs due to nonlinear boundary conditions is challenging. In this paper, we mainly investigate the chaotic oscillation of a two-dimensional non-strictly hyperbolic equation due to an energy-injecting boundary condition and a distributed self-regulating boundary condition. By using the method of characteristics, we give a rigorous proof of the onset of the chaotic vibration phenomenon of the zD non-strictly hyperbolic equation. We have also found a regime of the parameters when the chaotic vibration phenomenon occurs. Numerical simulations are also provided.

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