A Nonlocal Boundary-Value Problem for Differential Equations with Fractional Time Derivative with Variable Coefficients

Conditions for the existence and uniqueness of a classical solution of a nonlocal boundary-value problem for a differential equation with a regularized Riemann–Liouville fractional time derivative with variable coefficients are investigated.     

A boundary-value problem for parabolic equations with general nonlocal conditions

We study a boundary-value problem with general two-point conditions with respect to the time coordinate, and periodic conditions on the spatial coordinates for Shilov-parabolic equations with constant coefficients. We construct the solution in the form of a Fourier series. We establish conditions for existence and uniqueness of a classical solution of the problem. We prove quantitative theorems on a lower bound for the small denominators that arise in solving the problem. Translated fromMatematichni Methody i Fiziko-mekhanichni Polya, Vol. 38, 1995.     

One-Sided Nonlocal Boundary-Value Problem for Singular Parabolic Equations

In spaces of classical functions with power weight, we prove the existence and uniqueness of a solution of a one-sided nonlocal boundary-value problem for parabolic equations with an arbitrary power order of degeneracy of coefficients. We obtain an estimate for the solution of this problem in the corresponding spaces.     

Nonlocal boundary-value problem for parabolic equations with variable coefficients

We study the boundary-value problem for Petrovskii parabolic equations of arbitrary order with variable coefficients with conditions nonlocal in time. We establish conditions for the existence and uniqueness of a classical solution of this problem and prove metric theorems on lower bounds of small denominators appearing in the construction of a solution of the problem.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 7, pp. 915–921, July, 1995.This work was partially supported by the Foundation for Fundamental Research of the Ukrainian State Committee on Science and Technology.     

Global solutions and invariant tori of difference-differential equations

We prove the existence of an m-parameter family of global solutions of a system of difference-differential equations. For difference-differential equations on a torus, we introduce the notion of rotation number. We also consider the problem of perturbation of an invariant torus of a system of difference-differential equations and study the problem of the existence of periodic and quasiperiodic solutions of second-order difference-differential equations.     

关于耦合型对流-扩散方程组的奇异摄动边值问题

本文考虑下述耦合型对流-扩散方程组的奇异摄动边值问题:本文提出两种方法:一种是初值化解法,用这种方法,原始问题转化成一系列没有扰动的一阶微分方程或方程组的初值问题,从而得到一个渐近展开式;第二种是边值化解法,用这种方法,原始问题转化成一组没有边界层现象的边值问题,从而可以得到精确解和使用经典的数值方法去得到具有关于摄动参数ε一致的高精度数值解.     

Solvability of nonlinear boundary-value problems arising in modeling plasma diffusion across a magnetic field and its equilibrium configurations

We study the simplest one-dimensional model of plasma density balance in a tokamak type system, which can be reduced to an initial boundary-value problem for a second-order parabolic equation with implicit degeneration containing nonlocal (integral) operators. The problem of stabilizing nonstationary solutions to stationary ones is reduced to studying the solvability of a nonlinear integro-differential boundary-value problem. We obtain sufficient conditions for the parameters of this boundary-value problem to provide the existence and the uniqueness of a classical stationary solution, and for this solution we obtain the attraction domain by a constructive method.     

Improved scales of spaces and elliptic boundary-value problems. III

We study elliptic boundary-value problems in improved scales of functional Hilbert spaces on smooth manifolds with boundary. The isotropic Hörmander-Volevich-Paneyakh spaces are elements of these scales. The local smoothness of a solution of an elliptic problem in an improved scale is investigated. We establish a sufficient condition under which this solution is classical. Elliptic boundary-value problems with parameter are also studied.     

The Dirichlet Problem for an Elliptic System of Second-Order Equations with Constant Real Coefficients in the Plane

A solution of the Dirichlet problem for an elliptic systemof equations with constant coefficients and simple complex characteristics in the plane is expressed as a double-layer potential. The boundary-value problem is solved in a bounded simply connected domain with Lyapunov boundary under the assumption that the Lopatinskii condition holds. It is shown how this representation is modified in the case of multiple roots of the characteristic equation. The boundary-value problem is reduced to a system of Fredholm equations of the second kind. For a Hölder boundary, the differential properties of the solution are studied.     

Approximate controllability and homogenization of a semilinear elliptic problem

The L²- and H¹-approximate controllability and homogenization of a semilinear elliptic boundary-value problem is studied in this paper. The principal term of the state equation has rapidly oscillating coefficients and the control region is locally distributed. The observation region is a subset of codimension 1 in the case of L²-approximate controllability or is locally distributed in the case of H¹-approximate controllability. By using the classical Fenchel-Rockafellar's duality theory, the existence of an approximate control of minimal norm is established by means of a fixed point argument. We consider its asymptotic behavior as the rapidly oscillating coefficients H-converge. We prove its convergence to an approximate control of minimal norm for the homogenized problem.     

On the steady propagation of a semi-infinite crack

We consider the rectilinear propagation of a semi-infinite crack with constant velocity in a crystal structure. We obtain the solutions of homogeneous boundary-value problems for the corresponding difference-differential operators in spaces of one and two dimensions. We give a justification of the computational aspect of the problem. Bibliography: 8 titles.Translated fromProblemy Matematicheskogo Analiza, No. 12, 1992, pp. 127–153.     

On the Global-in-Time Existence of a Generalized Solution to a Free-Boundary Problem

A problem with free (unknown) boundary for a one-dimensional diffusion-convection equation is considered. The unknown boundary is found from an additional condition on the free boundary. By the extension of the variables, the problem in an unknown domain is reduced to an initial boundary-value problem for a strictly parabolic equation with unknown coefficients in a known domain. These coefficients are found from an additional boundary condition that enables the construction of a nonlinear operator whose fixed points determine a solution of the original problem.

     

The b and h functions for integral equations with displacement kernels: A computational method and an application to radiative transfer

Many problems in the theory of radiative transfer reduce to the solution of Fredholm integral equations with displacement kernels. Frequently, we are interested in the solutions of the Fredholm integral equations as well as certain functionals on the solution (reflection and transmission coefficients, etc.). Earlier it was shown that these functionals can be expressed algebraically in terms of the basic functions b and h. Normally, these functions are computed as solutions of an initial-value problem. Since they represent internal interactions due to isotropic illuminations, they are also solutions of a linear two-point boundary-value problem, which, unfortunately, s unstable. The purpose of this paper is to show that this unstable problem can be solved using a Gram-Schmidt orthogonalization scheme. This is demonstrated by making comparisons against earlier calculations using the initial-value method. In addition, the process is ideally suited to take advantage of multitasking on parallel processors.     

Asymptotically Equivalent Singular Perturbation Problems

A study is made of a two-point nonlinear boundary-value problem with a small parameter multiplying the highest derivative. It is shown that under certain circumstances the asymptotic solution to the problem is expressible in terms of the solution to a linear boundary-value problem—in which case the two problems are said to be asymptotically equivalent. The coefficients of the linear problem necessarily satisfy certain conditions, and these conditions are shown to bear a close relationship to the equations obtained in constructing a solution to the nonlinear problem by standard matching methods.     

Diffraction of light by supersonic waves: The solution of the raman-nath equations—I

In the problem of the diffraction of light by a supersonic wave, at normal incidence of the light, the solution of the system of difference-differential equations of Raman and Nath, for the amplitudes of the diffracted light beams, is reduced to the integration of a partial differential equation. The coefficients of the Laurent expansion of the solution of the latter equation yield the expressions for the amplitudes of the diffracted light waves. The partial differential equation has been integrated for two approximations. (1) Forρ=0, the well-known results of Raman and Nath’s preliminary theory are re-established. (2) Forρ?1 a power series inρ, the terms of which are calculated as far as the third one, leads to the solution of Mertens and Berry obtained by a perturbation method.     

Solvability of nonlinear boundary-value problems arising in modeling plasma diffusion across a magnetic field and its equilibrium configurations

We study the simplest one-dimensional model of plasma density balance in a tokamak type system, which can be reduced to an initial boundary-value problem for a second-order parabolic equation with implicit degeneration containing nonlocal (integral) operators. The problem of stabilizing nonstationary solutions to stationary ones is reduced to studying the solvability of a nonlinear integro-differential boundary-value problem. We obtain sufficient conditions for the parameters of this boundary-value problem to provide the existence and the uniqueness of a classical stationary solution, and for this solution we obtain the attraction domain by a constructive method.Translated from Matematicheskie Zametki, vol. 77, no. 2, 2005, pp. 219–234.Original Russian Text Copyright © 2005 by G. A. Rudykh, A. V. Sinitsyn.This revised version was published online in April 2005 with a corrected issue number.     

Criteria of unique solvability of nonlocal boundary-value problem for systems of hyperbolic equations with mixed derivatives

We consider nonlocal boundary-value problem for a system of hyperbolic equations with two independent variables. We investigate questions of existence of unique classical solution to problem under consideration. In terms of initial data we propose criteria of unique solvability and suggest algorithms of finding of solutions to nonlocal boundary-value problem. As an application we give conditions of solvability of periodic boundary-value problem for a system of hyperbolic equations.     

Diffraction of light by supersonic waves: The solution of the raman-nath equations

The partial differential equation associated with the system of difference-differential equations of Raman-Nath for the amplitudes of the diffracted light-waves is solved exactly by the method of the separation of the variables. The solution is presented as a double infinite series containing the Fourier coefficients of the even periodic Mathieu functions with periodπ and the corresponding eigenvalues. Considering this solution as a Laurent series in one of the variables, the Laurent coefficients immediately give the exact expressions for the amplitudes of the diffracted light-waves, from which the formulae for the intensities are calculated. The connection between the Raman-Nath method and Brillouin’s Mathieu function method has thus been achieved.     

Boundary-value problems for an elliptic equation with complex coefficients and a certain moment problem

Elliptic systems of two second-order equations, which can be written as a single equation with complex coefficients and a homogeneous operator, are studied. The necessary and sufficient conditions for the connection of traces of a solution are obtained for an arbitrary bounded domain with a smooth boundary. These conditions are formulated in the form of a certain moment problem on the boundary of a domain; they are applied to the study of boundary-value problems. In particular, it is shown that the Dirichlet problem and the Neumann problem are solvable only together. In the case where the domain is a disk, the indicated moment problem is solved together with the Dirichlet problem and the Neumann problem. The third boundary-value problem in a disk is also investigated.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 11, pp. 1476–1483, November, 1993.     

Diffraction of light by superposed parallel supersonic waves,one being then-th harmonic of the other. Solution of the system of difference-differential equations for the amplitudes by a series expansion method

In the problem of the diffraction of light by two parallel supersonic waves, consisting of a fundamental tone and itsn-th harmonic, the solution of the system of difference-differential equations for the amplitudes has been reduced to the integration of a partial differential equation. The expressions for the amplitudes of the diffracted light waves are obtained as the coefficients of the Laurent expansion of the solution of this partial differential equation. The latter has been integrated for two approximations:

1. Forρ = 0, the results of Murty’s elementary theory are reestablished.
2. Forρ ≤ 1, a power series inρ, the terms of which are calculated as far as the third one, leads to a new expression for the intensities of the diffracted light waves, verifying the general symmetry properties obtained by Mertens.