Diffraction of light revisited

The diffraction of monochromatic light is considered for a plane screen with an open infinite slit by solving the vectorial Maxwell–Helmholtz system in the upper half‐space with the Fourier method. With this approach we can represent each solution satisfying an appropriate energy condition by its boundary fields in the Sobolev spaces H^±1/2. We show that Sommerfeld's theory using a boundary integral equation with Hankel kernels for the so‐called B‐polarization is covered by our approach, but in general it violates a necessary energy condition. Our representation includes also solutions which are not covered by Sommerfeld's theory. Copyright © 2007 John Wiley & Sons, Ltd.     

Numerical Simulations of Pile Integrity Tests Using a Coupled FEM SBFEM Approach

Piles are widely used to build a proper foundation for various buildings. The pile's quality in situ can be tested by a so called pile integrity test. In order to apply this test, an acceleration sensor is attached to the pile's head which than receives an impulse. Due to this impulse a p-wave runs through the pile. The major part of this wave is reflected from the pile's toe and is measured by the attached acceleration sensor on top of the pile. This yields an acceleration-time plot which has to be analysed to determine the pile's condition. Sometimes the interpretation of these plots is difficult, specially when the cross-section of the pile is changing or is influenced by the surrounding soil. For a better understanding of this kind of measurements, numerical simulations can be performed. For these simulations a coupled finite element method (FEM) and scaled boundary finite element method (SBFEM) approach is used. This approach satisfies Sommerfeld's radiation condition and allows simulating an infinite half-space. This ensures that the applied impulse will not to be reflected at the artificial boundary which is introduced by the boundary of the numerical discretisation. The coupled approach proposed here requires discretisation of a small domain only in contrast to a purely FEM-based approach. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stabilization of the trial method for the Bernoulli problem in case of prescribed Dirichlet data

We apply the trial method for the solution of Bernoulli's free boundary problem when the Dirichlet boundary condition is imposed for the solution of the underlying Laplace equation, and the free boundary is updated according to the Neumann boundary condition. The Dirichlet boundary value problem for the Laplacian is solved by an exponentially convergent boundary element method. The update rule for the free boundary is derived from the linearization of the Neumann data around the actual free boundary. With the help of shape sensitivity analysis and Banach's fixed‐point theorem, we shed light on the convergence of the respective trial method. Especially, we derive a stabilized version of this trial method. Numerical examples validate the theoretical findings.Copyright © 2014 John Wiley & Sons, Ltd.     

Optimal dynamic law enforcement

In this paper we present an intertemporal extension of Becker's [Journal of Political Economy 76 (1968) 169] static economic approach to crime and punishment. For a dynamic supply of offenders we determine the optimal dynamic trade-off between damages caused by offenders, law enforcement expenditures and cost of imprisonment. By using Pontryagin's maximum principle we obtain interesting insight into the dynamical structure of optimal law enforcement policies. It is found that inherently multiple steady states are generated which can be saddle points, unstable points and boundary solutions. As in other non-linear control models there exists a threshold (a so-called Skiba point) which makes the optimal enforcement policy dependent on the initial conditions. It turns out that above the Skiba point the optimal trade-off between social costs implies a steady state with a high level of offences, while below the threshold the optimal law enforcement should eradicate crime.     

Zaremba's problem for elliptic equations in the neighborhood of a singular point or in the infinity

In the present paper the behavior of solutions of the mixed Zaremba's problem in the neighborhood of a boundary point and at infinity is studied. In part I of this paper[4] the concept of Wiener's generalized solution of Zaremba's problem was introduced and the so called Growth Lemma for the class of domains, satisfying isoperimetric condition, was proven. In part II regularity criterion for joining points of Neumann's and Dirichlet's boundary conditions is formulated. Generalized solution in unlimited domains as a limit of Zaremba's problem's solutions in a sequence of limited domains is introduced and a regularity condition allowed to obtain an analogue of Phragmen-Lindeloeff theorem for the solutions of Zaremba's problem. Main results of the present paper are formulated in terms of divergence of Wiener's type series.     

Green's functions for elliptic and parabolic systems with Robin-type boundary conditions

The aim of this paper is to investigate Green's function for parabolic and elliptic systems satisfying a possibly nonlocal Robin-type boundary condition. We construct Green's function for parabolic systems with time-dependent coefficients satisfying a possibly nonlocal Robin-type boundary condition assuming that weak solutions of the system are locally Hölder continuous in the interior of the domain, and as a corollary we construct Green's function for elliptic system with a Robin-type condition. Also, we obtain Gaussian bound for Robin Green's function under an additional assumption that weak solutions of Robin problem are locally bounded up to the boundary. We provide some examples satisfying such a local boundedness property, and thus have Gaussian bounds for their Green's functions.     

On the diffraction of Poincaré waves

The diffraction of tidal waves (Poincaré waves) by islands and barriers on water of constant finite depth is governed by the two-dimensional Helmholtz equation. One effect of the Earth's rotation is to complicate the boundary condition on rigid boundaries: a linear combination of the normal and tangential derivatives is prescribed. (This would be an oblique derivative if the coefficients were real.) Corresponding boundary-value problems are treated here using layer potentials, generalizing the usual approach for the standard exterior boundary-value problems of acoustics. Singular integral equations are obtained for islands (scatterers with non-empty interiors) whereas hypersingular integral equations are obtained for thin barriers. Copyright © 2001 John Wiley & Sons, Ltd.     

Existence and asymptotic behavior of global smooth solution for p‐system with nonlinear damping and fixed boundary effect

This paper is concerned with the initial boundary value problem for the p‐system with nonlinear damping and fixed boundary condition. We show that the corresponding problem admits a unique global solution, and such a solution tends time asymptotically to the corresponding nonlinear diffusion wave governed by the classical Darcy's law provided that the corresponding prescribed initial error function is sufficiently small. Copyright © 2013 John Wiley & Sons, Ltd.     

Numerical study of the flow in a three-dimensional thermally driven cavity

Solutions for the fully compressible Navier–Stokes equations are presented for the flow and temperature fields in a cubic cavity with large horizontal temperature differences. The ideal-gas approximation for air is assumed and viscosity is computed using Sutherland's law. The three-dimensional case forms an extension of previous studies performed on a two-dimensional square cavity. The influence of imposed boundary conditions in the third dimension is investigated as a numerical experiment. Comparison is made between convergence rates in case of periodic and free-slip boundary conditions. Results with no-slip boundary conditions are presented as well. The effect of the Rayleigh number is studied.     

A numerical PDE approach for pricing callable bonds

Many debt issues contain an embedded call option that allows the issuer to redeem the bond at specified dates for a specified price. The issuer is typically required to provide advance notice of a decision to exercise this call option. The valuation of these contracts is an interesting numerical exercise because discontinuities may arise in the bond value or its derivative at call and/or notice dates. Recently, it has been suggested that finite difference methods cannot be used to price callable bonds requiring notice. Poor accuracy was attributed to discontinuities and difficulties in handling boundary conditions. As an alternative, a semi-analytical method using Green's functions for valuing callable bonds with notice was proposed. Unfortunately, the Green's function method is limited to special cases. Consequently, it is desirable to develop a more general approach. This is provided by using more advanced techniques such as flux limiters to obtain an accurate numerical partial differential equation method. Finally, in a typical pricing model an inappropriate financial condition is required in order to properly specify boundary conditions for the associated PDE. It is shown that a small perturbation of such a model is free from such artificial conditions.     

Warped MPDAE Models including Minimisation Criteria for the Simulation of RF Signals

In radio frequency (RF) applications, electric circuits produce signals including widely separated time scales. A multidimensional representation yields an efficient model by decoupling the time scales. Consequently, a warped multirate partial differential algebraic equation (MPDAE) describes the circuit's behaviour. The appropriate determination of an arising local frequency function is crucial for the efficiency of this approach. Variational calculus implies a necessary condition to a specific solution, which exhibits a minimal amount of oscillations in the whole domain of dependence. We apply a similar strategy to minimise oscillatory performance in some boundary values only. Now variational calculus yields a boundary condition, which can easily be used in numerical methods. We compare the results of both minimisation criteria in a simulation of a warped MPDAE model. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

ℋ︁‐matrices for the convection‐diffusion equation

Hierarchical matrices (ℋ︁‐matrices) provide a technique for the sparse approximation of large, fully populated matrices. This technique has been shown to be applicable to stiffness matrices arising in boundary element method applications where the kernel function displays certain smoothness properties. The error estimates for an approximation of the kernel function by a separable function can be carried over directly to error estimates for an approximation of the stiffness matrix by an ℋ︁‐matrix, using a certain standard partitioning and admissibility condition for matrix blocks. Similarly, ℋ︁‐matrix techniques can be applied in the finite element context where it is the inverse of the stiffness matrix that is fully populated. Here one needs a separable approximation of Green's function of the underlying boundary value problem in order to prove approximability by matrix blocks of low rank. Unfortunately, Green's function for the convection‐diffusion equation does not satisfy the required smoothness properties, hence prohibiting a straightforward generalization of the separable approximation through Taylor polynomials. We will use Green's function to motivate a modification in the (hierarchical) partitioning of the index set and as a consequence the resulting hierarchy of block partitionings as well as the admissibility condition. We will illustrate the effect of the proposed modifications by numerical results.     

Stochastic immersed boundary method incorporating thermal fluctuations

We give a brief introduction to the stochastic immersed boundary method which allows for simulation of small length-scale physical systems in which elastic structures interact with a fluid flow in the presence of thermal fluctuations. The conventional immersed boundary method is extended to account for thermal fluctuations by introducing stochastic forcing terms in the fluid equations. This gives a system of stiff SPDE's for which standard numerical approaches perform poorly. We discuss a numerical method derived using stochastic calculus to overcome the stiff features of the equations. We then discuss results which indicate that the method captures physical features predicted by statistical mechanics for small length-scale systems. The stochastic immersed boundary method holds promise as a numerical approach in simulating microscopic mechanical systems in which thermal fluctuations play a fundamental role. A more detailed discussion of this work is given in [1, 2, 3]. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Initial Boundary Value Problem of an Equation from Mathematical Finance

Consider the initial boundary value problem of the strong degenerate parabolic equation ?_(xx)u + u?_yu-?_tu = f(x, y, t, u),(x, y, t) ∈ Q_T = Ω×(0, T)with a homogeneous boundary condition. By introducing a new kind of entropy solution, according to Oleinik rules, the partial boundary condition is given to assure the well-posedness of the problem. By the parabolic regularization method, the uniform estimate of the gradient is obtained, and by using Kolmogoroff 's theorem, the solvability of the equation is obtained in BV(Q_T) sense. The stability of the solutions is obtained by Kruzkov's double variables method.     

A mixed problem of linear elastodynamics

This paper is devoted to a semigroup approach to an initial-boundary value problem of linear elastodynamics in the case where the boundary condition is a regularization of the genuine mixed displacement-traction boundary condition. More precisely, it is a smooth linear combination of displacement and traction boundary conditions, but is not equal to the pure traction boundary condition. Some previous results with mixed displacement-traction boundary condition are due to Inoue and Ito. The crucial point in our semigroup approach is to generalize the classical variational approach to the degenerate case, by using the theory of fractional powers of analytic semigroups.     

Stability results of a distributed problem involving Bresse system with history and/or Cattaneo law under fully Dirichlet or mixed boundary conditions

In this paper, we study the stability of a 1‐dimensional Bresse system with infinite memory‐type control and/or with heat conduction given by Cattaneo's law acting in the shear angle displacement. When the thermal effect vanishes, the system becomes elastic with memory term acting on one equation. We consider the interesting case of fully Dirichlet boundary conditions. Indeed, under equal speed of propagation condition, we establish the exponential stability of the system. However, in the natural physical case when the speeds of propagation are different, using a spectrum method, we show that the Bresse system is not uniformly exponentially stable. In this case, we establish a polynomial energy decay rate. Our study is valid for all other mixed boundary conditions.     

The challenges of statistical patterns of language: The case of Menzerath's law in genomes

The importance of statistical patterns of language has been debated over decades. Although Zipf's law is perhaps the most popular case, recently, Menzerath's law has begun to be involved. Menzerath's law manifests in language, music and genomes as a tendency of the mean size of the parts to decrease as the number of parts increases in many situations. This statistical regularity emerges also in the context of genomes, for instance, as a tendency of species with more chromosomes to have a smaller mean chromosome size. It has been argued that the instantiation of this law in genomes is not indicative of any parallel between language and genomes because (a) the law is inevitable and (b) noncoding DNA dominates genomes. Here mathematical, statistical, and conceptual challenges of these criticisms are discussed. Two major conclusions are drawn: the law is not inevitable and languages also have a correlate of noncoding DNA. However, the wide range of manifestations of the law in and outside genomes suggests that the striking similarities between noncoding DNA and certain linguistics units could be anecdotal for understanding the recurrence of that statistical law. © 2012 Wiley Periodicals, Inc. Complexity, 2012     

Boundary value problems in the theory of thermoporoelasticity for materials with double porosity

In this paper the linear quasi-static theory of thermoelasticity for solids with double porosity is considered. The system of equations of this theory is based on the equilibrium equations for solids with double porosity, conservation of fluid mass, constitutive equations, Darcy's law for materials with double porosity and Fourier's law for heat conduction. The basic internal and external boundary value problems (BVPs) of steady vibrations are formulated. The uniqueness and existence theorems for classical solutions of the above mentioned BVPs are proved by means of the potential method (boundary integral equation method) and the theory of singular integral equations. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Equilibrium problem for thermoelectroconductive body with the Signorini condition on the boundary

We investigate a boundary value problem for a thermoelectroconductive body with the Signorini condition on the boundary, related to resistance welding. The mathematical model consists of an energy‐balance equation coupled with an elliptic equation for the electric potential and a quasistatic momentum balance with a viscoelastic material law. We prove the existence of a weak solution to the model by using the Schauder fixed point theorem and classical results on pseudomonotone operators. Copyright © 2001 John Wiley & Sons, Ltd.     

On the application of the nonstationary form of the Tappert equation as an artificial boundary condition

In this paper, a nonstationary analog of the range refraction parabolic equation is derived. A new approach to the derivation of Tappert’s operator asymptotic formula with the use of noncommutative analysis is presented. The obtained nonstationary equation is proposed as an artificial boundary condition for the wave equation in underwater acoustics. This form of artificial boundary condition has low computational cost and systematically takes into account variations of sound speed. This is confirmed by various numerical experiments, including propagation of normal modes and wave fields produced by point source.