Reflected pulses from a refractive random medium at grazing incidence

A problem of acoustic pulse reflection by a one-dimensionalrefractive random medium is considered in the case of grazingangle incidence. The material parameters of the medium are assumedto vary with a random microscale and a deterministic macroscale.A system of stochastic equations for random scattering variablesis derived based upon the random modelling of three separatescales of variations. The statistical properties of the reflectedpulses are characterized by an asymptotic diffusion limit theoremof stochastic differential equations with multiple scales. Thetransport equations governing the limiting stochastic distributionsof the random reflection coefficient are obtained in the propagatingregime, which leads to the power spectral densities of the reflectedpressure and particle velocity fields.     

Inverse problem for differential pencils on a hedgehog graph

We study boundary value problems on a hedgehog graph for second-order ordinary differential equations with a nonlinear dependence on the spectral parameter. We establish properties of spectral characteristics and consider the inverse spectral problem of reconstructing the coefficients of a differential pencil on the basis of spectral data. For this inverse problem, we prove a uniqueness theorem and obtain a procedure for constructing its solution.     

Inverse spectral problems for differential systems on a finite interval

Inverse spectral problems are studied for non-selfadjoint systems of ordinary differential equations on a finite interval. We establish properties of the spectral characteristics, and provide a procedure for constructing the solution of the inverse problem of recovering the coefficients of differential systems from the given spectral characteristics.     

Recovering differential pencils on compact graphs

We study boundary value problems on compact graphs without circles (i.e. on trees) for second-order ordinary differential equations with nonlinear dependence on the spectral parameter. We establish properties of the spectral characteristics and investigate the inverse spectral problem of recovering the coefficients of the differential equation from the so-called Weyl vector which is a generalization of the Weyl function (m-function) for the classical Sturm-Liouville operator. For this inverse problem we prove the uniqueness theorem and obtain a procedure for constructing the solution by the method of spectral mappings.     

On the general nonlinear self-adjoint spectral problem for differential-algebraic systems

We consider a general self-adjoint spectral problem, nonlinear with respect to the spectral parameter, for linear differential-algebraic systems of equations. Under some assumptions, we present a method for reducing such a problem to a general self-adjoint nonlinear spectral problem for a system of differential equations. In turn, this permits one to pass to a problem for a Hamiltonian system of ordinary differential equations. In particular, in this way, one can obtain a method for computing the number of eigenvalues of the original problem lying in a given range of the spectral parameter.     

Inverse spectral problem for differential operator pencils on noncompact spatial networks

We study boundary value problems on noncompact cycle-free graphs (i.e., trees) for second-order ordinary differential equations with a nonlinear dependence on the spectral parameter. We establish properties of the spectrum and analyze the inverse problem of reconstructing the coefficients of a differential equation on the basis of the so-called Weyl functions. For this inverse problem, we prove a uniqueness theorem and obtain a procedure for constructing the solution by the method of spectral mapping.     

On the optimal control of integral-functional equations

The problem of the optimal control of stochastic integral-functional equations of neutral type with an intergral quality functional is considered. For the case of a linear quadratic problem an explicit form of the optimal control is presented.

A class of equations which originated in the synthesis of Volterra equations, and stochastic differential equations with after-effects of neutral type are discussed. The problem of the optimal control of such systems is an essential development of the theory of controlled differential equations /1–8/. Examples of real objects whose mathematical models contain equations with an after-effect are discussed in /9/. A study of integral equations of neutral type is essential in controlling the motion of bodies in a continuous medium, /10/. Volterra equations first arose in the theory of creep and form the basis of this theory /11, 12/.     

On the self-adjoint nonlinear eigenvalue problem for Hamiltonian systems of ordinary differential equations with singularities

Certain properties of the nonlinear self-adjoint eigenvalue problem for Hamiltonian systems of ordinary differential equations with singularities are examined. Under certain assumptions on the way in which the matrix of the system and the matrix specifying the boundary condition at a regular point depend on the spectral parameter, a numerical method is proposed for determining the number of eigenvalues lying on a prescribed interval of the spectral parameter.     

Determination of the number of an eigenvalue of a singular nonlinear self-adjoint spectral problem for a linear Hamiltonian system of differential equations

We suggest a method for determining the number of an eigenvalue of a self-adjoint spectral problem nonlinear with respect to the spectral parameter, for some class of Hamiltonian systems of ordinary differential equations on the half-line. The standard boundary conditions are posed at zero, and the solution boundedness condition is posed at infinity. We assume that the matrix of the system is monotone with respect to the spectral parameter. The number of an eigenvalue is determined by the properties of the corresponding nontrivially solvable homogeneous boundary value problem. For the considered class of systems, it becomes possible to compute the numbers of eigenvalues lying in a given range of the spectral parameter without finding the eigenvalues themselves.     

On Differential Operators with Singularity and Discontinuity Conditions inside an Interval

We investigate boundary-value problems for differential equations with singularity and discontinuity conditions inside an interval. We describe properties of the spectrum, prove a theorem on the completeness of eigenfunctions and associated functions, and study the inverse spectral problem.     

Implicit operator differential equations and applications to electrodynamics

A cylindrical waveguide with layered dispersive medium is considered. An implicit operator differential equation of second order is obtained by separating variables in the corresponding boundary‐value problem for Maxwell's equations. Existence and uniqueness theorems are proved and explicit formulae for solutions are given. The spectral theory of operators and operator sheaves are used. Copyright © 2000 John Wiley & Sons, Ltd.     

On recovering differential systems on a finite interval from spectra

Inverse spectral problems are studied for non-self-adjoint systems of ordinary differential equations on a finite interval. We establish properties of spectral characteristics and provide a procedure for constructing the solution of the inverse problem of recovering the coefficients of differential systems from given spectra. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2006, Vol. 255, pp. 273–287.     

Nonlinear eigenvalue problem for second-order Hamiltonian systems

The nonlinear self-adjoint eigenvalue problem for a Hamiltonian system of two ordinary differential equations is examined under the assumption that the matrix of the system is a monotone function of the spectral parameter. Certain properties of eigenvalues that were previously established by the authors for Hamitonian systems of arbitrary order are now worked out in detail and made more precise for the above system. In particular, a single second-order ordinary differential equation is analyzed.     

On two-parameter spectral problems and their application

We study a spectral problem with two complex parameters for a normal linear system of second-order ordinary differential equations on a closed interval with splitting or nonlocal boundary conditions. The results of this study are used to prove the existence and uniqueness of a generalized solution of a boundary value problem in a cylinder for a class of partial differential equations.     

Recovery of differential equations with nonlinear dependence on the spectral parameter

The inverse spectral problem of recovering pencils of second-order differential operators on the half-line is studied. We give a formulation of the inverse problem, prove the uniqueness theorem and provided a procedure for constructing the solution of the inverse problem. We also establishe connections with inverse problems for partial differential equations.     

一类分数阶多项延迟微分方程的Jacobi谱配置方法

本文利用Jacobi谱配置方法数值求解了一类分数阶多项延迟微分方程,并证明了该方法是收敛的,通过若干数值算例验证了相应的理论结果,结果表明Jacobi谱配置方法求解这类方程是非常高效的,同时也为这类分数阶延迟微分方程的数值求解提供了新的选择,对分数阶泛函方程的数值方法的研究有一定的指导意义.     

A method for solving nonlinear spectral problems for a class of systems of differential algebraic equations

A method for calculating eigenvalues of a nonlinear spectral problem for one class of linear differential algebraic equations is proposed under the assumption of an analytical dependence on spectral parameter of the matrices appearing in the system of equations and the matrices determining boundary conditions.     

Hyperbolicity of periodic solutions of functional differential equations with several delays

We study conditions for the hyperbolicity of periodic solutions to nonlinear functional differential equations in terms of the eigenvalues of the monodromy operator. The eigenvalue problem for the monodromy operator is reduced to a boundary value problem for a system of ordinary differential equations with a spectral parameter. This makes it possible to construct a characteristic function. We prove that the zeros of this function coincide with the eigenvalues of the monodromy operator and, under certain additional conditions, the multiplicity of a zero of the characteristic function coincides with the algebraic multiplicity of the corresponding eigenvalue.     

Low-order spectral models of the atmospheric circulation: A survey

A quasi-geostrophic potential vorticity equation is derived from the Navier-Stokes equations for atmospheric motions. It describes the evolution of a quasi-horizontally flow on time scales of a few days and more. The associated boundary-value problem is analyzed by projection of the equation onto orthonormal eigenfunctions (modes) of a Sturm-Liouville operator. The result is a spectral model, consisting of an infinite number of nonlinear ordinary differential equations for the evolution of the mode amplitudes. Low-order spectral models, in which only a few modes are resolved, appear to have properties which agree with observations of the atmospheric circulation. However, little justification is available for truncating the spectral expansion at low resolution numbers. It is argued that stochastic forcing terms should be added to the equations, but it is not a priori clear how they should be specified.A derivation is presented of a specific low-order spectral model of the quasi-geostrophic potential vorticity equation. Some of its subsystems are analyzed for their physical and mathematical properties. It appears that topography can act as a triggering mechanism to generate multiple equilibria. The corresponding flow patterns resemble preference states of the atmospheric circulation. The systems can vacillate between three characteristic regimes with transitions provided either by external or internal mechanisms. A discussion is presented on the validity of stochastically forced spectral models and deterministic chaotic models for the atmospheric circulation.Present affiliation: Institute of Meteorology and Oceanography, Princetonplein 5, 3584 CC, Utrecht, The Netherlands.     

Solution of three-dimensional problems of the theory of elasticity for a picewise homogeneous medium with constant poisson''s ratio : PMM vol. 43, no. 6, 1979, pp. 1122–1125

Singular integral equations of the theory of elasticity are studied for a piecewise homogeneous medium with the same Poisson's ratio. It is shown that a solution can be obtained using the method of successive approximations. Use of the potential method for the fundamental problems of the theory of elasticity leads to singular integral equations of second kind [1], In the case of the second internal and external problems, and of the first internal problem, the spectral properties of the integral operators allow the use of the method of successive approximations to obtain a solution.