Asymptotic analysis of aircraft wing model in subsonic air flow

This paper is the first in a series of several works devotedto the asymptotic and spectral analysis of an aircraft wingin a subsonic air flow. This model has been developed in theFlight Systems Research Center of UCLA and is presented in theworks of Balakrishnan. The model is governed by a system oftwo coupled integro-differential equations and a two parameterfamily of boundary conditions modelling the action of the self-strainingactuators. The unknown functions (the bending and the torsionangle) depend on time and one spatial variable. The differentialparts of the above equations form a coupled linear hyperbolicsystem; the integral parts are of convolution type. The systemof equations of motion is equivalent to a single operator evolution–convolutiontype equation in the state space of the system equipped withthe so-called energy metric. The Laplace transform of the solutionof this equation can be represented in terms of the so-calledgeneralized resolvent operator. The generalized resolvent operatoris an operator-valued function of the spectral parameter. Thisgeneralized resolvent operator is a finite meromorphic functiondefined on the complex plane having the branch cut along thenegative real semi-axis. The poles of the generalized resolventare precisely the aeroelastic modes, and the residues at thesepoles are the projectors on the generalized eigenspaces. Inthis paper, our main object of interest is the dynamics generatorof the differential parts of the system. It is a non-selfadjointoperator in the state space with a pure discrete spectrum. Inthe present paper, we show that the spectrum consists of twobranches, and we derive their precise spectral asymptotics.Based on these results, in the next paper we will derive theasymptotics of the aeroelastic modes and approximations forthe mode shapes.     

Solution of one class of systems of dual summation equations for associated Legendre functions

We have obtained analytical solutions of one class of systems of dual summation equations for associated Legendre functions with fractional indices. Such equations appear in studying the interaction of vector electromagnetic fields with the circular edge of a conductive open cone in the low-frequency region. We have derived formulas for the reexpansion of Legendre functions, which are used for passage from summation equations to infinite systems of linear algebraic equations, containing convolution-type matrix operators. The operators inverse to them are applied for finding a solution in the required class of sequences. We give an example of determining the effect of interaction of TM- and TE-waves with the edge of a finite cone.     

A new form of the redfield equation for dissipative systems related to the matrix of correlation functions

In the Redfield theory framework, we consider the problem of the vibrational dynamics in dissipative systems. We decompose the Hamiltonian of interaction of the observed system with a thermal bath into terms that are products of system transition operators and bath transition operators. Using the decomposition, we construct a correlation function matrix containing all the information about the interaction of the system with the bath and obtain a new operator form of the Redfield equation. We consider the procedure for factoring the interaction operator and constructing the correlation function matrix. Using the diagonalization of the obtained matrix, we give correlated dissipation operators, whose introduction simplifies the form of the Redfield equations. We show that for several problems in which fundamental transition frequencies can be chosen, this procedure significantly reduces the dimensionality of the dissipative dynamics problem.     

Functional calculus of Dirac operators and complex perturbations of Neumann and Dirichlet problems

We prove that Neumann, Dirichlet and regularity problems for divergence form elliptic equations in the half-space are well posed in L₂ for small complex L_∞ perturbations of a coefficient matrix which is either real symmetric, of block form or constant. All matrices are assumed to be independent of the transversal coordinate. We solve the Neumann, Dirichlet and regularity problems through a new boundary operator method which makes use of operators in the functional calculus of an underlaying first order Dirac type operator. We establish quadratic estimates for this Dirac operator, which implies that the associated Hardy projection operators are bounded and depend continuously on the coefficient matrix. We also prove that certain transmission problems for k-forms are well posed for small perturbations of block matrices.     

Nonlocal symmetries and recursion operators: Partial differential and differential-difference equations

A systematic method to derive the nonlocal symmetries for partial differential and differential-difference equations with two independent variables is presented and shown that the Korteweg-de Vries (KdV) and Burger's equations, Volterra and relativistic Toda (RT) lattice equations admit a sequence of nonlocal symmetries. An algorithm, exploiting the obtained nonlocal symmetries, is proposed to derive recursion operators involving nonlocal variables and illustrated it for the KdV and Burger's equations, Volterra and RT lattice equations and shown that the former three equations admit factorisable recursion operators while the RT lattice equation possesses (2×2) matrix factorisable recursion operator. The existence of nonlocal symmetries and the corresponding recursion operator of partial differential and differential-difference equations does not always determine their mathematical structures, for example, bi-Hamiltonian representation.     

Asymptotic regularity of solutions to Hartree–Fock equations with Coulomb potential

We study the asymptotic regularity of solutions to Hartree–Fock (HF) equations for Coulomb systems. To deal with singular Coulomb potentials, Fock operators are discussed within the calculus of pseudo‐differential operators on conical manifolds. First, the non‐self‐consistent‐field case is considered, which means that the functions that enter into the nonlinear terms are not the eigenfunctions of the Fock operator itself. We introduce asymptotic regularity conditions on the functions that build up the Fock operator, which guarantee ellipticity for the local part of the Fock operator on the open stretched cone ?₊ × S². This proves the existence of a parametrix with a corresponding smoothing remainder from which it follows, via a bootstrap argument, that the eigenfunctions of the Fock operator again satisfy asymptotic regularity conditions. Using a fixed‐point approach based on Cancès and Le Bris analysis of the level‐shifting algorithm, we show via another bootstrap argument that the corresponding self‐consistent‐field solutions to the HF equation have the same type of asymptotic regularity. Copyright © 2008 John Wiley & Sons, Ltd.     

Asymptotic properties of the heat kernel on conic manifolds

We derive asymptotic properties for the heat kernel of elliptic cone (or Fuchs type) differential operators on compact manifolds with boundary. Applications include asymptotic formulas for the heat trace, counting function, spectral function, and zeta function of cone operators. The author was supported in part by a Ford Foundation Fellowship.     

Method of perturbations in vector electromagnetic problems of diffraction by a wedge

A vector electromagnetic problem of diffraction by a wedge-shaped region is reduced to a system of coupled functional equations by using Sommerfeld integrals. This system of functional equations is solved by the perturbation method, and the convergence of the related series is analyzed. The system of functional equations is further reduced to linear equations with contracting operators, and the solution is represented in the form of a Neumann series. Reduction to a system with compact operators is also considered. Bibliography: 13 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 230, 1995, pp. 138–156. Translated by M. A. Lyalinov.     

Pseudodifferential operator method in a problem on the diffraction of an electromagnetic wave on a dielectric body

We study the problem on the diffraction of electromagnetic waves on a solid body in free space. To analyze the integro-differential equations describing this phenomenon, we use the theory of pseudodifferential operators. We evaluate the asymptotic expansion of the symbol and prove the ellipticity and Fredholmness with index zero of the problem operator.     

A boundary integral equation method for three-dimensional crack problems in elasticity

This paper presents a solution procedure for three-dimensional crack problems via first kind boundary integral equations on the crack surface. The Dirichlet (Neumann) problem is reduced to a system of integral equations for the jump of the traction (of the field) across the crack surface. The calculus of pseudodifferential operators is used to derive existence and regularity of the solutions of the integral equations. With the concept of the principal symbol and the Wiener-Hopf technique we derive the explicit behavior of the densities of the integral equations near the edge of the crack surface. Based on the detailed regularity results we show how to improve the boundary element Galerkin method for our integral equations. Quasi-optimal asymptotic estimates for the Galerkin error are given.     

Morozov's discrepancy principle for tikhonov

In this paper severely ill-posed problems are studied which are represented in the form of linear operator equations with infinitely smoothing operators but with solutions having only a finite smoothness. It is well known, that the combination of Morozov's discrepancy principle and a finite dimensional version of the ordinary Tikhonov regularization is not always optimal because of its saturation property. Here it is shown, that this combination is always order-optimal in the case of severely ill-posed problems.     

Perturbation of the Drazin inverse for closed linear operators

We investigate the perturbation of the Drazin inverse of a closed linear operator recently introduced by second author and Tran, and derive explicit bounds for the perturbations under certain restrictions on the perturbing operators. We give applications to the solution of perturbed linear equations, to the asymptotic behaviour ofC ₀-semigroups of linear operators, and to perturbed differential equations. As a special case of our results we recover recent perturbation theorems of Wei and Wang.     

Banach algebra generated by a finite number of bergman polykernel operators,continuous coefficients,and a finite group of shifts

We study the Banach algebra generated by a finite number of Bergman polykernel operators with continuous coefficients that is extended by operators of weighted shift that form a finite group. By using an isometric transformation, we represent the operators of the algebra in the form of a matrix operator formed by a finite number of mutually complementary projectors whose coefficients are Toeplitz matrix functions of finite order. Using properties of Bergman polykernel operators, we obtain an efficient criterion for the operators of the algebra considered to be Fredholm operators.     

Spectral analysis of operator polynomials and higher-order difference operators

The study of the spectral properties of operator polynomials is reduced to the study of the spectral properties of the operator specified by the operator matrix. The results obtained are applied to higher-order difference operators. Conditions for their invertibility and for them to be Fredholm, as well as the asymptotic representation for bounded solutions of homogeneous difference equations are obtained.     

On the numerical analysis of finite element and dirichlet-to-neumann methods for nonlinear exterior transmission problems

We provide the numerical analysis of the combination of finite elements and Dirichlet-to-Neumann mappings (based on boundary integral operators) for a class of nonlinear exterior transmission problems whose weak formulations reduce to Lipschitz-continuous and strongly monotone operator equations. As a model we consider a nonlinear second order elliptic equation in divergence form in a bounded inner region of the plane, coupled with the Laplace equation in the corresponding unbounded exterior part. A discrete Galerkin scheme is presented by using linear finite elements on a triangulation of the domain, and then applying numerical quadrature and analytical formulae to evaluate all the linear, bilinear and semilinear forms involved. We prove the unique solvability of the discrete equations, and show the strong convergence of the approximate solutions. Furthermore, assuming additional regularity on the solution of the continuous operator equation, the asymptotic rate of convergence O(h) is also derived. Finally, numerical experiments are presented, which confirm the convergence results.     

板几何中奇异迁移方程解的渐近稳定性

在L~p(1p+∞)空间,研究了板几何中一类具反射边界条件的各向异性、连续能量、均匀介质的奇异迁移方程,通过构造算子,利用比较算子方法,证明了奇异迁移算子A相应的奇异迁移半群V(t)(t≥0)的Dyson-Phillips展开式的一阶余项R_1(t)的紧性,得到了半群V(t)与U(t)(streaming算子B产生)本质谱相同,本质谱型一致;迁移算子A的谱在区域T中由有限个具有限代数重数的离散本征值组成;迁移方程解的渐近稳定性.     

A numerical study of electromagnetic waves in periodic waveguides

Here are considered time‐harmonic electromagnetic waves in a quadratic waveguide consisting of a periodic dielectric core enclosed by conducting walls. The permittivity function may be smooth or have jumps. The electromagnetic field is given by a magnetic vector potential in Lorenz gauge, and defined on a Floquet cell. The Helmholtz operator is approximated by a Chebyshev collocation, Fourier–Galerkin method. Laurent's rule and the inverse rule are employed for the representation of Fourier coefficients of products of functions. The computations yield, for known wavenumbers, values of the first few eigenfrequencies of the field. In general, the dispersion curves exhibit band gaps. Field patterns are identified as transverse electric, TE, transverse magnetic, TM, or hybrid modes. Maxwell's equations are fulfilled. A few trivial solutions appear when the permittivity varies in the guiding direction and across it. The results of the present method are consistent with exact results and with those obtained by a low‐order finite element software. The present method is more efficient than the low‐order finite element method. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 490–513, 2014     

Special operator equations

The study addresses the matrix operator equations of a special form which are used in the theory of Markov chains. Solving the operator equations with stochastic transition probability matrices of large finite or even countably infinite size reduces to the case of stochastic matrices of small size. In particular, the case of ternary chains is considered in detail. A Markov model for crack growth in a composite serves as an example of application.     

Operator kernel estimates for functions of generalized Schrödinger operators

We study the decay at large distances of operator kernels of functions of generalized Schrödinger operators, a class of semibounded second order partial differential operators of mathematical physics, which includes the Schrödinger operator, the magnetic Schrödinger operator, and the classical wave operators (i.e., acoustic operator, Maxwell operator, and other second order partial differential operators associated with classical wave equations). We derive an improved Combes-Thomas estimate, obtaining an explicit lower bound on the rate of exponential decay of the operator kernel of the resolvent. We prove that for slowly decreasing smooth functions the operator kernels decay faster than any polynomial.

     

Scattering for Schrödinger operators in a class of domains with non-compact boundaries

Schrödinger operators in a class of domains with asymptotic cones are considered. A generalized Fourier transform representing the absolutely continuous part of the Schrödinger operator as multiplication by $\text{¦ξ¦}{}^2$ in the asymptotic cone is constructed. Wave operators relating the free Laplacian to Schrödinger operators are computed using the generalized Fourier transform. The wave operators relating Schrödinger operators acting in domains with the same asymptotic cone are computed and shown to be complete.