A method of solving a nonlinear boundary value problem with a parameter for a loaded differential equation

A nonlinear loaded differential equation with a parameter on a finite interval is studied. The interval is partitioned by the load points, at which the values of the solution to the equation are set as additional parameters. A nonlinear boundary value problem for the considered equation is reduced to a nonlinear multipoint boundary value problem for the system of nonlinear ordinary differential equations with parameters. For fixed parameters, we obtain the Cauchy problems for ordinary differential equations on the subintervals. Substituting the values of the solutions to these problems into the boundary condition and continuity conditions at the partition points, we compose a system of nonlinear algebraic equations in parameters. A method of solving the boundary value problem with a parameter is proposed. The method is based on finding the solution to the system of nonlinear algebraic equations composed.     

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

A nonlinear self-adjoint eigenvalue problem for the general linear system of ordinary differential equations is examined on an unbounded interval. A method is proposed for the approximate reduction of this problem to the corresponding problem on a finite interval. Under the assumption that the initial data are monotone functions of the spectral parameter, a method is given for determining the number of eigenvalues lying on a prescribed interval of this parameter. No direct calculation of eigenvalues is required in this method.     

A nonlocal problem for singular linear systems of ordinary differential equations

A system of linear ordinary differential equations is examined on an infinite half-interval. This system is supplemented by the boundedness condition for solutions and a nonlocal linear condition specified by the Stieltjes integral. A method for approximating the resulting problem by a problem posed on a finite interval is proposed, and the properties of the latter are investigated. A numerically stable method for solving this problem is examined. This method uses an auxiliary boundary value problem with separated boundary conditions.     

Solving some problems for systems of linear ordinary differential equations with redundant conditions

Numerical methods are proposed for solving some problems for a system of linear ordinary differential equations in which the basic conditions (which are generally nonlocal ones specified by a Stieltjes integral) are supplemented with redundant (possibly nonlocal) conditions. The system of equations is considered on a finite or infinite interval. The problem of solving the inhomogeneous system of equations and a nonlinear eigenvalue problem are considered. Additionally, the special case of a self-adjoint eigenvalue problem for a Hamiltonian system is addressed. In the general case, these problems have no solutions. A principle for constructing an auxiliary system that replaces the original one and is normally consistent with all specified conditions is proposed. For each problem, a numerical method for solving the corresponding auxiliary problem is described. The method is numerically stable if so is the constructed auxiliary problem.     

On the well-posedness of periodic problems for the system of hyperbolic equations with finite time delay

A periodic problem for the system of hyperbolic equations with finite time delay is investigated. The investigated problem is reduced to an equivalent problem, consisting the family of periodic problems for a system of ordinary differential equations with finite delay and integral equations using the method of a new functions introduction. Relationship of periodic problem for the system of hyperbolic equations with finite time delay and the family of periodic problems for the system of ordinary differential equations with finite delay is established. Algorithms for finding approximate solutions of the equivalent problem are constructed, and their convergence is proved. Criteria of well-posedness of periodic problem for the system of hyperbolic equations with finite time delay are obtained.     

A stability estimate for a solution to an inverse problem of electrodynamics

We consider the integrodifferential system of equations of electrodynamics which corresponds to a dispersive nonmagnetic medium. For this system we study the problem of determining the spatial part of the kernel of the integral term. This corresponds to finding the part of dielectric permittivity depending nonlinearly on the frequency of the electromagnetic wave. We assume that the support of dielectric permittivity lies in some compact domain Ω ⊂ ℝ³. In order to find it inside Ω we start with known data about the solution to the corresponding direct problem for the equations of electrodynamics on the whole boundary of Ω for some finite time interval. On assuming that the time interval is sufficiently large we estimate the conditional stability of the solution to this inverse problem.     

Systems of convolution equations of the first and second kind on a finite interval and factorization of matrix-functions

Systems of n convolution equations of the first and second kind on a finite interval are reduced to a Riemann boundary value problem for a vector function of length 2n. We prove a theorem about the equivalence of the Riemann problem and the initial system. Sufficient conditions are obtained for the well-posedness of a system of the second kind. Also under study is the case of the periodic kernel of the integral operator of a system of the first and second kind.     

Global solution to a singular integro-differential system related to the entropy balance

This paper is devoted to the mathematical analysis of a thermodynamic model describing phase transitions with thermal memory in terms of an entropy equation and a momentum balance for the microforces. The initial and boundary value problem is addressed for the related integro-differential system of partial differential equations (PDEs). Existence and uniqueness, continuous dependence on the data, and regularity results are proved for the global solution, in a finite time interval.     

Optimal finite-dimensional solution for a class of nonlinear observation problems

The filtering problem in a differential system with linear dynamics and observations described by an implicit equation linear in the state is solved in finite-dimensional recursive form. The original problem is posed as a deterministic fixed-interval optimization problem (FIOP) on a finite time interval. No stochastic concepts are used. Via Pontryagin's principle, the FIOP is converted into a linear, two-point boundary-value problem. The boundary-value problem is separated by using a linear Riccati transformation into two initial-value problems which give the equations for the optimal filter and filter gain. The optimal filter is linear in the state, but nonlinear with respect to the observation. Stability of the filter is considered on the basis of a related properly linear system. Three filtering examples are given.     

Inversion of convolution equations on a finite interval and realization triples

This paper develops further the connections between convolution equations and realization triples. Here the emphasis is on equations on a finite interval. For each system of equations an operator (called indicator) is introduced which can be used to describe the inversion properties of the system. This indicator may be of simpler form than the convolution operator defined by the equations even for the case when the corresponding symbol is non-rational     

The problem of package guidance by a given time for a linear control system with delay

The problem of guaranteed closed-loop guidance by a given time under incomplete information on the initial state is studied for a dynamical control system with delay by means of the method of open-loop control packages. A solvability criterion is proved for this problem in the case of a finite set of admissible initial states. The proposed technique is illustrated by a specific linear control system of differential equations with delay.     

A hybrid numerical model for multiphase fluid flow in a deformable porous medium

In this paper, a fully coupled finite volume-finite element model for a deforming porous medium interacting with the flow of two immiscible pore fluids is presented. The basic equations describing the system are derived based on the averaging theory. Applying the standard Galerkin finite element method to solve this system of partial differential equations does not conserve mass locally. A non-conservative method may cause some accuracy and stability problems. The control volume based finite element technique that satisfies local mass conservation of the flow equations can be an appropriate alternative. Full coupling of control volume based finite element and the standard finite element techniques to solve the multiphase flow and geomechanical equilibrium equations is the main goal of this paper. The accuracy and efficiency of the method are verified by studying several examples for which analytical or numerical solutions are available. The effect of mesh orientation is investigated by simulating a benchmark water-flooding problem. A representative example is also presented to demonstrate the capability of the model to simulate the behavior in heterogeneous porous media.     

On problem of the dynamics of a viscoelastic medium with memory on an infinite interval

The existence of a weak solution of a boundary value problem for a viscoelasticity model with memory on an infinite time interval is proved. The proof relies on an approximation of the original boundary value problem by regularized ones on finite time intervals and makes use of recent results concerning the solvability of Cauchy problems for systems of ordinary differential equations in the class of regular Lagrangian flows.     

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.     

一个与硅氧化有关的粘性可压缩流体的自由边界问题

本文研究一个描述硅的氧化过程的自由边界问题．它的数学模型是一个可压缩的Ｎａｖｉｏｒ－Ｓｔｏｋｅｓ方程与一个抛物方程以及一个双曲方程的耦合，其中在自由边界上存在表面张力并且密度方程是非齐次的．本文将证明只要已知数据满足相容性条件，则上述问题有唯一局部强解．     

Can a solution of a linear delay differential equation have an infinite number of isolated zeros on a finite interval?

A solution of a linear delay differential equation can have an infinite number of isolated zeros on a finite interval. As is well known, for ordinary differential equations this is impossible.     

Synthesis of a rational filter in the presence of complete alternance

The construction of a rational function that is nonnegative on two intervals of which one is infinite is considered. It is assumed that the maximum deviation of the function from zero on the infinite interval takes the minimum possible value under the condition that the values of the function on the finite interval are within the given bounds. It is assumed that the rational function (fraction) has the complete alternance. In this case, the original problem is reduced to solving a system of nonlinear equations. For solving this system, a two-stage method is proposed. At the first stage, a subsystem is selected and used to find a good approximation for the complete system. At the second stage, the complete system of nonlinear equations is solved. The solution is explained in detail for the case when the order of the fraction is between one and four. Numerical results for a fraction of order ten are presented.     

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.     

Guaranteed-accuracy approximation of reachable sets for a linear dynamic system subject to impulse actions

A method is proposed for approximating the reachable set of a dynamic system with a state space dimension no higher than six-eight considered on a finite time interval. The system is governed by linear differential equations with piecewise constant coefficients and impulse actions specified at prescribed times. The method is based on guaranteed-accuracy polyhedral approximations of reachable sets at researcher-specified times. Every approximation is constructed using the preceding one. A procedure is described for choosing parameters of the method that ensure the required accuracy with close-to-minimal time costs.     

On the Maxwell system under impedance boundary conditions with memory

We consider an initial boundary value problem for the system of the Maxwell equations in a bounded domain with smooth boundary on a finite time interval with new boundary conditions with memory. In appropriate function spaces, we define and study the nonselfadjoint operator that is generated by the Maxwell operator under a boundary condition with memory. Using the operator method, we prove an existence and uniqueness theorem for a solution to the initial boundary value problem.