Optimal processes in the model of two-sector economy with an integral utility function

An infinite-horizon two-sector economy model with a Cobb–Douglas production function is studied for different depreciation rates, the utility function being an integral functional with discounting and a logarithmic integrand. The application of the Pontryagin maximum principle leads to a boundary value problem with special conditions at infinity. The presence of singular modes in the optimal solution complicates the search for a solution to the boundary value problem of the maximum principle. To construct the solution to the boundary value problem, the singular modes are written in an analytical form; in addition, a special version of the sweep algorithm in continuous form is proposed. The optimality of the extremal solution is proved.     

Existence of solutions for linear hyperbolic initial-boundary value problems in a rectangle

In a recent article, we achieved the well-posedness of linear hyperbolic initial and boundary value problems (IBVP) in a rectangle via semigroup method, and we found that there are only two elementary modes called hyperbolic and elliptic modes in the system. It seems that, there is only one set of boundary conditions for the hyperbolic mode, while there are infinitely many sets of boundary conditions for the elliptic mode, which can lead to well-posedness. In this article, we continue to consider linear hyperbolic IBVP in a rectangle in the constant coefficients case and we show that there are also infinitely many sets of boundary conditions for hyperbolic mode which will lead to the existence of a solution. We also have uniqueness in some special cases. The boundary conditions satisfy the reflection conditions introduced in Section 3, which turn out to be equivalent to the strictly dissipative conditions.     

The stability of boundary conditions for an angled-derivative difference scheme

Tidal forcing of the shallow water equations is typical of a class of problems where an approximate equilibrium solution is sought by long time integration of a differential equation system. A combination of the angled-derivative scheme with a staggered leap-frog scheme is sometimes used to discretise this problem. It is shown here why great care then needs to be taken with the boundary conditions to ensure that spurious solution modes do not lead to numerical instabilities. Various techniques are employed to analyse two simple model problems and display instabilities met in practical computations; these are then used to deduce a set of stable boundary conditions.Dedicated to Professor J. Crank on the occasion of his 80th birthday     

On barotropic trapped wave solutions with no-slip boundary conditions

Barotropic trapped wave solutions of a linearized system of the ocean dynamics equations are described for a semi-infinite, f-plane model basin of constant depth bordering a straight, vertical coast, for some “typical” values of the model parameters. No-slip boundary conditions are considered. When the wave length is shorter than the Rossby deformation radius, the main features of the wave solutions are as follows: the Kelvin wave exponential offshore decay scale essentially decreases as the wave length decreases, and an additional wave solution propagating in the opposite direction appears.     

Reaction-diffusion model for combustion with fuel consumption: II. Robin boundary conditions

A model, derived in a previous paper, for the reaction betweena gaseous oxidant and a solid porous fuel is analysed furtherfor general Robin boundary conditions. Numerical solutions areobtained and the effects of varying the dimensionless parameters,particularly the Frank-Kamenetskii parameter and the Lewisnumber L, are discussed in detail and compared with resultsobtained previously when Dirichlet boundary conditions are applied.Analytic solutions are obtained for the small-time developmentand for large values of . This latter solution shows the existenceof a propagating reaction-diffusion burning wave, and has featureswhich are qualitatively different to those derived earlier.     

Strong Interactions between Solitary Waves Belonging to Different Wave Modes

Strong interactions between weakly nonlinear long waves are studied. Strong interactions occur when the linear long wave phase speeds are nearly equal although the waves belong to different modes. Specifically we study this situation in the context of internal wave modes propagating in a density stratified fluid. The interaction is described by two coupled Korteweg-deVries equations, which possess both dispersive and nonlinear coupling terms. It is shown that the coupled equations possess an exact analytical solution involving the characteristic “sech²” profile of the Korteweg-deVries equation. It is also shown that when the coefficients satisfy some special conditions, the coupled equations possess an n-solition solution analogous to the Korteweg-deVries n-solition solution. In general though the coupled equations are found not to be amenable to solution by the inverse scattering transform technique, and thus a numerical method has been employed in order to find solutions. This method is described in detail in Appendix A. Several numerical solutions of the coupled equations are presented.     

A radiation condition arising from the limiting absorption principle for a closed full‐ or half‐waveguide problem

In this paper, we consider the propagation of waves in a closed full or half waveguide where the index of refraction is periodic along the axis of the waveguide. Motivated by the limiting absorption principle, proven in the Appendix by a functional analytic perturbation theorem, we formulate a radiation condition that assures uniqueness of a solution and allows the existence of propagating modes. Our approach is quite different to the known one as, eg, considered recently by Fliss and Joly and allows an extension to open wave guides. After application of the Floquet‐Bloch transform, we consider the Bloch variable α as a parameter in the resulting quasiperiodic boundary value problem and study the behaviour of the solution when α tends to an exceptional value by a singular perturbation result, which goes back to Colton and Kress.     

Evolution of nonparametric surfaces with speed depending on curvature II. The mean curvature case

We consider an evolution which starts as a flow of smooth surfaces in nonparametric form propagating in space with normal speed equal to the mean curvature of the current surface. The boundaries of the surfaces are assumed to remain fixed. G. Huisken has shown that if the boundary of the domain over which this flow is considered satisfies the “mean curvature” condition of H. Jenkins and J. Serrin (that is, the boundary of the domain is convex “in the mean”) then the corresponding initial boundary value problem with Dirichlet boundary data and smooth initial data admits a smooth solution for all time. In this paper we consider the case of arbitrary domains with smooth boundaries not necessarily satisfying the condition of Jenkins-Serrin. In this case, even if the flow starts with smooth initial data and homogeneous Dirichlet boundary data, singularities may develop in finite time at the boundary of the domain and the solution will not satisfy the boundary condition. We prove, however, existence of solutions that are smooth inside the domain for all time and become smooth up to the boundary after elapsing of a sufficiently long period of time. From that moment on such solutions assume the boundary values in the classical sense. We also give sufficient conditions that guarantee the existence of classical solutions for all time t ≧ 0. In addition, we establish estimates of the rate at which solutions tend to zero as t → ∞.     

Generating Boundary Conditions for the Calculation of Tsunami Propagation on Nested Grids

Some boundary conditions used to numerically simulate tsunami generation and propagation are studied. Special attention is given to generating boundary conditions thatmake it possible to simulate tsunami waves with desired characteristics (amplitude, time period and, in general, waveform). Since the water flow velocity in a propagating tsunami wave is uniquely defined by its height and ocean depth, one can simulate a wave propagating from the boundary into the simulation area. This can be done by specifying the wave height and water flow velocity on the boundary. This method is used to numerically simulate the propagation of a tsunami from the source to the coast on a sequence of refined grids. In this numerical experiment the wave parameters are transferred from the larger area to the subarea via boundary conditions. This method can also generate a wave that has certain characteristics on a specified line.     

On the necessity of the solvable conditions of the typical boundary value problems for quasilinear hyperbolic systems

Some solvable conditions have been derived to ensure the existence and the uniqueness of the C^∞solution for the typical boundary problem on a local angular region for quasilinear hyperbolic systems in two variables[1]. These solvables conditions mean that, under the formulation of the typical boundary problem, the all order derivatives of the solution can be determined uniquely at the vertex. The main purpose of this paper is to show that these solvable conditions are also necessary. In other words, if these solvable conditions fail to hold, then the boundary value problem will either have no solution or have infinite number of solutions.     

Pseudo‐reflection phenomena for singularities in thin elastic shells

We consider problems of statics of thin elastic shells with hyperbolic middle surface subjected to boundary conditions ensuring the geometric rigidity of the surface. The asymptotic behaviour of the solutions when the relative thickness tends to zero is then given by the membrane approximation. It is a hyperbolic problem propagating singularities along the characteristics. We address here the reflection phenomena when the propagated singularities arrive to a boundary. As the boundary conditions are not the classical ones for a hyperbolic system, there are various cases of reflection. Roughly speaking, singularities provoked elsewhere are not reflected at all at a free boundary, whereas at a fixed (or clamped) boundary the reflected singularity is less singular than the incident one. Reflection of singularities provoked along a non‐characteristic curve C are also considered. Copyright © 2003 John Wiley & Sons, Ltd.     

Transparent boundary conditions based on the pole condition for time‐dependent,two‐dimensional problems

The pole condition approach for deriving transparent boundary conditions is extended to the time‐dependent, two‐dimensional case. Nonphysical modes of the solution are identified by the position of poles of the solution's spatial Laplace transform in the complex plane. By requiring the Laplace transform to be analytic on some problem‐dependent complex half‐plane, these modes can be suppressed. The resulting algorithm computes a finite number of coefficients of a series expansion of the Laplace transform, thereby providing an approximation to the exact boundary condition. The resulting error decays super‐algebraically with the number of coefficients, so relatively few additional degrees of freedom are sufficient to reduce the error to the level of the discretization error in the interior of the computational domain. The approach shows good results for the Schrödinger and the drift‐diffusion equation but, in contrast to the one‐dimensional case, exhibits instabilities for the wave and Klein–Gordon equation. Numerical examples are shown that demonstrate the good performance in the former and the instabilities in the latter case. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

All-angle-negative-refraction and ultra-refraction for liquid surface waves in 2D phononic crystals

We analyse transport properties of linear liquid waves propagating within arrays of immersed rigid circular cylindrical obstacles fixed to a rough bottom. A comparison between Multipole and Finite Element methods is drawn in the case of Robin boundary conditions coupled with Floquet-Bloch boundary conditions. We find that the first band is concave yet nearly flat (associated waves of small negative group velocity) and it displays a cut-off (zero-frequency stop band associated with a singular perturbation). Thanks to this anomalous dispersion in such fluid filled structures, we achieve both ultra-refraction and negative refraction for waves propagating at their surface. Potential applications lie in a omnidirective ‘water antenna’ and a convergent flat ‘water lens’. The latter one is demonstrated experimentally.     

Dissipative Boundary Conditions for Finite-Difference Problems in Elasticity Theory

Dissipative differential-difference boundary conditions for elasticity equations are derived and investigated. These conditions completely eliminate the longitudinal and transverse waves reflected from the boundary in case of normal incidence and substantially reduce the reflected waves in a wide range of other incidence angles. The dissipative boundary conditions reduce computer resource requirements for simulation of seismic wave propagation in unbounded regions. Grid analogues of the dissipative boundary conditions are constructed for a two-layer finite-difference scheme. Comparative results are reported for the solution of two-dimensional seismic problems using these boundary conditions for various incidence angles. __________ Translated from Prikladnaya Matematika i Informatika, No. 18, pp. 83–92, 2004.     

Discretized Energy Minimization in a Wave Guuide with point Sources

An anti-noise problem on a finite time interval is solved by minimization of a quadratic functional on the Hilbert space of square integrable controls. To this end, the one-dimensional wave equation with point sources and pointwise reflecting boundary conditions is decomposed into a system for the two propagating components of waves. Wellposedness of this system is proved for a class of data that includes piecewise linear initial conditions and piecewise constant forcing functions. It is shown that for such data the optimal piecewise constant control is the solution of a sparse linear system. Methods for its computational treatment are presented as well as examples of their applicability. The convergence of discrete approximations to the general optimization problem is demonstrated by finite element methods.     

Numerical stabilization from the boundary for solutions of a model one-dimensional of a model one-Dimensional RBMK reactor

The problem of construction of first kind boundary conditions providing an asymptotic change of the trivial solution of a model one-dimensional RBMK reactor to the required stationary state is numerically studied according to specific features of this model. Results of calculations are presented for different admissible modes. The principal feasibility of efficient stabilization of the dynamics of occurring processes by boundary control of fast and slow neutrons is shown as well as its essential slow-down in the control of only fast neutrons.     

An approximate inertial manifolds approach to postprocessing the Galerkin method for the Navier-Stokes equations

In a recent paper we have introduced a postprocessing procedure for the Galerkin method for dissipative evolution partial differential equations with periodic boundary conditions. The postprocessing technique uses approximate inertial manifolds to approximate the high modes (the small scale components) in the exact solutions in terms of the Galerkin approximations, which in this case play the role of the lower modes (large scale components). This procedure can be seen as a defect-correction technique. But contrary to standard procedures, the correction is computed only when the time evolution is completed. Here we extend these results to more realistic boundary conditions. Specifically, we study in detail the two-dimensional Navier-Stokes equations subject to homogeneous (nonslip) Dirichlet boundary conditions. We also discuss other equations, such as reaction-diffusion systems and the Cahn-Hilliard equations.

     

Nonlinear Spatial Evolution of Small Disturbances from Boundary Conditions in Flat Inclined-Channel Flow

The asymptotic behavior of small disturbances as they evolve spatially from boundary conditions in a flat inclined channel is determined. These small disturbances develop into traveling waves called roll waves, first discussed by Dressler in 1949. Roll waves exist if the Froude number F exceeds 2, which consist of a periodic pattern of bores, or discontinuities. After confirming the instability condition for   F > 2  for the linearized equations in the boundary value case, the nonlinear boundary value problem for the weakly unstable region of F slightly larger than 2 is studied. Multiple scales and the Fredholm alternative theorem are applied to determine the evolution of the solution in space. It is found that the solution is dominated by the evolution of the disturbance along one characteristic. The shock conditions governing the asymptotic solution are determined and these conditions are used to determine the approximate shape of the resulting traveling wave from the solution. Both asymptotic and numerical results for periodic disturbances are presented.     

Exact modes for post-buckling characteristics of nonlocal nanobeams in a longitudinal magnetic field

An exact mode solution that investigates the prebuckling and postbuckling characteristics of nonlocal nanobeams with fixed–fixed, hinged–hinged, and fixed–hinged boundary conditions in a longitudinal magnetic field is determined. The geometric nonlinearity arising from mid-plane stretching is considered to obtain the nonlinear governing equation of motion by virtue of Hamilton's principle. The influences of the nonlocal and magnetic parameters on the prebuckling and postbuckling dynamics of nanobeams with various boundary conditions are evaluated, indicating that the critical buckling force can be decreased with the increase of the nonlocal parameter while can be increased with increasing the magnetic parameter. It is demonstrated that the first natural frequency of the nanobeam with fixed–fixed and fixed–hinged conditions in postbuckling configuration is increased from zero to a constant value for higher values of the nonlocal parameter with increasing the axial force. The second natural frequency of the buckled nanobeam is always decreased with an increase of the nonlocal parameter. The results show that the internal resonance between the first and second modes of the postbuckling nanobeams can be quickly and easily activated by increasing the nonlocal parameters, especially for fixed–fixed and hinged–hinged boundary conditions. In addition, the results obtained by exact mode solution are compared those obtained by classical mode solution. It is found that the classical mode is valid only for nonlocal nanobeams with the hinged–hinged boundary conditions.     

Symmetric and anti-symmetric vibrations in micropolar thermoelastic materials plate with voids

The problem of symmetric and anti-symmetric vibrations in micropolar thermoelastic plate with voids has been investigated. The dispersive frequency equations are obtained for different surface waves propagating in the plate. The velocity curves are depicted for the symmetric and anti-symmetric vibrations, plate, Rayleigh and flexural waves. It is found that there exist two modes in the solution of frequency equation for the surface waves in micropolar thermoelastic plate with voids. We have observed that the first modes of velocity ratios of corresponding surface waves are lesser than those of second mode of vibration.