Alternative Integral Representation Research Articles

Painlevé V and confluent Heun equations associated with a perturbed Gaussian unitary ensemble

We discuss the monic polynomials of degree n orthogonal with respect to the perturbed Gaussian weight w(z,t)=|z|α(z2+t)λe−z2,z∈R,t>0,α>−1,λ>0, which arises from a symmetrization of a semi-classical Laguerre weight wLag(z,t)=zγ(z+t)ρe−z,z∈R+,t>0,γ>−1,ρ>0. The weight wLag(z) has been widely investigated in multiple-input multi-output antenna wireless communication systems in information theory. Based on the ladder operator method, two auxiliary quantities, Rn(t) and rn(t), which are related to the three-term recurrence coefficients βn(t), are defined, and we show that they satisfy coupled Riccati equations. This turns to be a particular Painlevé V (PV, for short), i.e., PVλ22,−(1−(−1)nα)28,−2n+α+2λ+12,−12. We also consider the quantity σn(t)≔2tddtlnDn(t), which is allied to the logarithmic derivative of the Hankel determinant Dn(t). The difference and differential equations satisfied by σn(t), as well as an alternative integral representation of Dn(t), are obtained. The asymptotics of the Hankel determinant under a suitable double scaling, i.e., n → ∞ and t → 0 such that s ≔ 4nt is fixed, are established. Finally, by using the second order difference equation satisfied by the recurrence coefficients, we obtain the large n full asymptotic expansions of βn(t) with the aid of Dyson’s Coulomb fluid approach. By employing these results, the second differential equations satisfied by the orthogonal polynomials will be reduced to a confluent Heun equation.

Some integral representations for the Faxén function

In this paper, we consider the Faxén integral and present some alternative integral representations using the suitable changing of functions in the integrand. These integral representations are derived in terms of the Wright, generalized Wright and Mittag–Leffler functions. An identity is also given for the addition formula of this function.

Integral Representations for the Hartman–Watson Density

This paper concerns the density of the Hartman--Watson law. Yor (1980) obtained an integral formula that gives a closed-form expression of the Hartman--Watson density. In this paper, based on Yor's formula, we provide alternative integral representations for the density. As an immediate application, we recover in part a Dufresne's result (2001) that exhibits remarkably simple representations for the laws of exponential additive functionals of Brownian motion.

Adapting free-space fast multipole method for layered media Green's function: Algorithm and analysis

In this paper, we present a numerical algorithm for the accurate and efficient computation of the convolution of the frequency domain layered media Green's function with a given density function. Instead of compressing the convolution matrix directly as in the classical fast multipole method, fast direct solvers, and fast H-matrix algorithms, the new algorithm considers a translated form of the original matrix so that many existing building blocks from the highly optimized free-space fast multipole method can be easily adapted to the Sommerfeld integral representations of the layered media Green's function. An asymptotic analysis is performed on the Sommerfeld integrals for large orders to provide an estimate of the decay rate in the new "multipole" and "local" expansions. In order to avoid the highly oscillatory integrand in the original Sommerfeld integral representations when the source and target are close to each other, or when they are both close to the interface in the scattered field, mathematically equivalent alternative direction integral representations are introduced. The convergence of the multipole and local expansions and formulas and quadrature rules for the original and alternative direction integral representations are numerically validated.

Poroelastic response of spherical indentation into a half space with a drained surface via step displacement

Indentation of a poroelastic solid by a spherical-tip tool is analyzed within the framework of Biot’ s theory. We seek the response of the indentation force and the field quantities as functions of time when a rigid pervious indenter is loaded instantaneously to a fixed depth. The particular case where the surface of the semi-infinite domain is permeable and under a drained condition is considered. Compressibility of both the fluid and solid phases is taken into account. The solution procedure based on the McNamee–Gibson displacement function method is adopted in this work. One of the difficulties in solving poroelastic contact problems theoretically is in evaluating integrals with kernels that oscillate rapidly. We show that such issues can be overcome by using alternative integral representations with exponentially decaying functions in the kernels. Special functions, such as the modified Struve functions and the modified Bessel functions, and the method of contour integration can be utilized to aid the removal of the oscillation. Problem formulation and the solution procedure are first introduced. Implications of the poroelastic solution for incipient failure in form of tensile crack initiation and onset of plastic deformation are then discussed. An interesting outcome from this analysis is that the transient response of the dimensionless indentation force shows only weak dependence on one derived material constant and can in fact be fitted by a simple elementary function, which can then be conveniently used for material characterization in the laboratory.

An integral representation, complete monotonicity, and inequalities of the Catalan numbers

In the paper, by virtue of the Cauchy integral formula in the theory of complex functions, the authors establish an integral representation for the generating function of the Catalan numbers in combinatorics. From this, the authors derive an alternative integral representation, complete monotonicity, determinantal and product inequalities for the Catalan numbers.

An alternative integral representation for the product of two parabolic cylinder functions

ABSTRACTRecently, [Veestraeten D. An integral representation for the product of parabolic cylinder functions. Integral Transforms Spec Funct. 2017;28(1):15–21] derived an integral representation for with that was expressed in terms of the Gaussian hypergeometric function. This paper obtains an alternative expression for in which the integrand contains the parabolic cylinder function itself with the condition for convergence being at . The latter property is subsequently used to generate a new integral representation for in which restrictions on the order μ are absent.

Dynamics of local symmetry correlators for interacting many-particle systems.

Recently [P. A. Kalozoumis et al. Phys. Rev. Lett. 113, 050403 (2014)] the concept of local symmetries in one-dimensional stationary wave propagation has been shown to lead to a class of invariant two-point currents that allow to generalize the parity and Bloch theorem. In the present work, we establish the theoretical framework of local symmetries for higher-dimensional interacting many-body systems. Based on the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy, we derive the equations of motion of local symmetry correlators which are off-diagonal elements of the reduced one-body density matrix at symmetry related positions. The natural orbital representation yields equations of motion for the convex sum of the local symmetry correlators of the natural orbitals as well as for the local symmetry correlators of the individual orbitals themselves. An alternative integral representation with a unique interpretation is provided. We discuss special cases, such as the bosonic and fermionic mean field theory, and show in particular that the invariance of two-point currents is recovered in the case of the non-interacting one-dimensional stationary wave propagation. Finally we derive the equations of motion for anomalous local symmetry correlators which indicate the breaking of a global into a local symmetry in the stationary non-interacting case.

Integral Equations and the Scattering Diagram in the Problem of Diffraction by Two Shifted Contacting Wedges with Polygonal Boundary

The acoustic problem of diffraction by two wedges with different wave velocities is studied. It is assumed that the wedges with parallel edges have a common part of the boundary and the second wedge is shifted with respect of the first one in the orthogonal to the edges direction along the common part of the boundary. The wave field is governed by the Helmholtz equations. On the polygonal boundary, separating these shifted wedges from the exterior, the Dirichlet boundary condition is satisfied. The wave field is excited by an infinite filamentary source, which is parallel to the edges. In these conditions, the problem is effectively two-dimensional. The Kontorovich–Lebedev transform is applied to separate the radial and angular variables and to reduce the problem at hand to integral equations of the second kind for so-called spectral functions. The kernel of the integral equations given in the form of an integral of the product of Macdonald functions is analytically transformed to a simplified expression. For the problem at hand, some reductions of the equations are also discussed for the limiting or degenerate values of parameters. Making use of an alternative integral representation of the Sommerfeld type, expressions for the scattering diagram are then given in terms of spectral functions. Bibliography: 24 titles.

Bicharacteristics and Fourier integral operators in Kasner spacetime

The scalar wave equation in Kasner spacetime is solved, first for a particular choice of Kasner parameters, by relating the integrand in the wave packet to the Bessel functions. An alternative integral representation is also displayed, which relies upon the method of integration in the complex domain for the solution of hyperbolic equations with variable coefficients. In order to study the propagation of wave fronts, we integrate the equations of bicharacteristics which are null geodesics, and we are able to express them, for the first time in the literature, with the help of elliptic integrals for another choice of Kasner parameters. For generic values of the three Kasner parameters, the solution of the Cauchy problem is built through a pair of integral operators, where the amplitude and phase functions in the integrand solve a coupled system of partial differential equations. The first is the so-called transport equation, whereas the second is a nonlinear equation that reduces to the Eikonal equation if the amplitude is a slowly varying function. Remarkably, the analysis of such a coupled system is proved to be equivalent to building first an auxiliary covariant vector having vanishing divergence, while all nonlinearities are mapped into solving a covariant generalization of the Ermakov–Pinney equation for the amplitude function. Last, from a linear set of equations for the gradient of the phase one recovers the phase itself. This is the parametrix construction that relies upon Fourier–Maslov integral operators, but with a novel perspective on the nonlinearities in the dispersion relation. Furthermore, the Adomian method for nonlinear partial differential equations is applied to generate a recursive scheme for the evaluation of the amplitude function in the parametrix. The resulting formulas can be used to build self-dual solutions to the field equations of noncommutative gravity, as has been shown in the recent literature.

Fast Evaluation of Far-Field Signals for Time-Domain Wave Propagation

Time-domain simulation of wave phenomena on a finite computational domain often requires a fictitious outer boundary. An important practical issue is the specification of appropriate boundary conditions on this boundary, often conditions of complete transparency. Attention to this issue has been paid elsewhere, and here we consider a different, although related, issue: far-field signal recovery. Namely, from smooth data recorded on the outer boundary we wish to recover the far-field signal which would reach arbitrarily large distances. These signals encode information about interior scatterers and often correspond to actual measurements. This article expresses far-field signal recovery in terms of time-domain convolutions, each between a solution multipole moment recorded at the boundary and a sum-of-exponentials kernel. Each exponential corresponds to a pole term in the Laplace transform of the kernel, a finite sum of simple poles. Greengard, Hagstrom, and Jiang have derived the large-$$\ell $$l (spherical-harmonic index) asymptotic expansion for the pole residues, and their analysis shows that, when expressed in terms of the exact sum-of-exponentials, large-$$\ell $$l signal recovery is plagued by cancellation errors. Nevertheless, through an alternative integral representation of the kernel and its subsequent approximation by a smaller number of exponential terms (kernel compression), we are able to alleviate these errors and achieve accurate signal recovery. We empirically examine scaling relations between the parameters which determine a compressed kernel, and perform numerical tests of signal teleportation from one radial value $$r_1$$r1 to another $$r_2$$r2, including the case $$r_2=\infty $$r2=?. We conclude with a brief discussion on application to other hyperbolic equations posed on non-flat geometries where waves undergo backscatter.

Laplace transforms of the logarithmic functions and their applications

Abstract This paper deals with theorems and formulas using the technique of Laplace and Steiltjes transforms expressed in terms of interesting alternative logarithmic and related integral representations. The advantage of the proposed technique is illustrated by logarithms of integrals of importance in certain physical and statistical problems.

Axisymmetric Stokes' flow in a spherical shell revisited via the Fokas method. Part I: Irrotational flow

The axisymmetric irrotational Stokes' flow for a spherical shell is analysed by means of the recently developed Fokas method via the use of global relations. Alternative series and new integral representations concerning a system of concentric spheres, yielding, by a limiting procedure, the Dirichlet or Neumann problems for the interior and the exterior of a sphere, are presented. The boundary value problems considered can be classically solved using either the finite Gegenbauer transform or the Mellin transform. Application of the Gegenbauer transform yields a series representation which is uniformly convergent at the boundary, but not convenient for many applications. The Mellin transform, on the other hand, furnishes an integral representation which is not uniformly convergent at the boundary. Here, by algebraic manipulations of the global relation: (i) a Gegenbauer series representation is derived in a simpler manner, instead of solving ODEs and (ii) an alternative integral representation, different from the Mellin transform representation is derived which is uniformly convergent at the boundary. Copyright © 2011 John Wiley & Sons, Ltd.

Performance analysis of coherent equal gain combining over Nakagami-m fading channels

We study the exact average output signal-to-noise ratio (SNR) and symbol error rate (SER) of M-ary phaseshift-keying (PSK) signals with coherent equal gain combining reception. The analysis assumes independent Nakagami-m (1960) fading paths, which are not necessarily identically distributed. On one hand, we use geometric summations to obtain closed-form expressions for the average output SNR over diversity paths with an exponentially decaying power delay profile. On the other hand, capitalizing on an alternative integral representation of the conditional SER along with Gauss-Hermite quadrature integration, we derive an average SER expression in the form of a single finite-range integral and an integrand composed of tabulated functions. We also present a simpler but approximate approach for the closed-form evaluation of the SER of these signals over independent identically distributed Nakagami-m fading paths.

Plate bending boundary element formulation considering variable thickness

In this article the plate bending boundary element method formulation, based on the Kirchhoff’s hypothesis, is extended to take into account possible variation in the plate thickness or stiffness. For this particular case, an appropriate form of the Betti’s reciprocal work theorem was taken to derive integral representations of displacements and internal forces, starting by assuming that the plate domain exhibits variable thickness or stiffness. Some alternative integral representations to treat this problem are discussed pointing out the basic differences among them. The domain integrals, remaining in all integral equations and required to take into account the stiffness variation, are properly handled leading to small and reduced number of internal unknowns and their corresponding integral representations. As usual for this problem, only deflection integral representations related to boundary collocations are taken to write the necessary algebraic relations, written for singular and nonsingular points.

Identification and control of distributed parameter systems by means of the singular value decomposition

In this paper the basic properties of the singular value decomposition (SVD) for integral equation models of distributed parameter systems (DPS) are presented in the context of process identification and model-based control. In addition, new methods of analysis and computation are described in which SVD-based techniques are used to provide a practical solution for the DPS identification and control problem. Once developed, these novel procedures are applied to a class of linear DPS which are conventionally described by parabolic partial differential equation (PDE) models. An alternative integral equation representation for these processes is used to provide a general and readily identifiable pseudomodal input/output model for a wide variety of such DPS, including both spatially self-adjoint (diffusive) and non-self-adjoint (convective/diffusive) systems It is demonstrated that this SVD-based internal equation model of DPS is a sound and convenient basis for pseudo-modal feedback control system designs. A later paper will extend these methods to allow for model-predictive controller designs and nonlinear DPS

A novel approach for the determination of asymptotic expansions of certain oscillatory integrals

A standard method for deriving asymptotic expansion consists of applying integration by parts to standard integral representation. In this paper we consider cases in which this method is ineffective. To deal with these, we derive alternative integral representations which are amenable to this and other standard methods.

Evaluation of the Wave-Resistance Green Function: Part 1—The Double Integral

Analytical and numerical studies are made of the source potential for steady motion beneath a free surface. Various alternative integral representations are reviewed, and attention is focused on the component which usually is expressed as a double integral. A particular form is selected for numerical applications, where the double integral represents a symmetrical nonradiating disturbance, and the far-field waves are accounted for separately in the complementary single integral. Systematic expansions are derived for the singularity of the double integral at the origin, and for its asymptotic behavior far from the origin. Guided by these expansions, numerical approximations of the double integral are derived in terms of three-dimensional polynomials, which greatly facilitate the computation of the double integral. Tables of the coefficients in these approximations are presented, permitting the double integral to be evaluated throughout the three-dimensional domain with an accuracy of five to six decimal places. Greater accuracy can be achieved by using extended tables of the same coefficients. Algorithms for evaluating the Chebyshev polynomial approximations and a description of the computational methods used to derive the coefficients are included in the Appendices.

Alternative integral representations for the Green function of the theory of ship wave resistance

Three alternative single-integral representations for the Green function of the theory of ship wave resistance are derived in a unified manner from a basic double-integral representation. These alternative single-integral representations, which essentially are modifications of well-known double-internal representations due to Michell, Havelock, and Peters, are compared and discussed. Another object of this study is to examine the field equation and the boundary condition satisfied by the Green function in the limiting case when the singular point is exactly at the free surface.

The influence of solid bodies on low Mach number vortex sound

The mechanisms of sound generation by unsteady, subsonic flows in the presence of solid boundaries are investigated. For this purpose an alternative integral representation for the radiated pressure field is applied which is different from the generally used integral representation introduced by Lighthill and Curle. The main advantage of the method is that there is a linear dependence of the integrand on the time derivative of the vorticity fluctuations in the hydrodynamic near field; instead of the ordinary Green function a “vector Green function” is used. This vector Green function can be chosen for a given flow field in such a way that surface integrals do not appear. Finally, the theory is illustrated by two- and three-dimensional model flows. Analytical solutions are determined by applying the method of matched asymptotic expansions.