The solvability of integral lightguide equations

Integral equations of a three-dimensional periodic lightguide are studied for arbitrary values of the frequency parameter. The solution to the homogeneous integral equation vanishes for those values of the spectral parameter that are different from some isolated points. Bibliography: 5 titles. Translated fromProblemy Matematicheskogo Analiza, No. 19, 1999, pp. 89–100.     

Some non‐local problems for the parabolic‐hyperbolic type equation with complex spectral parameter

In the present paper the unique solvability of two non‐local problems for the mixed parabolic‐hyperbolic type equation with complex spectral parameter is proved. Sectors for values of the spectral parameter where these problems have unique solutions are shown. Uniqueness of the solution is proved by the method of energy integral and existence is proved by the method of integral equations. In particular cases, eigenvalues and corresponding eigenfunctions of the studied problems are found. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Spectral properties of a fourth-order differential operator with integrable coefficients

The aim of this paper is to study spectral properties of differential operators with integrable coefficients and a constant weight function. We analyze the asymptotic behavior of solutions to a differential equation with integrable coefficients for large values of the spectral parameter. To find the asymptotic behavior of solutions, we reduce the differential equation to a Volterra integral equation. We also obtain asymptotic formulas for the eigenvalues of some boundary value problems related to the differential operator under consideration.     

Unitary monodromy of Lamé differential operators

The classical second order Lamé equation contains a so-called accessory parameter B. In this paper we study for which values of B the Lamé equation has a monodromy group which is conjugate to a subgroup of SL(2,ℝ) (unitary monodromy with indefinite hermitian form). We reformulate the problem as a spectral problem and give an asymptotic expansion for the spectrum.      

N-group neutron transport theory: A criticality problem in slab geometry

The steady-state equation for N-group neutron transport in slab geometry is written as an integral equation. A spectral analysis is made of the integral operator and related to the criticality problem. The method depends on a representation for the resolvent kernel for a subcritical slab and on analytic continuation in a complex parameter to characterize eigenvalues in terms of singularities of the resolvent. The analytic continuation is based on a bifurcation analysis of some nonlinear matrix integral equations whose solutions provide a matrix Wiener-Hopf factorization of the Fourier transform of the kernel of the transport operator.     

Generalized Dirichlet‐to‐Neumann Map in Time‐Dependent Domains

We study the heat, linear Schrödinger (LS), and linear KdV equations in the domain l(t) < x < ∞ , 0 < t < T , with prescribed initial and boundary conditions and with l(t) a given differentiable function. For the first two equations, we show that the unknown Neumann or Dirichlet boundary value can be computed as the solution of a linear Volterra integral equation with an explicit weakly singular kernel. This integral equation can be derived from the formal Fourier integral representation of the solution. For the linear KdV equation we show that the two unknown boundary values can be computed as the solution of a system of linear Volterra integral equations with explicit weakly singular kernels. The derivation in this case makes crucial use of analyticity and certain invariance properties in the complex spectral plane. The above Volterra equations are shown to admit a unique solution.     

About the Tricomi problem for the mixed parabolic–hyperbolic type equation with complex spectral parameter

The unique solvability of the Tricomi problem for the parabolic–hyperbolic equation with complex spectral parameter is proved. Uniqueness of the solution is shown by the method of energy integral and existence by the method of integral equations.     

The exterior dirichlet problem for the Laplace equation

Using a standard application of Green's theorem, the exterior Dirichlet problem for the Laplace equation in three dimensions is reduced to a pair of integral equations. One integral equation is of the second kind and the other is of the first kind. It is known that the integral equation of the second kind is not uniquely solvable, however, it has been demonstrated that the pair of integral equations has a unique solution. The present approach is based on the observation that the known function appearing in the integral equation of the second kind lies in a certain Banach space E which is a proper subspace of the Banach space of continuous complex-valued functions equipped with the maximum norm. Furthermore, it can be shown that the related integral operator when restricted to E has spectral radius less than unity. Consequently, a particular solution to the integral equation of the second kind can be obtained by the method of successive approximations and the unique solution to the problem is then obtained by using the integral equation of the first kind. Comparisons are made between the present algorithm and other known constructive methods. Finally, an example is supplied to illustrate the method of this paper.     

On a particular second kind Volterra integral equation with a spectral parameter

We study the solvability questions for a particular second kind Volterra integral equation with a spectral parameter λ arising in the theory of boundary value problems for the spectrally loaded parabolic equations in unbounded domains when the order of a derivative in a loaded summand agrees with that in the differential part of the equation.     

On Some Integral Equations in Hilbert Space with an Application to the Theory of Elasticity

A Volterra type integral equation in a Hilbert space with an additional linear operator L and a spectral parameter depending on time is considered. If the parameter does not belong to the spectrum of L unconditional solvability of the considered problem is proved. In the case where the initial value of the parameter coincides with some isolated point of the spectrum of the operator L sufficient conditions for solvability are established. The obtained results are applied to the partial integral equations associated with a contact problem of the theory of elasticity.     

Polynomial solutions of q-Heun equation and ultradiscrete limit

We study polynomial-type solutions of the q-Heun equation, which is related with quasi-exact solvability. The condition that the q-Heun equation has a non-zero polynomial-type solution is described by the roots of the spectral polynomial, whose variable is the accessory parameter E. We obtain sufficient conditions that the roots of the spectral polynomial are all real and distinct. We consider the ultradiscrete limit to clarify the roots of the spectral polynomial and the zeros of the polynomial-type solution of the q-Heun equation.     

Polarized light transfer above a reflecting surface

The existence and uniqueness are established for the solution of the equation of transfer of polarized light in a homogeneous atmosphere of finite optical thickness, assuming reflection by the planetary surface. A general L_p-space formulation is adopted. The boundary value problem is first written as a vector-valued integral equation. Using monotonicity properties of the spectral radii of the integral operators involved as well as recent half-range completeness results for kinetic equations with reflective boundary conditions, the present results follow as a corollary.     

Splitting and the spectrum of a linear differential equation with quasiperiodic coefficients

A linear differential equation with quasiperiodic coefficients for large values of the spectral parameter is split into an exponentially dichotomous system and a two-dimensional] system with the properties of the one-dimensional Schrödinger equation. The set of the values of the parameter, for which the equation has solution of the form A(t) epx (iat) with a real number a and a quasiperiodic function A(t), is a single out.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 10, pp. 1383–1388, October, 1990.     

On the spline collocation method for the single layer equation related to time-fractional diffusion

The boundary element spline collocation method is studied for the time-fractional diffusion equation in a bounded two-dimensional domain. We represent the solution as the single layer potential which leads to a Volterra integral equation of the first kind. We discretize the boundary integral equation with the spline collocation method on uniform meshes both in spatial and time variables. In the stability analysis we utilize the Fourier analysis technique developed for anisotropic pseudodifferential equations. We prove that the collocation solution is quasi-optimal under some stability condition for the mesh parameters. We have to assume that the mesh parameter in time satisfies (h_t=c h^\frac2a)(h_t=c h^{\frac{2}{\alpha}}), where (h) is the spatial mesh parameter.     

Inverse scattering problem for Sturm–Liouville operator with nonlinear dependence on the spectral parameter in the boundary condition

The inverse problem of the scattering theory for Sturm–Liouville operator on the half line with boundary condition depending quadratic on the spectral parameter is considered. Scattering data are defined, some properties of the scattering data are examined, the main equation is obtained, solvability of the integral equation is proved and uniqueness of algorithm to the potential with given scattering data is studied. Copyright © 2010 John Wiley & Sons, Ltd.     

Propagation of Gaussian wave packets in thin periodic quantum waveguides with a nonlocal nonlinearity

We consider the nonlinear Schrödinger equation with an integral Hartree-type nonlinearity in a thin quantum waveguide and study the propagation of Gaussian wave packets localized in the spatial variables. In the case of periodically varying waveguide walls, we establish the relation between the behavior of wave packets and the spectral properties of the auxiliary periodic problem for the one-dimensional Schrödinger equation. We show that for a positive value of the nonlinearity parameter, the integral nonlinearity prevents the packet from spreading as it propagates. In addition, we find situations such that the packet is strongly focused periodically in time and space.     

Three-dimensional singularities of elastic fields near vertices

For the computation of the singular behavior of an elastic field near a three-dimensional vertex subject to displacement boundary conditions we use a boundary integral equation of the first kind whose unknown is the boundary stress. Localization at the vertex and Mellin transformation yield a one-dimensional integral equation on a piecewise circular curve γ in IR³ depending holomorphically on the complex Mellin parameter. The corresponding spectral points and packets of generalized eigenvectors characterize the desired stress field and are computed by a spline-Galerkin method with graded meshes at the corner points of the curve γ. © 1993 John Wiley & Sons, Inc.     

On an inverse scattering problem for a class Dirac operator with discontinuous coefficient and nonlinear dependence on the spectral parameter in the boundary condition

On the positive semi‐infinite interval, we obtained a generalization of the Marchenko method for a Dirac equation system with a discontinuous coefficient and a quadratic polynomial on a spectral parameter in the boundary condition. In this connection, we use an new integral representation of the Jost solution of equation systems, which does not have a ‘triangular’ form. The scattering function of the problem is defined, and its properties are examined. The Marchenko‐type main equation is obtained, and it is shown that the potential is uniquely recovered in terms the scattering function. Copyright © 2012 John Wiley & Sons, Ltd.     

On the Inverse Scattering Transform for the Kadomtsev-Petviashvili Equation

The initial value problem of the Kadomtsev-Petviashvili equation for one choice of sign in the equation has been recently investigated in the literature. Here we consider the other choice of sign. We introduce suitable eigenfunctions which though bounded are not analytic in the spectral parameter. This, in contrast to the known case, prevents us from formulating the inverse problem as a nonlocal Riemann-Hilbert boundary value problem. Nevertheless a suitable formulation is given and a formal solution is constructed via a linear integral equation.     

ON THE TRANSPORT EQUATION WITH INTEGRALBOUNDARY CONDITIONS AND ITS CONSERVATIVE LAW

Abstract. The well-posedness for the time-dependent neutron transport equation with integral boundary conditions is established in LI space. Some spectral properties of the transport operator are discussed, the dorninant dgenvalue is proved existhlg,and furthermore, the conservative law of migrating particle numbers is established.