On the absence of logarithmic singularities in the solutions of Lamé-type equations

The object of this research is linear differential equations of the second order with regular singularities. We extend the concept of a regular singularity to linear partial differential equations. The general solution of a linear differential equation with a regular singularity is a linear combination of two linearly independent solutions, one of which in the general case contains a logarithmic singularity. The well-known Lamé equation, where the Weierstrass elliptic function is one of the coefficients, has only meromorphic solutions. We consider such linear differential equations of the second order with regular singularities, for which as a coefficient instead of the Weierstrass elliptic function we use functions that are the solutions to the first Painlevé or Korteweg – de Vries equations. These equations will be called Lamé-type equations. The question arises under what conditions the general solution of Lamé-type equations contains no logarithms. For this purpose, in the present paper, the solutions of Lamé-type equations are investigated and the conditions are found that make it possible to judge the presence or absence of logarithmic singularities in the solutions of the equations under study. An example of an equation with an irregular singularity having a solution with an logarithmic singularity is given, since the equation, defining it, has a multiple root.

MATHEMATICAL MODEL OF WAVE DIFFRACTION BY THE SYSTEM OF STRIPES WITH DIFFERENT VALUES OF SURFACE IMPEDANCE

A mathematical model of diffraction of E-polarized and H-polarized waves on a finite system of not perfectly conducting tapes is obtained. The value of the surface impedance on the two sides of the stripes is different. The initial boundary value problem for the Helmholtz equation with boundary conditions of the third kind was reduced to a system of boundary integral equations. This system of boundary integral equations consists of singular integral equations of the first kind and integral equations of the second kind with a logarithmic singularity. The method of parametric representation of integral operator was used to perform transformations. The values of the physical characteristics of the process are expressed through the solutions of the obtained systems of integral equations. Numerical solution of these equations is performed using a computational scheme based on the discrete singularities method.

Effects of Filling Material Roughness on Scattering from a Rectangular Groove

This study investigated the effects of filling material roughness on the H-polarized scattering signatures of a two-dimensional (2D) rectangular groove embedded on an infinite ground plane. Under the assumption of a weakly rough surface on the groove, simplifying suppositions can be used to estimate Green’s functions inside and outside the groove. Enforcing continuity in the tangential magnetic field on the rough surface enables the construction and resolution of a Fredholm’s integral equation with logarithmic singularity. The results were verified via two full-wave electromagnetic field simulations carried out using the finite element method and the method of moments. The findings showed that even a weakly rough surface, such as a polished exterior with weak residual roughness, can considerably affect and change 2D scattering patterns.

Calculation of a key function in the asymptotic description of moving contact lines

Summary An important element of the asymptotic description of flows having a moving liquid/gas interface which intersects a solid boundary is a function denoted $Q_i \left( \alpha \right)$ by Hocking and Rivers (The spreading of a drop by capillary action, J. Fluid Mech. 121 (1982) 425–442), where $0 < \alpha < \pi$ is the contact angle of the interface with the wall. $Q_i \left( \alpha \right)$ arises from matching of the inner and intermediate asymptotic regions introduced by those authors and is required in applications of the asymptotic theory. This article describes a new numerical method for the calculation of $Q_i \left( \alpha \right)$, which, because it explicitly allows for the logarithmic singularity in the kernel of the governing integral equation and uses quadratic interpolation of the non-singular factor in the integrand, is more accurate than that employed by Hocking and Rivers. Nonetheless, our results show good agreement with theirs, with, however, noticeable departures near $\alpha = \pi $. We also discuss the limiting cases $\alpha \to 0$ and $\alpha \to \pi $. The leading-order terms of $Q_i \left( \alpha \right)$ in both limits are in accord with the analysis of Hocking (A moving fluid interface. Part 2. The removal of the force singularity by a slip flow, J. Fluid Mech. 79 (1977) 209–229). The next-order terms are also considered. Hocking did not go beyond leading order for $\alpha \to 0$, and we believe his results for the next order as $\alpha \to \pi $ to be incorrect. Numerically, we find that the next-order terms are $O\left( {\alpha ^2} \right)$ for $\alpha \to 0$ and $O\left( 1 \right)$ as $\alpha \to \pi $. The latter result agrees with Hocking, but the value of the $O\left( 1 \right)$ constant does not. It is hoped that giving details of the numerical method and more precise information, both numerical and in terms of its limiting behaviour, concerning $Q_i \left( \alpha \right)$ will help those wanting to use the asymptotic theory of contact-line dynamics in future theoretical and numerical work.

Ising-like models for stacking faults in a free electron metal

We propose an extension of the axial next nearest neighbour Ising (ANNNI) model to a general number of interactions between spins. We apply this to the calculation of stacking fault energies in magnesium—particularly challenging due to the long-ranged screening of the pseudopotential by the free electron gas. We employ both density functional theory (DFT) using highest possible precision, and generalized pseudopotential theory (GPT) in the form of an analytic, long ranged, oscillating pair potential. At the level of first neighbours, the Ising model is reasonably accurate, but higher order terms are required. In fact, our ‘ AN N NI model’ is slow to converge—an inevitable feature of the free electron-like electronic structure. In consequence, the convergence and internal consistency of the AN N NI model is problematic within the most precise implementation of DFT. The GPT shows the convergence and internal consistency of the DFT bandstructure approach with electron temperature, but does not lead to loss of precision. The GPT is as accurate as a full implementation of DFT but carries the additional benefit that damping of the oscillations in the AN N NI model parameters are achieved without entailing error in stacking fault energies. We trace this to the logarithmic singularity of the Lindhard function.

A Mode-Matching Solution for TE-Backscattering from an Arbitrary 2D Rectangular Groove in a PEC

In this paper, the simple yet effective mode-matching technique is utilized to compute TE-backscattering from a 2D filled rectangular groove in an infinite perfect electric conductor (PEC). The tangential magnetic fields inside and outside of the groove are represented as the sums of infinite series of cosine harmonics (half-range Fourier cosine series). By applying the continuity of the tangential magnetic field, these modes are matched on the groove to obtain the series coefficients by solving a system of linear equations. For this purpose, some oscillatory logarithmic singular integrals involving Hankel and trigonometric functions are solved numerically, starting by removing the logarithmic singularity via integration by parts. In the following, the new well-behaved highly oscillatory integrals are computed using efficient methods, and several comparisons are made to demonstrate the validity and ability of the presented procedure.

An approximate solution of one singular integro-differential equation using the method of orthogonal polynomials

Two computational schemes for solving boundary value problems for a singular integro-differential equation, which describes the scattering of H-polarized electromagnetic waves by a screen with a curved boundary, are constructed. This equation contains three types of integrals: a singular integral with the Cauchy kernel, integrals with a logarithmic singularity and with the Helder type kernel. The integrands, along with the solution function, contain its first derivative. The proposed schemes for an approximate solution of the problem are based on the representation of the solution function in the form of a linear combination of the Chebyshev orthogonal polynomials and spectral relations that allows to obtain simple analytical expressions for the singular component of the equation. The expansion coefficients of the solution in terms of the Chebyshev polynomial basis are calculated by solving a system of linear algebraic equations. The results of numerical experiments show that on a grid of 20 –30 points, the error of the approximate solution reaches the minimum limit due to the error in representing real floating-point numbers.