On the solution of the inhomogeneous polyharmonic equation and the inhomogeneous helmholtz equation

We present formulas that simplify finding the solutions of the Poisson equation, the inhomogeneous polyharmonic equation, and the inhomogeneous Helmholtz equation in the case of a polynomial right-hand side. They are based on the representation of an analytic function by harmonic functions. The resulting formulas remain valid for some analytic right-hand sides for which the corresponding operator series converge.     

Multi‐periodic eigensolutions to the Dirac operator and applications to the generalized Helmholtz equation on flat cylinders and on the n‐torus

In this paper, we study the solutions to the generalized Helmholtz equation with complex parameter on some conformally flat cylinders and on the n‐torus. Using the Clifford algebra calculus, the solutions can be expressed as multi‐periodic eigensolutions to the Dirac operator associated with a complex parameter λ∈?. Physically, these can be interpreted as the solutions to the time‐harmonic Maxwell equations on these manifolds. We study their fundamental properties and give an explicit representation theorem of all these solutions and develop some integral representation formulas. In particular, we set up Green‐type formulas for the cylindrical and toroidal Helmholtz operator. As a concrete application, we explicitly solve the Dirichlet problem for the cylindrical Helmholtz operator on the half cylinder. Finally, we introduce hypercomplex integral operators on these manifolds, which allow us to represent the solutions to the inhomogeneous Helmholtz equation with given boundary data on cylinders and on the n‐torus. Copyright © 2009 John Wiley & Sons, Ltd.     

Completeness theorem for singular differential pencils

A theorem completeness theorem of special vector functions induced by the products of the so-called Weyl solutions of a fourth-order differential equation and by their derivatives on the semiaxis is presented. We prove that such nonlinear combinations of Weyl solutions and their derivatives constitute a linear subspace of decreasing (at infinity) solutions of a linear singular differential system of Kamke type. We construct and study the Green function of the corresponding singular boundary-value problems on the semiaxis for operator pencils defining differential systems of Kamke type. The required completeness theorem is proved by using the analytic and asymptotic properties of the Green function, operator spectral theory methods, and analytic function theory.     

A nonconforming domain decomposition approximation for the Helmholtz screen problem with hypersingular operator

We present and analyze a nonconforming domain decomposition approximation for a hypersingular operator governed by the Helmholtz equation in three dimensions. This operator appears when considering the corresponding Neumann problem in unbounded domains exterior to open surfaces. We consider small wave numbers and low‐order approximations with Nitsche coupling across interfaces. Under appropriate assumptions on mapping properties of the weakly singular and hypersingular operators with Helmholtz kernel, we prove that this method converges almost quasioptimally, that is, with optimal orders reduced by an arbitrarily small positive number. Numerical experiments confirm our error estimate. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 125–141, 2017     

Explicit Formulas for the Green’s Function and the Bergman Kernel for Monogenic Functions in Annular Shaped Domains in {\mathbb{R}^{n+1}}

By applying a reflection principle we set up fully explicit representation formulas for the harmonic Green’s function for orthogonal sectors of the annulus of the unit ball of ${\mathbb{R}^n}$ . From the harmonic Green’s function we then can determine the Bergman kernel function of Clifford holomorphic functions by applying an appropriate vector differentiation. As a concrete application we give an explicit analytic representation formula of the solutions to an n-dimensional Dirichlet problem in annular shaped domains that arises in the context of heat conduction.     

Generalized Integral Representations for Functions with Values in C（V_（3,3））

By using the solution to the Helmholtz equation u-λu = 0(λ ≥ 0),the explicit forms of the so-called kernel functions and the higher order kernel functions are given.Then by the generalized Stokes formula,the integral representation formulas related with the Helmholtz operator for functions with values in C(V3,3) are obtained.As application of the integral representations,the maximum modulus theorem for function u which satisfies Hu = 0 is given.     

A singular integral equation with the Cauchy kernel on a closed interval in a class of distributions

In the present paper, we introduce the notion of a singular integral with the Cauchy kernel for distributions and consider a singular integral equation with the Cauchy kernel on a closed interval for the case in which the right-hand side is a distribution that admits a representation in the form of the sum of a distribution vanishing in neighborhoods of the endpoints and an ordinary function satisfying the Hölder condition. The solution is also sought in the form of a distribution. Distributions are treated as linear functionals on some test functions. We analyze the solvability of the equation in the class of distributions and obtain explicit formulas for the inversion of this equation, similar to formulas for ordinary solutions. To analyze the solvability of the singular integral equation, we use an approach based on the consideration of the Riemann boundary value problem for analytic functions with a generalized boundary condition. When stating and studying this problem, we use the results in [1, 2].Translated from Differentsialnye Uravneniya, Vol. 40, No. 9, 2004, pp. 1208–1218.Original Russian Text Copyright © 2004 by Setukha.     

An Efficient Cartesian Grid-Based Method for Scattering Problems with Inhomogeneous Media

Boundary integral equations provide a powerful tool for the solution of scattering problems. However, often a singular kernel arises, in which case the standard quadratures will give rise to unavoidable deteriorations in numerical precision, thus special treatment is needed to handle the singular behavior. Especially, for inhomogeneous media, it is difficult if not impossible to find out an analytical expression for Green’s function. In this paper, an efficient fourth-order accurate Cartesian grid-based method is proposed for the two-dimensional Helmholtz scattering and transmission problems with inhomogeneous media. This method provides an alternative approach to indirect integral evaluation by solving equivalent interface problems on Cartesian grid with a modified fourth-order accurate compact finite difference scheme and a fast Fourier transform preconditioned conjugate gradient (FFT-PCG) solver. A remarkable point of this method is that there is no need to know analytical expressions for Green’s function. Numerical experiments are provided to demonstrate the advantage of the current approach, including its simplicity in implementation, its high accuracy and efficiency.     

Quadrature formulas for periodic functions and their application to the boundary element method

Two-dimensional and axisymmetric boundary value problems for the Laplace equation in a domain bounded by a closed smooth contour are considered. The problems are reduced to integral equations with a periodic singular kernel, where the period is equal to the length of the contour. Taking into account the periodicity property, high-order accurate quadrature formulas are applied to the integral operator. As a result, the integral equations are reduced to a system of linear algebraic equations. This substantially simplifies the numerical schemes for solving boundary value problems and considerably improves the accuracy of approximation of the integral operator. The boundaries are specified by analytic functions, and the remainder of the quadrature formulas decreases faster than any power of the integration step size. The examples include the two-dimensional potential inviscid circulation flow past a single blade or a grid of blades; the axisymmetric flow past a torus; and free-surface flow problems, such as wave breakdown, standing waves, and the development of Rayleigh-Taylor instability.     

Expansion of a Self-Adjoint Absolutely Continuous Singular Integral Operator in Generalized Eigenvectors and Its Diagonalization

We describe the relationship between the expansion of a self-adjoint operator in generalized eigenvectors and the direct integral of Hilbert spaces. We perform the explicit diagonalization of a self-adjoint absolutely continuous singular integral operator Y using an Hermitian nonnegative kernel consisting of boundary values of the determining function of the operator T = X + iY with respect to the resolvent of the imaginary part of Y.     

The Subelliptic Heat Kernels of the Quaternionic Hopf Fibration

The main goal of this work is to study the sub-Laplacian of the unit sphere which is obtained by lifting with respect to the Hopf fibration the Laplacian of the quaternionic projective space. We obtain in particular explicit formulas for its heat kernel and deduce an expression for the Green function of the conformal sub-Laplacian and small-time asymptotics. As a byproduct of our study we also obtain several results related to the sub-Laplacian of a projected Hopf fibration.     

<Emphasis Type="Italic">N</Emphasis>=1 supersymmetric conformal block recursion relations

We present explicit recursion relations for the four-point superconformal block functions that are essentially particular contributions of the given conformal class to the four-point correlation function. The approach is based on the analytic properties of the superconformal blocks as functions of the conformal dimensions and the central charge of the superconformal algebra. We compare the results with the explicit analytic expressions obtained for special parameter values corresponding to the truncated operator product expansion. These recursion relations are an efficient tool for numerically studying the four-point correlation function in superconformal field theory in the framework of the bootstrap approach, similar to that in the case of the purely conformal symmetry. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 3, pp. 476–487, September, 2007.     

The Helmholtz Operator on Higher Dimensional M?bius Strips Embedded in {\mathbb{R}^{4}}

In this paper we present explicit formulas for the fundamental solution to the Helmholtz operator on a higher-dimensional analogue of the M?bius strip in three real variables (embedded in ${\mathbb{R}^{4}}$ ) with values in distinct pinor bundles. Herefore we use an approach that uses classical harmonic analysis methods combined with some Clifford analysis tools and adapt it to this special geometry. The fundamental solution is described in terms of generalizations of the Weierstrass ${\wp}$ -function that are adapted to the context of these geometries. As our main result we present an analytic integral representation formula to express the solutions of the inhomogeneous time-independent Klein-Gordon problem on M?bius strips.     

Clifford分析中具有超正则核的T算子的性质

讨论了Clifford分析中具有超正则核的T(Ieodorescu)算子的基本性质.T算子是定义在区域上的奇异积分算子,它在广义解析函数理论和Vekua理论中起着重要的作用.在复分析中关于T算子的理论已经发展得很完善,但在Clifford分析中,具有超正则核的T算子的相关性质还没有得到研究.研究了Clifford分析中具有超正则核的T算子的基本性质,得到了这个算子在ΩR_+~(n+1)上的一致有界性,Hlder连续性以及这个算子的γ次可积性.     

Explicit Green operators for quantum mechanical Hamiltonians. I. The hydrogen atom

We study a new approach to determine the asymptotic behaviour of quantum many-particle systems near coalescence points of particles which interact via singular Coulomb potentials. This problem is of fundamental interest in electronic structure theory in order to establish accurate and efficient models for numerical simulations. Within our approach, coalescence points of particles are treated as embedded geometric singularities in the configuration space of electrons. Based on a general singular pseudo-differential calculus, we provide a recursive scheme for the calculation of the parametrix and corresponding Green operator of a nonrelativistic Hamiltonian. In our singular calculus, the Green operator encodes all the asymptotic information of the eigenfunctions. Explicit calculations and an asymptotic representation for the Green operator of the hydrogen atom and isoelectronic ions are presented.     

On the Green function in visco‐elastic media obeying a frequency power‐law

In this work, we present an explicit expression for the Green function in a visco‐elastic medium. We choose Szabo and Wu's frequency power law model to describe the visco‐elastic properties and derive a generalized visco‐elastic wave equation. We express the ideal Green function (without any viscous effect) in terms of the viscous Green function using an attenuation operator. By means of an approximation of the ideal Green function, we address the problem of reconstructing a small anomaly in a visco‐elastic medium from wavefield measurements. Copyright © 2011 John Wiley & Sons, Ltd.     

Factorizations and Singular Value Estimates of Operators with Gelfand–Shilov and Pilipović kernels

We prove that any linear operator with kernel in a Pilipovi? or Gelfand–Shilov space can be factorized by two operators in the same class. We also give links on numerical approximations for such compositions. We apply these composition rules to deduce estimates of singular values and establish Schatten–von Neumann properties for such operators.     

A spin integral equation for electromagnetic and acoustic scattering

We present a new integral equation for solving the Maxwell scattering problem against a perfect conductor. The very same algorithm also applies to sound-soft as well as sound-hard Helmholtz scattering, and in fact the latter two can be solved in parallel in three dimensions. Our integral equation does not break down at interior spurious resonances, and uses spaces of functions without any algebraic or differential constraints. The operator to invert at the boundary involves a singular integral operator closely related to the three-dimensional Cauchy singular integral, and is bounded on natural function spaces and depend analytically on the wave number. Our operators act on functions with pairs of complex two-by-two matrices as values, using a spin representation of the fields.     

Busy periods of discrete-time queues using the Lagrange implicit function theorem

In this paper, we study the busy-period distribution for discrete-time queues assuming a Bernoulli arrival process, with arbitrary service-time and batch-arrival distributions. We derive explicit analytic formulas for these distributions using the Lagrange Implicit Function Theorem applied to probability generating functions. The convenient coefficient operator notation used in these formulas leads to a computationally efficient method for obtaining the distributions in their entirety from these analytic formulas.