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.     

Numerical solution of the differential equation describing the behavior of the zeroth-order boundary function

The asymptotic solution of the integro-differential plasma-sheath equation is considered. This equation is singularly perturbed because of the small coefficient multiplying the highest order (second) derivative. The asymptotic solution is obtained by the boundary function method. Equations are derived for the first two coefficients in the form of both a regular series expansion and an expansion in boundary functions. The equation for the first coefficient of the regular series has only a trivial solution. A numerical algorithm is considered for the solution of the second-order differential equation describing the behavior of the zeroth-order boundary function. The proposed algorithm efficiently solves the boundary-value problem and produces a well-behaved solution of the Cauchy problem. __________ Translated from Prikladnaya Matematika i Informatika, No. 23, pp. 24–35, 2006.     

On applying the Laplace transform method to an equation describing an axially moving string

In this paper an initial‐boundary value problem for a linear, nonhomogeneous axially moving string equation will be considered. The velocity of the string is assumed to be constant, and the nonhomogeneous terms in the string equation are due to external forces acting on the string. The Laplace transform method will be used to construct the solution of the problem. It will turn out that the method has considerable, computational advantages compared to the usually applied method of modal analysis based on eigenfunction expansions. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

由导集运算定义拓扑的方法

对点集拓扑学中由导集运算决定拓扑的方法进行了讨论,主要结果是,设X是一个集合,d*:P(X)→P(X)是集值映射,若d*满足:A,B∈P(X),(1)d*()=,(2)d*(A∪B)=d*(A)∪d*(B),(3)d*(d*(A))A∪d*(A),(4)d*(A)={x∈X x∈d*(A-{x})},则存在X的唯一拓扑T,使得在拓扑空间(X,T)中,A∈P(X),d(A)=d*(A).     

A trace formula of a boundary value problem for the operator Sturm-Liouville equation

We obtain a regularized trace formula for the operator Sturm-Liouville equation with a boundary condition depending on a spectral parameter.     

Characteristic boundary value problem for a third-order functional-differential equation with the Bianchi operator

We consider a version of the Goursat problem for a previously unstudied third-order equation and prove the existence and uniqueness of the solution of this problem.     

The boundary quenching behavior of a semilinear parabolic equation

In this paper we consider the boundary quenching behavior of a semilinear parabolic problem in one-dimensional space, of which the nonlinearity appears both in the source term and in the Neumann boundary condition. First we proved that the solution quenches at somewhere in some finite time. Then we assert that the quenching can only occur on the left boundary if the given initial datum is monotone. Finally we derived the upper and lower bounds for the quenching rate of the solution near the quenching time. Thus we generalized our former results.     

Fredholm property of boundary value problems for a fourth-order elliptic differential-operator equation with operator boundary conditions

In a Hilbert space H, we study the Fredholm property of a boundary value problem for a fourth-order differential-operator equation of elliptic type with unbounded operators in the boundary conditions. We find sufficient conditions on the operators in the boundary conditions for the problem to be Fredholm. We give applications of the abstract results to boundary value problems for fourth-order elliptic partial differential equations in nonsmooth domains.     

A boundary function method for solving the model lighthill equation with a regular singular point

We prove that it is possible to apply a method similar to the Vishik-Lyusternik-Vasil’eva-Imanaliev boundary function method for constructing the asymptotics of the solution of the model Lighthill equation with a regular singular point.     

An iterative regularization method for the 2-constrained pseudoinversion of an operator equation

An iterative regularization algorithm is proposed for solving a special optimization problem, the so-called 2-constrained operator pseudoinversion. The convergence of the algorithm is examined in the case of perturbed input data. An error estimate is derived, and an a priori choice of the regularization parameters is described. The algorithm is applied to an optimal control problem with minimal costs.     

Expanding the asymptotic explosive boundary behavior of large solutions to a semilinear elliptic equation

The main goal of this paper is to study the asymptotic expansion near the boundary of the large solutions of the equation