有资格限制的指派问题的求解方法

在实际的指派工作中，常会遇到某个人有没有资格去承担某项工作的问题，因此，本建立了有资格限制的指派问题的数学模型。在此数学模型中，将效益矩阵转化为判定矩阵，由此给出了判定此种指派问题是否有解的方法；在有解的情况下，进一步将效益矩阵转化为求解矩阵，从而将有资格限制的指派问题化为传统的指派问题来求解。最后给出了一个数值例子来说明这样的处理方法是有效的。     

The solvability of a coupled thermoviscoelastic contact problem with small Coulomb friction and linearized growth of frictional heat

A coupled thermoviscoelastic frictional contact problem is investigated. The contact is modelled by the Signorini condition for the displacement velocities and the friction by the Coulomb law. The heat generated by friction is described by a non‐linear boundary condition with at most linear growth. The weak formulation of the problem consists of a variational inequality for the elasticity part and a variational equation for the heat conduction part. In order to prove the existence of a solution to this problem we first use an approximation of the Signorini condition by the penalty method. The existence of a solution for the approximate problem is shown using the fixed‐point theorem of Schauder. This theorem is applied to the composition of the solution operator for the contact problem with given temperature field and the solution operator for the heat equation problem with known displacement field. To obtain this proof, the unique solvability of both problems is necessary. Due to this reason it is necessary to introduce the penalty method. While the penalized contact problem has a unique solution, this is not clear for the original contact problem. The solvability of the original frictional contact problem is verified by an investigation of the limit for vanishing penalty parameter. Copyright © 1999 John Wiley & Sons, Ltd.     

Uniqueness of finding the coefficient of the derivative in a first order nonlinear equation

The problem of using an additional boundary condition to find a coefficient that depends on the spatial variable is considered. The existence and uniqueness of the solution to the direct problem is studied. The solution to the direct problem is proved to be stable with respect to the sought coefficient. Uniqueness conditions for the solution to the coefficient inverse problem are described.     

On the existence and uniqueness of a generalized solution of the mixed problem for the wave equation with the second and third boundary conditions

We prove the uniqueness of a generalized solution of an initial-boundary value problem for the wave equation with boundary conditions of the third and second kind. In addition, we find a closed-form expression for the analytic solution of that problem with zero initial data. The result plays an important role in the investigation of the boundary control problem. We show how to use the obtained solution for the investigation of the boundary control problem in the case of subcritical time intervals for which the solution of the boundary control problem, if it exists at all, is unique. We obtain necessary and sufficient conditions for the existence of a unique solution in a class admitting the existence of finite energy.     

The axisymmetric mixed problem of elasticity theory for a cone clamped along its side surface with an attached spherical segment

The axisymmetric mixed problem of the stress state of an elastic cone, with a spherical segment attached to the base, is considered. The side surface of the cone is rigidly clamped, while the surface of the spherical segment is under a load. By using a new integral transformation over the meridial angle the problem is reduced in transformant space to a vector boundary value problem, the solution of which is constructed using the solution of a matrix boundary value problem. The unknown function (the derivative of the displacements), which occurs in the solution, is determined from the approximate solution of a singular integral equation, for which a preliminary investigation is carried out of the nature of the singularity of the function at the ends of the integration interval. Subsequent use of inverse integral transformations leads to the final solution of the initial problem. The values of the stresses obtained are compared with those that arise in the cone for a similar load, when sliding clamping conditions are specified on the side surface of the cone (for this case an exact solution of this problem is constructed, based on the known result).     

Numerical differentiation for the second order derivatives of functions of two variables

A regularized optimization problem for computing numerical differentiation for the second order derivatives of functions with two variables from noisy values at scattered points is discussed in this article. We prove the existence and uniqueness of the solution to this problem, provide a constructive scheme for the solution which is based on bi-harmonic Green's function and give a convergence estimate of the regularized solution to the exact solution for the problem under a simple choice of regularization parameter. The efficiency of the constructive scheme is shown by some numerical examples.     

Stochastic calculus of variations and mechanics

The Cauchy problem for the heat equation with a potential term (imaginary time analogue of the Schrodinger equation) is considered. After a logarithmic transformation, a positive solution to this Cauchy problem turns into a solution to the dynamic programming equation for a problem of stochastic calculus of variations. The latter problem is one of least average action. The classical principle of least action is obtained as an appropriate parameter tends to zero.     

Continuity in the parameter of the minimum value of an integral functional over the solutions of an evolution control system

The problem of minimization of an integral functional with an integrand that is nonconvex with respect to the control is considered. We minimize our functional over the solution set of a nonlinear evolution control system with a time-dependent subdifferential operator in a Hilbert space. The control constraint is given by a nonconvex closed bounded set. The integrand, the control constraint, the initial conditions and the operators in the equation describing the control system all depend on a parameter. We consider, along with the original problem, the problem of minimizing an integral functional with an integrand convexified with respect to the control over the solution set of the same system, but now subject to the convexified control constraint. By a solution of the control system we mean a “trajectory–control” pair. We prove that for each value of the parameter the convexified problem has a solution, which is the limit of a minimizing sequence of the original problem, and the minimum value of the functional of the convexified problem is a continuous function of the parameter.     

热传导方程小波解的点态收敛

本文主要考虑热传导方程uxx=ut,0≤x<1,t≥0;u(1,t)=g(t),其中边界条件g(t)为已知函数．此定解问题为一不适定问题,也就是说当边界条件有微小扰动时,将会引起解大的扰动．本文将利用多分辨率分析构造一小波解,且证明此解是适定的,并给出所定义小波解与定解问题的真正解在点态意义下的误差估计．     

Inverse problem for a quasilinear system of partial differential equations with a nonlocal boundary condition

An initial boundary-value problem for a quasilinear system of partial differential equations with a nonlocal boundary condition involving a delayed argument is considered. The existence of a unique solution to this problem is proved by reducing it to a system of nonlinear integral-functional equations. The inverse problem of finding a solution-dependent coefficient of the system from additional information on a solution component specified at a fixed point of space as a function of time is formulated. The uniqueness of the solution of the inverse problem is proved. The proof is based on the derivation and analysis of an integral-functional equation for the difference between two solutions of the inverse problem.     

Characterizations of optimal solution sets of convex infinite programs

In this paper, several Lagrange multiplier characterizations of the solution set of a convex infinite programming problem are given. Characterizations of solution sets of cone-constrained convex programs are derived as well. The procedure is then adopted to a semi-convex problem with convex constraints. For this problem, we present firstly a necessary and sufficient condition for optimality and secondly a characterization of its optimal solution set, based on a Lagrange multiplier associated with a given solution and on directional derivatives of the objective function.      

A Stability Estimate for a Solution in the Inverse Problem of Finding the Coefficients of Lower Derivatives

We consider the problem of finding the coefficients of the first derivatives in a second-order hyperbolic equation. The additional information is the trace of a solution and its normal derivative on the lateral surface of the cylindrical domain of some direct problem for the original equation. The impulse point source lies outside the domain in which the sought coefficients are determined and is a parameter of the problem. We suppose that the number of sources for which the trace of a solution is given coincides with the number of the coefficients to be determined. The main result of this article is a stability estimate for a solution to the inverse problem under consideration.     

Integral-Functional Equation Arising in the Study of an Inverse Problem for a Quasilinear Hyperbolic Equation

A problem with data on the characteristics is considered for a quasilinear hyperbolic equation. The inverse problem of determining two unknown coefficients of the equation from some additional information about the solution is posed. One of the unknown coefficients depends on the independent variable, and the other, on the solution of the equation. Uniqueness theorems are proved for the solution of the inverse problem. The proof is based on the derivation of the integro-functional equation and the analysis of the uniqueness of its solution.     

Optimal processes in the model of two-sector economy with an integral utility function

An infinite-horizon two-sector economy model with a Cobb–Douglas production function is studied for different depreciation rates, the utility function being an integral functional with discounting and a logarithmic integrand. The application of the Pontryagin maximum principle leads to a boundary value problem with special conditions at infinity. The presence of singular modes in the optimal solution complicates the search for a solution to the boundary value problem of the maximum principle. To construct the solution to the boundary value problem, the singular modes are written in an analytical form; in addition, a special version of the sweep algorithm in continuous form is proposed. The optimality of the extremal solution is proved.     

Unique Solvability of the Inverse Problem with a Time-Dependent Boundary Condition for a Sorption Model

A mixed initial boundary-value problem is considered for nonequilibrium sorption dynamics with inner-diffusion kinetics. The problem allows for convection and longitudinal diffusion and has a time-dependent boundary condition. This condition contains the time derivative of a solution component and constitutes the balance equation for the absorbed mixture near the output boundary of the sorption region—inside the diffusion barrier. Bounds on the solution of the direct problem are obtained: nonnegativity of the solution and its first time derivatives, and boundedness of the solution by known functions. The inverse problem of estimating the nonlinear system parameter—the sorption isotherm—is considered and a uniqueness theorem is proved.     

Stability estimates for a class of semi-linear ill-posed problems

We study a final value problem for a nonlinear parabolic equation with positive self-adjoint unbounded operator coefficients. The problem is ill-posed. The regularized equation is given by a modified quasi-reversibility method. For this regularization solution, the Hölder type stability estimate between the regularization solution and the exact solution is obtained.     

Grid solution of problem with unilateral constraints

The present study deals with the solution of a problem, defined in a three-dimensional domain, arising in fluid mechanics. Such problem is modelled with unilateral constraints on the boundary. Then, the problem to solve consists in minimizing a functional in a closed convex set. The characterization of the solution leads to solve a time-dependent variational inequality. An implicit scheme is used for the discretization of the time-dependent part of the operator and so we have to solve a sequence of stationary elliptic problems. For the solution of each stationary problem, an equivalent form of a minimization problem is formulated as the solution of a multivalued equation, obtained by the perturbation of the previous stationary elliptic operator by a diagonal monotone maximal multivalued operator. The spatial discretization of such problem by appropriate scheme leads to the solution of large scale algebraic systems. According to the size of these systems, parallel iterative asynchronous and synchronous methods are carried out on distributed architectures; in the present study, methods without and with overlapping like Schwarz alternating methods are considered. The convergence of the parallel iterative algorithms is analysed by contraction approaches. Finally, the parallel experiments are presented.     

Verification of software codes for simulation of unsteady flows in a gas centrifuge

A simple semi-analytical solution is proposed for the problem of an unsteady gas flow in a gas centrifuge. The circulation in the centrifuge is driven by a source/sink of energy and by an external force (deceleration/acceleration of the gas rotation) acting on the gas at a given frequency. In the semi-analytical solution, the rotor is infinite, while the given forces vary harmonically with a given wave-length along the axial coordinate. As a result, the unsteady flow problem is reduced to a system of ordinary differential equations, which can be quickly solved to any prescribed accuracy. This problem is proposed for verifying numerical codes designed for the simulation of unsteady processes in gas centrifuges. A similar unsteady problem is solved numerically, in which case the cylinder is finite with the rotor length equal to the wavelength of the external force along the axis of rotation. The periodicity of the solution is set at end faces of the cylinder. As an example, the semi-analytical solution is compared with the numerical one obtained with these boundary conditions. The comparison confirms that the problem formulations are equivalent in both cases.     

On differential equations with interval coefficients

In this study, a new approach is developed to solve the initial value problem for interval linear differential equations. In the considered problem, the coefficients and the initial values are constant intervals. In the developed approach, there is no need to define a derivative for interval-valued functions. All derivatives used in the approach are classical derivatives of real functions. The reason for this is that the solution of the problem is defined as a bunch of real functions. Such a solution concept is compatible also with the robust stability concept. Sufficient conditions are provided for the solution to be expressed analytically. In addition, on a numerical example, the solution obtained by the proposed approach is compared with the solution obtained by the generalized Hukuhara differentiability. It is shown that the proposed approach gives a new type of solution. The main advantage of the proposed approach is that the solution to the considered interval initial value problem exists and is unique, as in the real case.     

On the nonuniqueness of the solution of the Tricomi problem with the generalized Frankl matching condition

We obtain an integral representation of the solution of the Tricomi problem for the Lavrent’ev-Bitsadze equation with mixed boundary conditions in the elliptic part of the domain and with zero posed on one characteristic of the equation. The gradient of the solution is not continuous but satisfies some condition referred to as the “generalized Frankl matching condition.” We state theorems implying that the inhomogeneous Tricomi problem either has a unique solution or is determined modulo a solution of the homogeneous Tricomi problem.