Numerical solution of the three-dimensional Dirichlet problem for inhomogeneous media by the method of integral equations

We consider the three-dimensional Dirichlet problem for equations of elliptic type in inhomogeneous media. The problem can be reduced to a system of loaded Fredholm integral equations of the second kind over the volume. We prove the uniqueness of a classical solution of the problem. We suggest a numerical solution algorithm of iterative type. An example of the numerical solution of the problem is considered, and the convergence of the iterative procedure is demonstrated numerically.     

Multiparameter spectral problem for some weakly coupled systems of Hamiltonian equations

We consider a multiparameter spectral problem for a weakly coupled system of ordinary differential equations in which every equation is Hamiltonian and contains two unknown functions. Using the notion of the number of an eigenvalue for a problem with one such equation, we give a statement of the problem of finding the desired eigentuple of values for the problem with several equations. We prove the existence and uniqueness of a solution of this problem and suggest and study a numerical solution method.     

Application of the Laplace transform to the analysis of a mixed problem describing the vibrations of a compound rod

We suggest a method for finding a closed-form expression for the solution of the mixed problem on the vibrations excited in a compound rod by boundary conditions of the first, second, and third kind. The method uses the Laplace transform. We also suggest one possible numerical method for finding the solution.     

Dirichlet problem for a third-order equation of mixed type in a rectangular domain

For a third-order differential equation of parabolic-hyperbolic type, we suggest a method for studying the first boundary value problem by solving an inverse problem for a second-order equation of mixed type with unknown right-hand side. We obtain a uniqueness criterion for the solution of the inverse problem. The solution of the inverse problem and the Dirichlet problem for the original equation is constructed in the form of the sum of a Fourier series.     

On the well-posedness of entropy solutions to conservation laws with a zero-flux boundary condition

We study a zero-flux type initial-boundary value problem for scalar conservation laws with a genuinely nonlinear flux. We suggest a notion of entropy solution for this problem and prove its well-posedness. The asymptotic behavior of entropy solutions is also discussed.     

Characteristic and mixed problems for second-order equations of the hyperbolic type

We prove a theorem on the reduction of the Goursat problem in integral form to an equivalent system of two Fredholm integral equations of the second kind and a uniqueness theorem for its solution. We suggest a method for reducing nonlocal characteristic and mixed problems, including the Samarskii problem for a second-order differential equation, to local problems for partial differential equations of higher order. We present examples illustrating the importance of the condition ensuring the uniqueness of the solution of the problems in question.     

Regularization of the Cauchy problem for the Laplace equation

We suggest a method for regularizing the solution of the Cauchy problem for the Laplace equation by introducing the biharmonic operator with a small parameter. We show that if there exists a solution of the original problem, then the difference between the spectral expansions of solutions of the original and regularized equations tends to zero in the space of square integrable functions as the regularization parameter tends to zero. If the original solution belongs to a Sobolev class, then we use results of Il’in’s spectral theory to derive an estimate for the rate of the convergence of the regularized solution to the exact solution.     

On a perturbation treatment of a model for MHD viscous flow

We discuss the solution of a nonlinear ordinary differential equation that appears in a model for MHD viscous flow caused by a shrinking sheet. We propose an accurate numerical solution and derive simple analytical expressions. Our results suggest that a recent perturbation treatment of the same problem exhibits a pathological behaviour and conjecture its probable cause.     

A Modified Regularization Method

We consider the Cauchy problem for the systems of the first-order ordinary differential equations with a small parameter E of degree q at the derivative. We study the possibility of solving this problem by means of the regularization method of the Lomov theory of singular perturbations. We show that, for q>1, an application of the procedure by Lomov leads only to the trivial solution to the problem in the class of resonance-free solutions. We suggest and describe a modification of the procedure which enables us to construct a nontrivial solution to the problem in the space of resonance-free solutions.     

精确罚函数和极小极大问题

在这篇文章中我们研究了对于不等式约束的非线性规划问题如何根据极小极大问题的鞍点来找精确罚问题的解。对于一个具有不等式约束的非线性规划问题,通过罚函数,我们构造出一个极小极大问题,应用交换“极小”或“极大”次序的策略,证明了罚问题的鞍点定理。研究结果显示极小极大问题的鞍点是精确罚问题的解。     

Input tracking for a parabolic equation on an infinite time interval

We study the input tracking problem for a parabolic equation on an infinite time interval on the basis of the measurement of phase coordinates. We suggest an algorithm stable under information noises and roundoff errors for the solution of the problem on the basis of constructions of dynamic inversion theory.     

Asymptotic solution method for a quasilinear minimum force control problem

For a transient process in a quasilinear system, we consider an optimization problem of finding a (multi-dimensional) control with minimum intensity. We suggest an algorithm for constructing asymptotic approximations to the solution of this problem. The main advantage of the algorithm is that an optimal control problem for a linear system is solved instead of the original essentially nonlinear problem.     

Numerical solution method for the electric impedance tomography problem in the case of piecewise constant conductivity and several unknown boundaries

We study the electrical impedance tomography problem with piecewise constant electric conductivity coefficient, whose values are assumed to be known. The problem is to find the unknown boundaries of domains with distinct conductivities. The input information for the solution of this problem includes several pairs of Dirichlet and Neumann data on the known external boundary of the domain, i.e., several cases of specification of the potential and its normal derivative. We suggest a numerical solution method for this problem on the basis of the derivation of a nonlinear operator equation for the functions that define the unknown boundaries and an iterative solution method for this equation with the use of the Tikhonov regularization method. The results of numerical experiments are presented.     

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.     

Adaptive symplectic conservative numerical methods for the Kepler problem

We suggest and substantiate a unified form of a family of adaptive conservative numerical methods for the Kepler problem. The family contains methods of the second, fourth, and sixth approximation order as well as an exact method. The methods preserve all the global properties of the exact solution of the problem. The variable time step is chosen automatically depending on the properties of the solution.     

Algorithm for shadowing the solution of a parabolic equation on an infinite time interval

For a parabolic equation, we consider the problem of constructing a feedback control synthesis algorithm ensuring that the solution of a given equation shadows the solution of another equation with unknown right-hand side. We suggest a noise-immune algorithm based on the extremal shift method well known in guaranteed control theory.     

On the solution of boundary value problems with nonseparated multipoint and integral conditions

We suggest a numerical method for solving systems of linear nonautonomous ordinary differential equations with nonseparated multipoint and integral conditions. By using this method, which is based on the operation of convolution of integral conditions into local ones, one can reduce the solution of the original problem to the solution of a Cauchy problem for systems of ordinary differential equations and linear algebraic equations. We establish bounded linear growth of the error of the suggested numerical schemes. Numerical experiments were carried out for specially constructed test problems.     

Convex optimization problems with arbitrary right-hand side perturbations

The problem of finding a solution to a system of mixed variational inequalities, which can be interpreted as a generalization of a primal–dual formulation of an optimization problem under arbitrary right-hand side perturbations, is considered. A number of various equilibrium type problems are particular cases of this problem. We suggest the problem to be reduced to a class of variational inequalities and propose a general descent type method to find its solution. If the primal cost function does not possess strengthened convexity properties, this descent method can be combined with a partial regularization method.     

Strongest equilibria for conflict problems and differential games

We suggest a new notion of equilibrium, which, together with its natural strengthening, permits one to find the unique strongest equilibrium (that is, the unique solution of the conflict problem) in cases where all known classical and modern theories do not provide such a solution.     

Multipoint boundary value problem for the Lyapunov equation in the case of strong degeneration of the boundary conditions

We obtain sufficient coefficient conditions for the unique solvability of a multipoint boundary value problem for the Lyapunov matrix differential equation in the case of strong degeneration of the boundary conditions. We suggest an efficient algorithm for constructing the solution.