Quantum motion over a finite one-dimensional domain: With and without dissipation

Quantum motion of a single particle over a finite one-dimensional spatial domain is considered for the generalized four parameter infinity of boundary conditions (GBC) of Carreauet al [1]. The boundary conditions permit complex eigenfunctions with nonzero current for discrete states. Explicit expressions are obtained for the eigenvalues and eigenfunctions. It is shown that these states go over to plane waves in the limit of the spatial domain becoming very large. Dissipation is introduced through Schrödinger-Langevin (SL) equation. The space and time parts of the SL equation are separated and the time part is solved exactly. The space part is converted to nonlinear ordinary differential equation. This is solved perturbatively consistent with the GBC. Various special cases are considered for illustrative purposes.     

Exact boundary conditions for the initial value problem of convex conservation laws

The initial value problem of convex conservation laws, which includes the famous Burgers’ (inviscid) equation, plays an important rule not only in theoretical analysis for conservation laws, but also in numerical computations for various numerical methods. For example, the initial value problem of the Burgers’ equation is one of the most popular benchmarks in testing various numerical methods. But in all the numerical tests the initial data have to be assumed that they are either periodic or having a compact support, so that periodic boundary conditions at the periodic boundaries or two constant boundary conditions at two far apart spatial artificial boundaries can be used in practical computations. In this paper for the initial value problem with any initial data we propose exact boundary conditions at two spatial artificial boundaries, which contain a finite computational domain, by using the Lax’s exact formulas for the convex conservation laws. The well-posedness of the initial-boundary problem is discussed and the finite difference schemes applied to the artificial boundary problems are described. Numerical tests with the proposed artificial boundary conditions are carried out by using the Lax–Friedrichs monotone difference schemes.     

On soliton structure of isospectral Vakhnenko equation: WKI-eigenvalue problem

In this Letter, an inverse scattering method is developed for the isospectral Vakhnenko equation, and the general N-solution is presented. Using this technique, a typical self-confined solitary wave hereafter named soliton, satisfying some vanishing boundary conditions is elicited. The detail on the scattering behavior of such structures including their phase shifts is outlined. This method is presented to be arguably more simple, tractable and straightforward than that recently investigated by Vakhnenko and Parkes [V.O. Vakhnenko, E.J. Parkes, Chaos Solitons Fractals 13 (2002) 1819] while solving the same equation. As a result, it is shown that when two single soliton solutions with ‘similar’ or ‘dissimilar’ amplitudes collide, there may be two types of features depending on the ratio of the two eigenvalues involved. It is then suggested an existence of some critical value for the ratio of the two eigenvalues at which the collision process changes its characteristic features.     

Boundary symmetries in linear differential and integral equation problems applied to the self-consistent Green’s function formalism of acoustic and electromagnetic scattering

Explicit symmetry relations for the Green’s function subject to homogeneous boundary conditions are derived for arbitrary linear differential or integral equation problems in which the boundary surface has a set of symmetry elements. For corresponding homogeneous problems subject to inhomogeneous boundary conditions implicit symmetry relations involving the Green’s function are obtained. The usefulness of these symmetry relations is illustrated by means of a recently developed self-consistent Green’s function formalism of electromagnetic and acoustic scattering problems applied to the exterior scattering problem. One obtains explicit symmetry relations for the volume Green’s function, the surface Green’s function, and the interaction operator, and the respective symmetry relations are shown to be equivalent. This allows us to treat boundary symmetries of volume-integral equation methods, boundary-integral equation methods, and the T matrix formulation of acoustic and electromagnetic scattering under a common theoretical framework. By specifying a specific expansion basis the coordinate-free symmetry relations of, e.g., the surface Green’s function can be brought into the form of explicit symmetry relations of its expansion coefficient matrix. For the specific choice of radiating spherical wave functions the approach is illustrated by deriving unitary reducible representations of non-cubic finite point groups in this basis, and by deriving the corresponding explicit symmetry relations of the coefficient matrix. The reducible representations can be reduced by group-theoretical techniques, thus bringing the coefficient matrix into block-diagonal form, which can greatly reduce ill-conditioning problems in numerical applications.     

Considerations to Rayleigh’s hypothesis

In 1907 Lord Rayleigh published a paper on the dynamic theory of gratings. In this paper he presented a rigorous approach for solving plane wave scattering on periodic surfaces. Moreover he derived explicit expressions for a perfectly conducting sinusoidal surface, and for perpendicular incidence of the electromagnetic plane wave. This paper was criticized by Lippmann in 1953 for he assumed Rayleigh’s approach to be incomplete. Since this time there have been published several arguments, proofs, and discussions concerning the correctness and the range of validity of Rayleigh’s approach not only for plane wave scattering on gratings but also for light scattering on nonspherical structures, in general. In the paper at hand we will discuss the different point of views on what is called “Rayleigh’s hypothesis” as well as the relevance of a found theoretical limit for its validity. Furthermore we present a numerical treatment of the original scattering problem of a p-polarized plane wave perpendicularly incident on a perfectly conducting sinusoidal surface (i.e., the scalar Dirichlet problem). In doing so we emphasizes the near-field solution especially within the grooves of the grating up to points on the surface, and below the surface. Two different Green’s function formulations of Huygens’ principle are used as starting points. One of this formulation results in the general T-matrix approach which is considered to be affected by Rayleigh’s hypothesis especially for near-field calculations. The other formulation provides a conventional boundary integral equation which is in accordance with Lippmann’s point of view and free of problems with Rayleigh’s hypothesis. But the obtained results show that Lippmann’s argumentation do not withstand a critical numerical analysis, and that the independence of least-squares approaches from Rayleigh’s hypothesis, as understood and proven by Millar, seems to hold also for certain methods which does not fit into such an approach.     

The zone boundary mode in periodic nonlinear electrical lattices

We study the existence and linear stability of the zone boundary mode in a nonlinear electrical lattice consisting of N inductors and N voltage-dependent capacitors with periodic boundary conditions. The inductances are allowed to alternate, while the capacitors are identical and each have a quadratic dependence on voltage. By block-diagonalizing a 2N×2N Floquet problem, we reduce the question of the stability of the mode to a single Hill’s equation that is analyzed using methods of perturbation theory and averaging. We show that periodicity of the lattice inductances degrades stability, and also show that the instability threshold is proportional to N⁻². Numerical computations validate the perturbative results.     

On determining the temperature field of axisymmetrical bodies with variable physical properties and mixed boundary conditions in a sliding coordinate system

The heat-conduction equation is solved with a view of determining the temperature field of axisymmetrical solids in the case of variable thermophysical coefficients in a sliding coordinate system with mixed boundary conditions.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 73–80, July, 1976.     

Mixed Convection in Viscoelastic Boundary Layer Flow and Heat Transfer over a Stretching Sheet

An investigation is carried out on mixed convection boundary layer flow of an incompressible and electrically conducting viscoelastic fluid over a linearly stretching surface in which the heat transfer includes the effects of viscous dissipation, elastic deformation, thermal radiation, and non-uniform heat source/sink for two general types of non-isothermal boundary conditions. The governing partial differential equations for the fluid flow and temperature are reduced to a nonlinear system of ordinary differential equations which are solved analytically using the homotopy analysis method (HAM). Graphical and numerical demonstrations of the convergence of the HAM solutions are provided, and the effects of various parameters on the skin friction coefficient and wall heat transfer are tabulated. In addition, it is demonstrated that previously reported solutions of the thermal energy equation given in [1] do not converge at the boundary, and therefore, the boundary derivatives reported are not correct.     

Size effects on Debye temperature, Einstein temperature, and volume thermal expansion coefficient of nanocrystals

Based on a size-dependent root of mean square amplitude (rms) model, the size-dependent Debye temperatures of nanocrystals are modeled without any adjustable parameter by considering both Lindemann’s criterion and Mott’s equation. In terms of this model, the Debye temperatures depend on both size and interface conditions, which lead to related applications on size effects of the Einstein temperature and the volume thermal expansion coefficient. It is found that the model’s predictions are in good agreement with available experimental and computer simulation results.     

A Poisson–Boltzmann solver on irregular domains with Neumann or Robin boundary conditions on non-graded adaptive grid

We introduce a second-order solver for the Poisson–Boltzmann equation in arbitrary geometry in two and three spatial dimensions. The method differs from existing methods solving the Poisson–Boltzmann equation in the two following ways: first, non-graded Quadtree (in two spatial dimensions) and Octree (in three spatial dimensions) grid structures are used; Second, Neumann or Robin boundary conditions are enforced at the irregular domain’s boundary. The irregular domain is described implicitly and the grid needs not to conform to the domain’s boundary, which makes grid generation straightforward and robust. The linear system is symmetric, positive definite in the case where the grid is uniform, nonsymmetric otherwise. In this case, the resulting matrix is an M-matrix, thus the linear system is invertible. Convergence examples are given in both two and three spatial dimensions and demonstrate that the solution is second-order accurate and that Quadtree/Octree grid structures save a significant amount of computational power at no sacrifice in accuracy.     

Sufficient uniqueness conditions for the solution of the time harmonic Maxwell’s equations associated with surface impedance boundary conditions

We consider the time-harmonic Maxwell’s equations for the scattering or radiating problem from a 3-D object that is either a metallic surface coated with material layers (MCS) or a dichroic structure (DS) made up of multiple frequency selective surfaces (FSS) embedded in materials. Low or high order impedance boundary conditions (IBC) are employed to reduce the numerical complexity of the solution of this problem via an integral equation or a finite element formulation. An IBC links the tangential components of the electric field to those of the magnetic field on the outer surface of the MCS, or on the FSSs, and avoids the solution of Maxwell’s equations inside the inhomogeneous domain for a MCS or, for a DS, the meshing of the numerous unit cells in a FSS. Sufficient uniqueness conditions (SUC) are established for the solutions of Maxwell’s equations associated with these IBCs, the performances of which, when constrained by the corresponding SUCs, are numerically evaluated for an infinite or finite planar structure.     

Analysis of car-following model considering driver’s physical delay in sensing headway

An extended car-following model is proposed by taking into account the delay of the driver’s response in sensing headway. The stability condition of this model is obtained by using the linear stability theory. The results show that the stability region decreases when the driver’s physical delay in sensing headway increases. The KdV equation and mKdV equation near the neutral stability line and the critical point are respectively derived by applying the reductive perturbation method. The traffic jams could be thus described by soliton solution and kink-antikink soliton solution for the KdV equation and mKdV equation respectively. The numerical results in the form of the space-time evolution of headway show that the stabilization effect is weakened when the driver’s physical delay increases. It confirms the fact that the delay of driver’s response in sensing headway plays an important role in jamming transition, and the numerical results are in good agreement with the theoretical analysis.     

Shear Bloch waves and coupled phonon–polariton in periodic piezoelectric waveguides

Coupled electro-elastic SH waves propagating in a periodic piezoelectric finite-width waveguide are considered in the framework of the full system of Maxwell’s electrodynamic equations. We investigate Bloch–Floquet waves under homogeneous or alternating boundary conditions for the elastic and electromagnetic fields along the guide walls. Zero frequency stop bands, trapped modes as well as some anomalous features due to piezoelectricity are identified. For mixed boundary conditions, by modulating the ratio of the length of the unit cell to the width of the waveguide, the minimum widths of the stop bands can be moved to the middle of the Brillouin zone. The dispersion equation has been investigated also for phonon–polariton band gaps. It is shown that for waveguides at acoustic frequencies, acousto-optic coupling gives rise to polariton behavior at wavelengths much larger than the length of the unit cell but at optical frequencies polariton resonance occurs at wavelengths comparable with the period of the waveguide.     

Scattering matrix for Feshbach-Villars equation for spin 0 and 1/2: Woods-Saxon potential

The Feshbach-Villars equation for spin 0 and 1/2 in the presence of Woods-Saxon potential is solved using an unified approach. The good boundary conditions for jumping potential are found and Klein tunneling and Klein paradox are discussed. The scattering matrix is constructed and the phase shifts, the transmission and reflection coefficients are deduced.     

Statistical characteristics of photon distributions in a semi-infinite turbid medium: Analytical expressions and their application to optical tomography

The paper focuses on the determination of statistical characteristics of photon distributions in a semi-infinite turbid medium, specifically the photon average trajectory and the root-mean-square deviation of photons from the average trajectory, with an approach based on the diffusion approximation to the radiative transfer equation. We show that the Dirichlet and Robin boundary conditions used for this purpose give close results. We derive exact analytical expressions for the case of the Dirichlet boundary condition. To demonstrate the practical value of our results we consider approximate solution of the inverse problem of time-domain diffuse optical tomography with the flat layer transmission geometry. The problem is solved with the method of photon average trajectories which are constructed with analytical expressions derived for a semi-infinite medium.     

A new integral representation for quasi-periodic fields and its application to two-dimensional band structure calculations

In this paper, we consider band structure calculations governed by the Helmholtz or Maxwell equations in piecewise homogeneous periodic materials. Methods based on boundary integral equations are natural in this context, since they discretize the interface alone and can achieve high order accuracy in complicated geometries. In order to handle the quasi-periodic conditions which are imposed on the unit cell, the free-space Green’s function is typically replaced by its quasi-periodic cousin. Unfortunately, the quasi-periodic Green’s function diverges for families of parameter values that correspond to resonances of the empty unit cell. Here, we bypass this problem by means of a new integral representation that relies on the free-space Green’s function alone, adding auxiliary layer potentials on the boundary of the unit cell itself. An important aspect of our method is that by carefully including a few neighboring images, the densities may be kept smooth and convergence rapid. This framework results in an integral equation of the second kind, avoids spurious resonances, and achieves spectral accuracy. Because of our image structure, inclusions which intersect the unit cell walls may be handled easily and automatically. Our approach is compatible with fast-multipole acceleration, generalizes easily to three dimensions, and avoids the complication of divergent lattice sums.     

Group Analysis of Nonlinear Heat-Conduction Problem for a Semi-Infinite Body

Abstract

The transformation group theoretic approach is applied to present an analysis of the nonlinear unsteady heat conduction problem in a semi–infinite body. The application of one–parameter group reduces the number of independent variables by one, and consequently the governing partial differential equation with the boundary and initial conditions to an ordinary differential equation with the appropriate corresponding boundary conditions. The ordinary differential equation is solved analytically for some special forms of the thermal parameters. The general analysis developed in this study corresponds to thermal parameters that has different forms with coordinates and time.     

Iterative Method for Solving a Problem with Mixed Boundary Conditions for Biharmonic Equation

The solution of boundary value problems (BVP) for fourth order differential equations by their reduction to BVP for second order equations, with the aim to use the available efficient algorithms for the latter ones, attracts attention from many researchers. In this paper, using the technique developed by the authors in recent works we construct iterative method for a problem with complicated mixed boundary conditions for biharmonic equation which is originated from nanofluidic physics. The convergence rate of the method is proved and some numerical experiments are performed for testing its dependence on a parameter appearing in boundary conditions and on the position of the point where a transmission of boundary conditions occurs.     

The Rayleigh-Plesset equation in terms of volume with explicit shear losses

The most common nonlinear equation of motion for the damped pulsation of a spherical gas bubble in an infinite body of liquid is the Rayleigh-Plesset equation, expressed in terms of the dependency of the bubble radius on the conditions pertaining in the gas and liquid (the so-called ‘radius frame’). However over the past few decades several important analyses have been based on a heuristically derived small-amplitude expansion of the Rayleigh-Plesset equation which considers the bubble volume, instead of the radius, as the parameter of interest, and for which the dissipation term is not derived from first principles. So common is the use of this equation in some fields that the inherent differences between it and the ‘radius frame’ Rayleigh-Plesset equation are not emphasised, and it is important in comparing the results of the two equations to understand that they differ both in terms of damping, and in the extent to which they neglect higher order terms. This paper highlights these differences. Furthermore, it derives a ‘volume frame’ version of the Rayleigh-Plesset equation which contains exactly the same basic physics for dissipation, and retains terms to the same high order, as does the ‘radius frame’ Rayleigh-Plesset equation. Use of this equation will allow like-with-like comparisons between predictions in the two frames.