Homogeneous Neumann Conditions Research Articles

The sine-Gordon equation in the finite line

A rather complete numerical study of the perturbed and nonperturbed sine-Gordon equation in one-dimensional Cartesian coordinates and finite domains is presented. The numerical study is based on five, three-point, linearly implicit finite difference methods of different spatial accuracy. For the unperturbed sGE, the accuracy of these five methods is found to be very sensitive to the implicitness parameter, less sensitive to the spatial order of accuracy, and almost no sensitive to the time step for second-order accurate, temporal discretizations. The largest errors of the five methods were found to occur at the front of the soliton solution of the unperturbed sGE, but the techniques were very accurate for long-time computations of the unperturbed sGE in both infinite domains and in finite domains upon many collisions of the kinks and antikinks with the boundaries, despite the fact that they do not preserve a discrete energy. It is shown that the largest values of the kinetic and total energies, and the smallest values of the strain and potential energies occur when solitons and antisolitons strike on the boundaries subject to homogeneous Neumann conditions, and the four energies remain constant between collisions with the boundaries. For the Malomed's perturbation, one of the fronts of the soliton–soliton doublet and soliton–antisoliton doublet is trapped at one boundary and the other one undergoes a change in amplitude and its front describes a curved trajectory; the soliton–antisoliton breather oscillates in a damped manner and slowly drifts towards one boundary. The soliton, antisoliton, soliton–soliton doublet and soliton–antisoliton doublet of the unperturbed sGE are robust under Kivshar and Malomed's perturbations, whereas the soliton–antisoliton breather loses its integrity, becomes a kink–antikink pair and then yields a chaotic behaviour in both space and time. Small amplifications proportional to the first-order temporal derivative of the amplitude do not alter substantially the initial dynamics of the soliton, antisoliton, soliton–soliton doublet and soliton–antisoliton doublet, but their amplitudes exhibit oscillatory behaviour of increasing amplitude at large times; however, small amplifications cause first a transition from the soliton–antisoliton breather to the kink–antikink solution. A linearized stability analysis of the spatially homogeneous solutions of the unperturbed sGE is performed and analytical solutions are obtained by means of the method of separation of variables for both homogeneous Dirichlet and Neumann boundary conditions, and the results show that the solution corresponding to a zero amplitude is oscillatory in both space and time, whereas that corresponding to an amplitude equal to π may exhibit exponential or linear growth depending on the length of the domain. If either of these growths is present, it is shown that the evolution of small, initial perturbations may result in homoclinic crossings and chaotic behaviour in both space and time in the absence of perturbations in the sGE and the existence of an attractor when perturbations of the Malomed's type are present. In either case, the solution may exhibit certain spatial symmetries or antisymmetries at large times. Finally, the results presented in this paper indicate that the accurate computation of homoclinic orbits of the sGE is not only controlled by the spatial accuracy of the numerical method, but also by its energy-preserving characteristics, and, perhaps, by the number of grid points used in the numerical technique.

Sharp estimates and finite speed of propagation for a Neumann problem in domains narrowing at infinity

We investigate the connection between the geometry of an unbounded domain $\Omega$ and the existence and qualitative behaviour of solutions to a degenerate doubly non linear parabolic equation posed in $\Omega$. The domain $\Omega$ is assumed to be ``narrowing'' at infinity in a suitable sense, so that it has infinite volume. On the boundary of $\Omega$ we prescribe a homogeneous Neumann condition. Among other results, we prove sharp estimates for the finite speed of propagation of the support of positive solutions originating from initial data with bounded support. This is done by means of a new approach, which is flexible enough to be applied to the geometry at hand, and to cover the case of initial data measures. We also show that even if the initial datum has finite mass, the solution need not be globally bounded over $\Omega$ for a fixed positive time. We provide sharp estimates for such solutions. Our main tool is a new embedding inequality connected with the geometry of $\Omega$.

Optimal bounds and blow up phenomena for parabolic problems in narrowing domains

We consider degenerate parabolic problems in domains with noncompact boundary and infinite volume, in any spatial dimension. The equation is of doubly nonlinear type. On the boundary we prescribe a homogeneous Neumann condition. The spatial domain is narrowing at infinity. We prove uniform convergence to 0 of solutions as time approaches ∞. To this end, due to the geometry of the domain, the requirement that the initial datum have finite mass is not enough, and we have to stipulate the further assumption that a certain moment of the initial datum (connected with the geometry of the domain) is finite. We prove optimal asymptotic estimates of the solution. Moreover, we apply our method to the investigation of blow-up problems in narrowing domains, obtaining a sharp condition, in integral form, for the existence of solutions defined for all positive times.

Compactness of support of solutions to nonlinear second-order elliptic and parabolic equations in a half-cylinder

We study equations of the form $$\begin{gathered} u_{tt} + Lu + b(x,t)u_t = a(x,t)\left| u \right|^{\sigma - 1} u, \hfill \\ - u_t + Lu = a(x,t)\left| u \right|^{\sigma - 1} u \hfill \\ \end{gathered}$$ , whereL is a uniformly elliptic operator and 0<σ<1. In the half-cylinder II0,∞={(x, t):x= (x 1,...,x n )∈ ω,t>0}, where Ή ⊂ ℝ n is a bounded domain, we consider solutions satisfying the homogeneous Neumann condition forx∈ϖω andt>0. We find conditions under which these solutions have compact support and prove statements of the following type: ifu(x, t)=o(t γ) ast→∞, then there exists aT such thatu(x, t)≡0 fort>T. In this case γ depends on the coefficients of the equation and on the exponent σ.

On the behavior of solutions to the Neumann problem in unbounded domains

The paper deals with the uniformly elliptic equation (aij(x)uxi)xj=f(x) in an unbounded domain Ω⊂ℝn and its solution u(x) that satisfies the homogeneous Neumann condition. The function f has a compact support. The domain Ω has the following structure: assume that {rm} is an increasing sequence of positive numbers, hm=rm+1−rm, and the ratio hm+1/hm lies between positive constants C1 and C2. The intersection of Ω with the spherical layer between the spheres of radius rm and rm+1 with center at the origin satisfies a certain inequality of isoperimetric type. It is shown in this paper that the set of solutions splits up into three classes: (i) the solutions for which $$\mathop {\overline {\lim } }\limits_{\left| x \right| \to \infty } u(x) = \infty $$ and $$\mathop {\underline {\lim } }\limits_{\left| x \right| \to \infty } U(x) = \infty $$ ; moreover, it is shown that these limits are attained with nearly the same speed (if C1/C2=1, then the speed is not less than the exponential one); (ii) the solutions for each of which a constant C exists such that $$\mathop {\lim }\limits_{\left| x \right| \to \infty } u(x) = C$$ and u(x)−C changes its sign for large |x|; here, the convergence to C is rapid (for C1/C2=1 this convergence is not slower than the exponential one); (iii) the solutions that do not change their sign for large |x| and increase or decrease to +∞ or −∞, respectively, with low speed (for C1/C2=1 with a linear speed) (one exception is possible here: a slow convergence to a constant). Bibliography: 7 titles.

An integral equation for the homogeneous Neumann condition1

AbstractModelling the theoretical response of several important geophysical systems involves the solution of Poisson's equation with homogeneous Neumann boundary conditions (i.e. a zero normal gradient) imposed over either open or closed surfaces.A simple integral equation solution to this problem is derived from first principles. It is applicable to both types of surface and in this respect represents an improvement on existing integral equation techniques. However, the present surface integral equation displays a strong singularity of order 1/R3 which requires an appropriate interpretation for its implementation.A comparison of some numerical results with analytical data taken from the literature demonstrates that the proposed integral equation technique is suitably robust, accurate and efficient for practical application in geophysical interpretation.

A duality theorem for an overdetermined eigenvalue problem

We prove an equivalent integral formulation of an eigenvalue problem for the Laplacian operator, with overdetermined boundary conditions. A special case of this problem, that with homogeneous Neumann condition, is known in the literature as Schiffer's problem, due to the still open conjecture, attributed to him, that the only domains possessing nontrivial solutions aren-balls. It is hoped that this integral formulation may provide a new tool with which to analyse Schiffer's conjecture.

A principle of images for absorbing boundary conditions

AbstractWhen one uses high‐order finite difference schemes for the wave equation, for instance fourth order schemes, the treatment of boundary conditions poses a real difficulty since one needs several additional equations (for the nodes close to the boundary), while one single scalar boundary condition is available. In the case of perfectly reflecting boundary conditions, namely the homogeneous Neumann or Dirichlet conditions, this difficulty can be overcomed by the use of the well‐known image principle, which permits the extension of the equation outside of the domain of calculation by an appropriate symmetrization of the data. We propose in this article a generalization of this principle to the absorbing boundary conditions. Through a symmetrization process, we are led to introduce a damped wave equation with a damping term supported by the boundary. The treatment of the boundary condition is then replaced by the approximation of this new damped wave equation in the whole space. The theoretical justification of our approach is based on new energy estimates for the wave equation (when high‐order absorbing boundary conditions are used), and constitutes an alternative to the use of the well‐known Kreiss criterion to prove the stability of the associated initial boundary value problems. © 1994 John Wiley &amp; Sons, Inc.

Polynomial time-marching for nonperiodic boundary value problems

A polynomial interpolation time-marching technique can efficiently provide balanced spectral accuracy in both the space and time dimensions for some PDEs. The Newton-form interpolation based on Fejer points has been successfully implemented to march the periodic Fourier pseudospectral solution in time. In this paper, this spectrally accurate time-stepping technique will be extended to solve some typical nonperiodic initial boundary value problems by the Chebyshev collocation spatial approximation. Both homogeneous Neumann and Dirichlet boundary conditions will be incorporated into the time-marching scheme. For the second order wave equation, besides more accurate timemarching, the new scheme numerically has anO(1/N 2) time step size limitation of stability, much larger thanO(1/N 4) stability limitation in conventional finitedifference time-stepping, Chebyshev space collocation methods.

BEHAVIOR OF SOLUTIONS OF A NONLINEAR VARIATIONAL PROBLEM IN A NEIGHBORHOOD OF SINGULAR POINTS OF THE BOUNDARY AND AT INFINITY

A study is made of the functions realizing a minimum for the functional where has power order of growth with respect to . The rate of decrease is established for the th-order derivatives of an extremal in the integral metric in a neighborhood of a singularity on the boundary of power cusp type and at infinity in domains having outside some ball the structure of a cylinder or layer, and also domains constricting or expanding at infinity in a power manner. Estimates are obtained under the assumption that homogeneous Dirichlet conditions or Neumann conditions are given on the indicated part of the boundary. The estimates depend on the geometry of the domain. The results obtained are new also for a broad class of nonlinear elliptic equations. Bibliography: 19 titles.

The Green function for potential flow in a rectangular channel

The evaluation of the Green function is considered for the three-dimensional Laplace equation, in the interior of a rectangular channel subject to homogeneous Neumann conditions on the boundaries. To complement the Fourier eigenfunction expansion which is effective in the far-field, a near-field algorithm is developed based on the simpler Green function for a channel of infinite width, using images to account for the channel sides. Examples are given of numerical applications including the added mass of a sphere in a square channel, and the interaction force between a ship and an adjacent canal wall.

Non-Linear Programming Applicable for the Control of Elastic Structures

This paper deals with the control of the von Mises stress at a boundary with homogeneous Neumann conditions. In order to solve this non-linear optimization problem in which the shape of the structural component must be treated as design variable, the variation of the objective and constraint functions with the change in shape is to be determined. Performing an accurate design sensitivity analysis is a difficult task because the shape does not appear explicitly in the formulation and because of the error associated with the approximation of the domain during discretization. The sensitivity analysis is done here using methods of theory of notches. The results will be applied to a FE-model of the structural component. The iterative method for changing the design based on this direction of search works efficiently even for a large number of design variables as shown in Schnack (1977a, 1978a, 1978b, 1979, 1980, 1982, 1983, 1985). Truncation and/or cancellation errors associated with numerical gradient computation are avoided. Using a dynamic programming formulation (Schnack and Spörl, 1986) the existence of a solution for this problem will be discussed. An example with this procedure is presented for a shaft in bending.

Stabilization of Wave Equation by Periodical Moving Actuators

The actuator B(t) is a moving object in D and we consider the wave equation in the non cylindrical domain U = tx(D\\B(t)). The boundary S(t) of B(t) is decomposed in three parts : S(t) = S_(t) U S_(t) U S0(t) with respect to the sign of the normal speed on S(t). Homogeneous Neumann and Dirichlet conditions are imposed respectively on S_(t.) and S_(t.) U S0(t). By penalisation and Galerkin technique we prove the existence and the uniqueness of solutions. We discuss only the energy dissipation associated to that, passive actuator.

Evolution systems in semiconductor device modeling: A cyclic uncoupled line analysis for the gummel map

AbstractThe initial/boundary‐value problem for isothermal, lattice, semiconductor device modeling is described and analyzed. This nonlinear elliptic/parabolic system of reaction/diffusion/convection type is determined by a Maxwell equation, relating space/charge and the electric field, and by two continuity equations for the free electron and hole carrier concentrations. The Einstein relations for Brownian motion are not assumed in this analysis, so that the electrostatic potential, u, and the carrier concentrations, n and p, are the fundamental dependent variables of the system. The boundary conditions are Dirichlet conditions for dependent variable values on the contact portions of the device, and homogeneous Neumann conditions, expressing insulation, on the complement. Complicating the analysis are the transition singularity points between the mixed boundary conditions, and the field dependence of the mobility and diffusion coefficients. By means of a physically motivated analysis of the convective current component, we are able to uncouple the system by a cyclic horizontal line analysis, without an unreasonable time step restriction. The corresponding linear equations are solved by a contractive inner iteration. The outer iteration is shown to converge to a unique solution of the system, under singularity classification at the transition points. The definition of this outer iteration follows the steady‐state Gummel iteration at discrete time steps. An existence theory is a by‐product of the analysis, and is separated from uniqueness theory.

A 3-component system of competition and diffusion

Abstract : This report studies the existence of non-constant solutions of certain two-point boundary value problems for 3-component systems with a small parameter epsilon, under homogeneous Neumann conditions at the boundaries. This problem is related to the analysis of segregation patterns in population models of 3-competing and spatially dispersing species. It is shown that the reduced problem (epsilon = 0) has many non-constant solutions exhibiting spatial segregation. Only a few of these, however, can serve as valid lowest-order approximations to solutions of the original problem when epsilon is non-zero but small. A singular perturbation construction clarifies which are in this category. The results of numerical computations of solutions are also illustrated. (Author)

ON THE BEHAVIOR AT INFINITY OF SOLUTIONS OF SECOND ORDER ELLIPTIC EQUATIONS IN DOMAINS WITH NONCOMPACT BOUNDARY

In this paper the behavior at infinity of solutions of second order elliptic equations is studied. Here the solutions satisfy homogeneous Dirichlet conditions, Neumann conditions or periodicity conditions, in each case on the part of the boundary that belongs to some neighborhood of infinity. The authors obtain a priori estimates characterizing the behavior, as , of these solutions in domains with noncompact boundary, depending on the geometric properties of the domain and the behavior, again as , of the function on the right side of the equation. Bibliography: 13 titles.

Matched Asymptotic Expansions of the Green’s Function for the Electric Potential in an Infinite Cylindrical Cell

The potential is studied for a microelectrode current source inside a nerve fiber. The problem is represented by a point source in an infinitely long cylindrical conductor surrounded by a thin, low conductivity membrane bathed in a perfectly conducting medium. The potential satisfies Laplace’s equation with a mixed boundary condition containing a small parameter $\varepsilon $. As $\varepsilon \to 0$ it approaches a homogeneous Neumann condition and the problem becomes singular. Asymptotic expansions are obtained in terms of $\varepsilon $ by matching an inner expansion (valid at the source) to an outer expansion (valid away from the source). The inner expansion contains algebraic switchback terms whose orders are half-odd-integer powers of $\varepsilon $, as well as terms whose orders are integer powers of $\varepsilon $ which can be expressed as algebraic terms plus Fourier series with Fourier integral coefficients. The outer expansion is made uniform by introducing a strained axial coordinate, which ap...

Green's function for Laplace's equation in an infinite cylindrical cell

The Green's function for Laplace's equation in an infinite-length cylinder with a homogeneous mixed boundary condition is considered. Its eigenfunction expansion converges slowly when the axial separation between the source and observation points is small compared to the cylinder radius, and diverges when the axial separation is zero. Applying a modified form of a contour integral method of Watson to an integral representation of the Green's function, a more general expansion of the Green's function is derived. Watson's original method had previously been applied to the case when the source and observation points were both on the axis of the cylinder. The expansion contains a free parameter which may be adjusted to give rapid convergence for any axial separation. It fails, however, when the source and observation points are both near the surface of the cylinder. For two special values of the parameter, the general expansion reduces to the eigenfunction expansion or to the integral representation. The derivation is somewhat obscure, but the resulting formula has a simple interpretation as the superposition of the potential of two related boundary value problems in finite-length cylinders. Some numerical results are given in the spatial region which previously could not be calculated, for a boundary condition approaching a homogeneous Neumann condition, and for a homogeneous Dirichlet condition.

Density quantization of metallic hydrogen

A cellular model of atomic hydrogen is hypothesized in which the Fermi pressure due to a single foreign electron invading the cell is examined in lieu of the entire degenerate gas. The Schrodinger equation is then solved in a self-consistent fashion with Laplace's equation for the electronic cloud. The Wronskian of the two possible wave functions does not vanish because the homogeneous Neumann condition is applied to the surface of the cell, while the Coulomb singularity compels a solution to be generated from the origin. Two quantum numbers result, one for the energy and one for the density, thus there is a lower limit to the density at zero temperature. Further corrections for electron waves by the Born Von-Karmen conditions broadens the radial quantum number into a continuum, but with an absolute lower limit to the density given by the ground state solutions to the Schrodinger equation.

Off-Diagonal Long-Range Order and Generalized Bose Condensation

It is shown for boson systems with periodic boundary conditions that the existence of generalized Bose condensation, of which the simple type typified by the ideal Bose gas is a special case, is equivalent to the existence of off-diagonal long-range order (ODLRO) in the single-particle density matrix ρ1 provided that the two noncomuting limits involved in the criterion for ODLRO are taken in the proper order: first size of system → ∞, then interparticle separation → ∞. It is shown by means of an example that certain assumptions concerning the behavior of the single-particle momentum distribution function involved in proving this equivalence are actually satisfied in thermal equilibrium for some dynamical models. The analysis is generalized to box-enclosure boundary conditions by an extension of an argument due to Schafroth, according to which box-enclosure conditions are equivalent to homogeneous Neumann conditions except for a thermodynamically negligible surface effect, provided that one second-quantizes with respect to Hartree-Fock orbitals rather than free-particle orbitals. A general criterion for generalized Bose condensation in terms of eigenvalues of ρ1 is proposed. On the basis of the behavior of soluble models it is conjectured that for a boson system in thermal equilibrium subject to arbitrary boundary conditions, the existence of such Bose condensation is equivalent to the existence of ODLRO in ρ1. The two-particle density matrix ρ2 is discussed briefly. By means of a simplified model it is shown that for a Bose system generalized condensation implies large eigenvalues of ρ2 and ODLRO of ρ2, just as for ρ1. It is pointed out that the question of the existence or nonexistence of generalized Bose condensation of fermion pairs ought to be investigated.