Generalization of the Lagrange Method to the Case of Second-Order Linear Differential Equations with Constant Operator Coefficients in Locally Convex Spaces

The well-known Lagrange method for linear inhomogeneous differential equations is generalized to the case of second-order equations with constant operator coefficients in locally convex spaces. The solutions are expressed in terms of uniformly convergent functional vector-valued series generated by a pair of elements of a locally convex space. Sufficient conditions for the continuous dependence of solutions on the generating pair are obtained. The solution of the Cauchy problem for the equations under consideration is also obtained and conditions for its existence and uniqueness are given. In addition, under certain conditions, the so-called general solution of the equations (a function of most general form from which any particular solution can be derived) is obtained. The study is carried out using the characteristics (order and type) of an operator and of a sequence of operators. Also, the convergence of operator series with respect to equicontinuous bornology is used.     

An operator method of solving the Cauchy problem for a homogeneous system of partial differential equations

On the basis of a generalized separation-of-variables method we propose an operator method of constructing the solution of the Cauchy problem for a homogeneous system of partial differential equations of first order with respect to time and of infinite order with respect to the spatial variables. Translated fromMatematichni Metody i Fiziko-Mekhanichni Polya, Vol. 38, 1995.     

An analog of the method of continuation of solution with respect to the parameter for nonlinear operator equations

A process of second order is constructed for the solution of nonlinear operator equations which is an analog of the method of continuation of solution with respect to the parameter. For each value of the parameter the Newton-Kantorovich iteration formula is applied only once in all. The quadratic convergence of the process is ensured by the specification of the parameter by a special formula. The process under consideration enables us to avoid the singular points of the derivative of the nonlinear operator on the left-hand side of the operator equation.Translated from Matematicheskie Zametki, Vol. 23, No. 4, pp. 601–606, April, 1978.     

Homogeneous differential-operator equations in locally convex spaces

We describe a general method that allows us to find solutions to homogeneous differential-operator equations with variable coefficients by means of continuous vector-valued functions. The “homogeneity” is not interpreted as the triviality of the right-hand side of an equation. It is understood in the sense that the left-hand side of an equation is a homogeneous function with respect to operators appearing in that equation. Solutions are represented as functional vector-valued series which are uniformly convergent and generated by solutions to a kth order ordinary differential equation, by the roots of the characteristic polynomial and by elements of a locally convex space. We find sufficient conditions for the continuous dependence of the solution on a generating set. We also solve the Cauchy problem for the considered equations and specify conditions for the existence and the uniqueness of the solution. Moreover, under certain hypotheses we find the general solution to the considered equations. It is a function which yields any particular solution. The investigation is realized by means of characteristics of operators such as the order and the type of an operator, as well as operator characteristics of vectors, namely, the operator order and the operator type of a vector relative to an operator. We also use a convergence of operator series with respect to an equicontinuous bornology.     

On the asymptotic order of accuracy of Tikhonov regularization

In this paper, the rate of convergence and the order of accuracy (with respect to the error level in the data) of Tikhonov's method for approximating the minimal-norm least-square solution of an ill-posed operator equation is investigated. It is shown that, in general, this rate of convergence is arbitrarily small. It is further shown how this rate depends on some smoothness properties of the solution. All results describe optimal orders.     

An analytic method for the initial value problem of the electric field system in vertical inhomogeneous anisotropic media

The time-dependent system of partial differential equations of the second order describing the electric wave propagation in vertically inhomogeneous electrically and magnetically biaxial anisotropic media is considered. A new analytical method for solving an initial value problem for this system is the main object of the paper. This method consists in the following: the initial value problem is written in terms of Fourier images with respect to lateral space variables, then the resulting problem is reduced to an operator integral equation. After that the operator integral equation is solved by the method of successive approximations. Finally, a solution of the original initial value problem is found by the inverse Fourier transform.     

The numerical solution of an evolution problem of second order in time on a closed smooth boundary

We consider an initial value problem for the second-order differential equation with a Dirichlet-to-Neumann operator coefficient. For the numerical solution we carry out semi-discretization by the Laguerre transformation with respect to the time variable. Then an infinite system of the stationary operator equations is obtained. By potential theory, the operator equations are reduced to boundary integral equations of the second kind with logarithmic or hypersingular kernels. The full discretization is realized by Nyström's method which is based on the trigonometric quadrature rules. Numerical tests confirm the ability of the method to solve these types of nonstationary problems.     

On the Large-Time Behavior of Solutions to the Cauchy Problem for a 2-dimensional Discrete Kinetic Equation

Existence of global solution for a 2-dimensional discrete equation of kinetics and expansion with respect to smoothness are obtained, and the effect of progressing waves generated by the operator of interaction is investigated.     

Modified combined grid method for solving the Dirichlet problem for the Laplace equation on a rectangular parallelepiped

A modified combined grid method is proposed for solving the Dirichlet problem for the Laplace equation on a rectangular parallelepiped. The six-point averaging operator is applied at next-to-the-boundary grid points, while the 18-point averaging operator is used instead of the 26-point one at the remaining grid points. Assuming that the boundary values given on the faces have fourth derivatives satisfying the Hölder condition, the boundary values on the edges are continuous, and their second derivatives obey a matching condition implied by the Laplace equation, the grid solution is proved to converge uniformly with the fourth order with respect to the mesh size.     

On a combined grid method for solving the Dirichlet problem for the Laplace equation in a rectangular parallelepiped

A combined grid method for solving the Dirichlet problem for the Laplace equation in a rectangular parallelepiped is proposed. At the grid points that are at the distance equal to the grid size from the boundary, the 6-point averaging operator is used. At the other grid points, the 26-point averaging operator is used. It is assumed that the boundary values have the third derivatives satisfying the Lipschitz condition on the faces; on the edges, they are continuous and their second derivatives satisfy the compatibility condition implied by the Laplace equation. The uniform convergence of the grid solution with the fourth order with respect to the grid size is proved     

Local and global estimates of the solutions of the Cauchy problem for quasilinear parabolic equations with a nonlinear operator of Baouendi-Grushin type

In this paper, the qualitative properties of the solutions of the Cauchy problem for degenerate parabolic equations with a nonlinear operator of Baouendi-Grushin type are studied. Sharp local and global (with respect to the spatial and temporal variables) estimates of the solution are obtained. The property of the finiteness of the support of the solution is established.     

EXISTENCE OF SOLUTION AND APPROXIMATE CONTROLLABILITY OF A SECOND-ORDER NEUTRAL STOCHASTIC DIFFERENTIAL EQUATION WITH STATE DEPENDENT DELAY

This paper has two sections which deals with a second order stochastic neutral partial differential equation with state dependent delay. In the first section the existence and uniqueness of mild solution is obtained by use of measure of non-compactness. In the second section the conditions for approximate controllability are investigated for the distributed second order neutral stochastic differential system with respect to the approximate controllability of the corresponding linear system in a Hilbert space. Our method is an extension of co-author N. Sukavanam's novel approach in [22]. Thereby, we remove the need to assume the invertibility of a controllability operator used by authors in [5], which fails to exist in infinite dimensional spaces if the associated semigroup is compact. Our approach also removes the need to check the invertibility of the controllability Gramian operator and associated limit condition used by the authors in [20], which are practically difficult to verify and apply. An example is provided to illustrate the presented theory.     

An existence principle for a periodic solution of a differential equation in Banach space

The equation d²x/dt²=Ax +f(t, x) is considered in a Banach space E, where A is a fixed unbounded linear operator, andf(t, x) is a nonlinear operator which is periodic in t and satisfies a Lipschitz condition with respect to x E. Existence conditions have been obtained for a well defined generalized periodic solution of this equation, and also when this solution coincides with the true solution. Similar results have been obtained for the first order equation.Translated from Matematicheskie Zametki, Vol. 4, No. 1, pp. 105–112, July, 1968.     

Uniform operator continuity of the stationary Riccati equation in Hilbert space

In this paper the continuity in the uniform operator topology of the solution of the stationary Riccati equation in Hilbert space as a function of parameters is verified. The assumptions for this verification are the uniform operator continuity of the uncontrolled semigroup with respect to parameters, the uniform finiteness of the infimum of the quadratic cost functionals over the admissible controls, and uniform detectability. Some families of semigroups are described that satisfy the condition of continuity in the uniform operator topology with respect to parameters. The uniform operator continuity of the solution of the stationary Riccati equation with respect to parameters is important for applications to problems in adaptive control of stochastic evolution systems.This research was partially supported by NSF Grant ECS-8718026.     

On the torus automorphisms: analytic solution, computability and quantization

The exact solution of the classical torus automorphism, which partial case is Arnold Cat map is obtained and compared with the numerical solution. The torus, considered as the classical phase space admits the quantization in terms of the Weyl pair. The remarkable fact is that quantum map, as the evolution with respect to the discrete time, preserves the Weyl commutation relation. We have obtained also the operator solution of this quantum torus automorphism.     

Equations of anisotropic elastodynamics as a symmetric hyperbolic system: Deriving the time-dependent fundamental solution

The dynamic system of anisotropic elasticity from three second order partial differential equations is written in the form of the time-dependent first order symmetric hyperbolic system with respect to displacement velocity and stress components. A new method of deriving the time-dependent fundamental solution of the obtained system is suggested in this paper. This method consists of the following. The Fourier transform image of the fundamental solution with respect to a space variable is presented as a power series expansion relative to the Fourier parameters. Then explicit formulae for the coefficients of these power series are derived successively. Using these formulae the computer calculation of fundamental solution components (displacement velocity and stress components arising from pulse point forces) has been made for general anisotropic media (orthorhombic and monoclinic) and the simulation of elastic waves has been obtained. These computational examples confirm the robustness of the suggested method.     

Existence of smooth solutions of problems for parabolic systems with convex constraints on the boundary of the domain

The existence of a smooth solution for problems with a convex constraint on the boundary is proved. The parabolic operator has a diagonal form and a quadratic growth with respect to the gradient. The obtained solution has the maximal possible regularity for problems with boundary obstacles.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 171, pp. 5–11, 1989.     

Applying the dual operator formalism to derive the zeroth-order boundary function of the plasma-sheath equation

The article constructs the asymptotic solution of the Tonks-Langmuir integro-differential equation with an Emmert kernel, which describes the potential both in the bulk plasma and in a narrow boundary layer. Equations of this type are singularly perturbed, because the highest order (second) derivative is multiplied by a small coefficient. The asymptotic solution is obtained by the boundary function method. The second-order differential equation describing the behavior of the zeroth-order boundary function is investigated using the dual operator formalism — an analog of the conjugate operator in the linear theory. The application of this formalism has produced an asymptotic solution and has also made it possible to propose a number of homogeneous discrete three-point schemes for solving the equation. __________ Translated from Prikladnaya Matematika i Informatika, No. 22, pp. 76–90, 2005.     

GAUSS-SEIDEL-TYPE MULTIGRID METHODS

By making use of the Gauss-Seidel-type solution method, the procedure for computing the interpolation operator of multigrid methods is simplified. This leads to a saving of computational time. Three new kinds of interpolation formulae are obtained by adopting different approximate methods, to try to enhance the accuracy of the interpolatory oper-ator. A theoretical study proves the two-level convergence of these Gauss-Seidel-type MG methods. A series of numerical experiments is presented to evaluate the relative perfor-mance of the methods with respect to the convergence factor, CPU-time(for one V-cycle and the setup phase) and computational complexity.     

解第一类算子方程的一种新的正则化方法

对算子与右端都为近似给定的第一类算子方程提出一种新的正则化方法，依据广义Ａｒｃａｎｇｅｌｉ方法选取正则参数，建立了正则解的收敛性。这种新的正则化方法与通常的Ｔｉｋｈｏｎｏｖ正则化方法相比较，提高了正则解的渐近阶估计。