The planar Dirichlet problem for the Stokes equations

The Dirichlet problem for the Stokes equations is studied in a planar domain. We construct a solution of this problem in form of appropriate potentials and determine the unknown source densities via integral equation systems on the boundary of the domain. The solution is given explicitly in the form of a series. As a consequence we determine a solution of the Dirichlet problem for a compressible Stokes system and a solution of a boundary value problem on a domain with cracks. Copyright © 2011 John Wiley & Sons, Ltd.     

On the characteristic initial value problem for the wave equation in odd spatial dimensions with radial initial data

Summary The paper obtains an explicit solution of the characteristic initial value problem for the wave equation in odd spatial dimensions with radial initial data via solution of a characteristic boundary value problem involving a singular differential equation. The solution of the latter problem is obtained by a modified Riemann method. It is shown that on the time axis the solution of the original problem reduces to the solution that is obtainable by the use of Asgeirsson’s mean value theorem. Entrata in Redazione il 29 agosto 1971.     

An Iterative Method for Finding the Least Solution to the Tensor Complementarity Problem

In this paper, we are concerned with finding the least solution to the tensor complementarity problem. When the involved tensor is strongly monotone, we present a way to estimate the nonzero elements of the solution in a successive manner. The procedure for identifying the nonzero elements of the solution gives rise to an iterative method of solving the tensor complementarity problem. In each iteration, we obtain an iterate by solving a lower-dimensional tensor equation. After finitely many iterations, the method terminates with a solution to the problem. Moreover, the sequence generated by the method is monotonically convergent to the least solution to the problem. We then extend this idea for general case and propose a sequential mathematical programming method for finding the least solution to the problem. Since the least solution to the tensor complementarity problem is the sparsest solution to the problem, the method can be regarded as an extension of a recent result by Luo et al. (Optim Lett 11:471–482, 2017). Our limited numerical results show that the method can be used to solve the tensor complementarity problem efficiently.     

The Neumann problem for the Laplace equation on general domains

The solution of the weak Neumann problem for the Laplace equation with a distribution as a boundary condition is studied on a general open set G in the Euclidean space. It is shown that the solution of the problem is the sum of a constant and the Newtonian potential corresponding to a distribution with finite energy supported on ∂G. If we look for a solution of the problem in this form we get a bounded linear operator. Under mild assumptions on G a necessary and sufficient condition for the solvability of the problem is given and the solution is constructed. The research was supported by the Academy of Sciences of the Czech Republic, Institutional Research Plan No. AV0Z10190503.     

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.     

A uniqueness theorem in the inverse problem for the integrodifferential electrodynamics equations

Under study is the problem of finding the kernel and the index of dielectric permeability for the system of integrodifferential electrodynamics equations with wave dispersion. We consider a direct problem in which the external pulse current is a dipole located at a point y on the boundary ?B of the unit ball B. The point y runs over the whole boundary and is a parameter of the problem. The information available about the solution to the direct problem is the trace on ?B of the solution to the Cauchy problem given for the times close to the time when a wave from the dipole source arrives at a point x. The main result of the article consists in obtaining some theorems related to the uniqueness problems for a solution to the inverse problem.     

An exact algorithm for solving the bilevel facility interdiction and fortification problem

We present an exact approach for solving the $r$-interdiction median problem with fortification. Our approach consists of solving a greedy heuristic and a set cover problem iteratively that guarantees to find an optimal solution upon termination. The greedy heuristic obtains a feasible solution to the problem, and the set cover problem is solved to verify optimality of the solution and to provide a direction for improvement if not optimal. We demonstrate the performance of the algorithm in a computational study.     

Stability of a smooth flow in a Laval nozzle

A boundary-value problem for a non-linear second-order equation of mixed type in a cylindrical domain is considered. This problem simulates the development of small disturbances in a transonic flow of a chemical mixture in a Laval nozzle. The existence of a regular solution is proved with the help of a priori estimates for a corresponding linear problem and the contractive mapping theorem. The solution of the linear problem is constructed by the Galerkin method.     

Stability estimates for the solution to the problem of determining the kernel of a viscoelastic equation

For an integrodifferential equation corresponding to a two-dimensional viscoelastic problem, we study the problem of defining the spatial part of the kernel involved in the integral term of the equation. The support of the sought function is assumed to belong to a compact domain Ω. As information for solving this inverse problem, the traces of the solution to the direct Cauchy problem and its normal derivative are given for some finite time interval on the boundary of Ω. An important feature in the statement of the problem is the fact that the solution of the direct problem corresponds to the zero initial data and a force impulse in time localized on a fixed straight line disjoint with Ω. The main result of the article consists in obtaining a Lipschitz estimate for the conditional stability of the solution to the inverse problem under consideration.     

An inverse eigenvalue problem and an associated approximation problem for generalized K-centrohermitian matrices

A partially described inverse eigenvalue problem and an associated optimal approximation problem for generalized K-centrohermitian matrices are considered. It is shown under which conditions the inverse eigenproblem has a solution. An expression of its general solution is given. In case a solution of the inverse eigenproblem exists, the optimal approximation problem can be solved. The formula of its unique solution is given.     

The hyperbolic–elliptic equation with the nonlocal condition

The nonlocal boundary value problem for a hyperbolic–elliptic equation in a Hilbert space is considered. The stability estimate for the solution of the given problem is obtained. The first and second orders of difference schemes approximately solving this boundary value problem are presented. The stability estimates for the solution of these difference schemes are established. The theoretical statements for the solution of these difference schemes are supported by the results of numerical experiments. Copyright © 2013 John Wiley & Sons, Ltd.     

On the solution of Dirichlet’s problem of the complex Monge–Ampère equation for the Cartan–Hartogs domain of the first type

The complex Monge–Ampère equation is a nonlinear equation with high degree; therefore getting its solution is very difficult. In the present paper how to get the solution of Dirichlet’s problem of the complex Monge–Ampère equation on the Cartan–Hartogs domain of the first type is discussed, using an analytic method. Firstly, the complex Monge–Ampère equation is reduced to a nonlinear ordinary differential equation, then the solution of Dirichlet’s problem of the complex Monge–Ampère equation is reduced to the solution of a two-point boundary value problem for a nonlinear second-order ordinary differential equation. Secondly, the solution of Dirichlet’s problem is given as a semi-explicit formula, and in a special case the exact solution is obtained. These results may be helpful for a numerical method approach to Dirichlet’s problem of the complex Monge–Ampère equation on the Cartan–Hartogs domain of the first type.     

A new method for solving the nonlinear second-order boundary value differential equations

In this paper we use measure theory to solve a wide range of second-order boundary value ordinary differential equations. First, we transform the problem to a first order system of ordinary differential equations (ODE’s) and then define an optimization problem related to it. The new problem is modified into one consisting of the minimization of a linear functional over a set of Radon measures; the optimal measure is then approximated by a finite combination of atomic measures and the problem converted approximatly to a finite-dimensional linear programming problem. The solution to this problem is used to construct the approximate solution of the original problem. Finally we get the error functionalE (we define in this paper) for the approximate solution of the ODE’s problems.     

A Note on Fuzzy Relation Programming Problems with Max-Strict-t-Norm Composition

The fuzzy relation programming problem is a minimization problem with a linear objective function subject to fuzzy relation equations using certain algebraic compositions. Previously, Guu and Wu considered a fuzzy relation programming problem with max-product composition and provided a necessary condition for an optimal solution in terms of the maximum solution derived from the fuzzy relation equations. To be more precise, for an optimal solution, each of its components is either 0 or the corresponding component's value of the maximum solution. In this paper, we extend this useful property for fuzzy relation programming problem with max-strict-t-norm composition and present it as a supplemental note of our previous work.     

Behavior of the formal solution to a mixed problem for the wave equation

The behavior of the formal solution, obtained by the Fourier method, to a mixed problem for the wave equation with arbitrary two-point boundary conditions and the initial condition φ(х) (for zero initial velocity) with weaker requirements than those for the classical solution is analyzed. An approach based on the Cauchy–Poincare technique, consisting in the contour integration of the resolvent of the operator generated by the corresponding spectral problem, is used. Conditions giving the solution to the mixed problem when the wave equation is satisfied only almost everywhere are found. When φ(x) is an arbitrary function from L₂[0, 1], the formal solution converges almost everywhere and is a generalized solution to the mixed problem.     

Numerical solution of the inverse electrocardiography problem with the use of the Tikhonov regularization method

The inverse electrocardiography problem related to medical diagnostics is considered in terms of potentials. Within the framework of the quasi-stationary model of the electric field of the heart, the solution of the problem is reduced to the solution of the Cauchy problem for the Laplace equation in R ³. A numerical algorithm based on the Tikhonov regularization method is proposed for the solution of this problem. The Cauchy problem for the Laplace equation is reduced to an operator equation of the first kind, which is solved via minimization of the Tikhonov functional with the regularization parameter chosen according to the discrepancy principle. In addition, an algorithm based on numerical solution of the corresponding Euler equation is proposed for minimization of the Tikhonov functional. The Euler equation is solved using an iteration method that involves solution of mixed boundary value problems for the Laplace equation. An individual mixed problem is solved by means of the method of boundary integral equations of the potential theory. In the study, the inverse electrocardiography problem is solved in region Ω close to the real geometry of the torso and heart.     

Numerical treatment of integro-differential equations with a certain maximum property

The purpose of this article is to study the solution of an initial value problem of integro-differential type, where the differentiation is made for the timevariable and the integration for the space-variable. The problem has a propertyC, which implies that the solution is decreasing at the maximum-points. It is shown, that under rather general conditions, the solution tends to a constant, independent of both time and space. In the proofs, use is made of both the maximum norm and a weighted Euclidean norm.In order to study the numerical solution the problem is first discretized in the space-variable, preserving the propertyC. Finally, the problem is solved as a system of ordinary differential equations, making use of the propertyC to obtain not too pessimistic error estimations. The logarithmic norm, based on the maximum norm, is very useful in these final sections.     

The first fundamental problem of the theory of elasticity for a rectangular parallelepiped

In 1852 Lame [1] formulated the first fundamental problem of the theory of elasticity for a rectangular parallelepiped. An approximate solution to this problem was given by Filonenko-Borodich [2 and 3] who used Castigliano's variational principle. Later Mishonov [4] obtained an approximate solution to Lamé's problem in the form of divergent triple Fourier series. These series contain constants which are found from infinite systems of linear equations. Teodorescu [5] has considered a particular case of Lame's problem. Using his own method the author solves the problem in the form of double series analogous to those used in [6 to 8] and by Baida in [9 and 10] in solving problems on the equilibrium of a rectangular parallelepiped. The solution of the problem reduces to three infinite system of linear equations and the author asserts that these infinite systems are regular. It is shown in Section 5 that the infinite systems obtained by Teodorescu, on the other hand, will not be regular.

In the references mentioned above which investigate Lamé's problem the authors confine their attention either to obtaining a solution by an approximate method, or to reducing the solution process to one of obtaining infinite systems, leaving these uninvestigated. It must be emphasized that the main difficulty in solving this problem lies in investigating the infinite systems obtained which are significantly different from the infinite systems of the corresponding plane problem.

In this paper a solution is given to the first fundamental problem of the theory of elasticity for a rectangular parallelepiped with prescribed external stresses on the surface (Sections 2, 3 and 4). For the solution of this problem the author has used a form of the general solution of the homogeneous Lamé equations which contains five arbitrary harmonic functions and which constitutes a generalization of the familiar Papkovich-Neuber solution (Section 1). The solution is expressed in the form of double series containing four series of unknown constants which can be found from four infinite systems of linear algebraic equations. The infinite systems of linear equations obtained is studied for values of Poisson's ratio within the range 0 < σ ≤ 0.18. It is shown that for these values of Poisson's ratio the infinite systems are quasi-fully regular.     

A Hilbert space approach to maximum entropy reconstruction

We examine here the problem of reconstructing an X-ray attenuation function from measurements of its integrals. The approach that is taken is to maximize the difference of the entropy and the residual error in meeting the measurements. The solution of this optimization problem is constrained by requiring that the solution lie in a certain weakly compact subset of L², to be determined by physical information. We show that the constrained optimization problem is well-posed: there exists a unique solution (even when the measured data are inconsistent) and the solution depends continuously on the measurements. In the course of proving this, we show that the entropy functional is continuous on L². We further demonstrate that the solution of the optimization problem for a special case, must be piecewise constant.     

A method for solving the general parametric linear complementarity problem

This paper presents a solution method for the general (mixed integer) parametric linear complementarity problem pLCP(q(θ),M), where the matrix M has a general structure and integrality restriction can be enforced on the solution. Based on the equivalence between the linear complementarity problem and mixed integer feasibility problem, we propose a mixed integer programming formulation with an objective of finding the minimum 1-norm solution for the original linear complementarity problem. The parametric linear complementarity problem is then formulated as multiparametric mixed integer programming problem, which is solved using a multiparametric programming algorithm. The proposed method is illustrated through a number of examples.