Identification of a dissipation coefficient by a variational method

A generalized inverse problem for the identification of the absorption coefficient for a hyperbolic system is considered. The well-posedness of the problem is examined. It is proved that the regular part of the solution is an L ₂ function, which reduces the inverse problem to minimizing the error functional. The gradient of the functional is determined in explicit form from the adjoint problem, and approximate formulas for its calculation are derived. A regularization algorithm for the solution of the inverse problem is considered. Numerical results obtained for various excitation sources are displayed.     

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.     

Unsteady shear vibrations of a die for a nonuniform base

A dynamic shear contact problem is examined for uniform and nonuniform bases. A flexible layer on a rigid half-space is chosen as the nonuniform base. The problem is solved on the basis of applied equations for strain. By using this approach, it is possible to reduce the problem to a system of integrodifferential equations. A numerical algorithm is proposed for its solution. Attention is focused on the effect of the flexible layer on the displacement and reaction of the die. The numerical results are analyzed.Translated from Dinamicheskie Sistemy, No. 4, pp. 52–57, 1985.     

Existence and uniqueness for a nonlinear inverse reaction‐diffusion problem with a nonlinear source in higher dimensions

This paper analyzes the existence and the uniqueness problem for an n‐dimensional nonlinear inverse reaction‐diffusion problem with a nonlinear source. A transformation is used to obtain a new inverse coefficient problem. Then, a parabolic differential operator L_λ is defined to establish the relation between the solution of L_λ = 0 and the new inverse problem. Following this, it is shown that the inverse problem has at least one solution in the class of admissible coefficients. Furthermore, it is proved that this solution is the unique solution of the undertaken inverse problem. A numerical example is given to illustrate ill‐posedness of the inverse problem. Copyright © 2013 John Wiley & Sons, Ltd.     

Diffraction of a plane electromagnetic wave on a wedge-shaped inclusion

A technique is developed for reducing the problem of diffraction by a wedge-shaped inclusion to functional equations in the complex plane. An original formulation of the radiation conditions is proposed that enables one to formulate the diffraction problem in this case. An existence and uniqueness theorem is proven. For the case of a sufficiently small opening span of the angle a formula is derived which is a good approximation to the solution to the problem.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Stekova AN SSSR, Vol. 195, pp. 29–39, 1991.     

Finding a minimum path cover of a distance-hereditary graph in polynomial time

A path cover of a graph G=(V,E) is a set of pairwise vertex-disjoint paths such that the disjoint union of the vertices of these paths equals the vertex set V of G. The path cover problem is, given a graph, to find a path cover having the minimum number of paths. The path cover problem contains the Hamiltonian path problem as a special case since finding a path cover, consisting of a single path, corresponds directly to the Hamiltonian path problem. A graph is a distance-hereditary graph if each pair of vertices is equidistant in every connected induced subgraph containing them. The complexity of the path cover problem on distance-hereditary graphs has remained unknown. In this paper, we propose the first polynomial-time algorithm, which runs in O(|V|⁹) time, to solve the path cover problem on distance-hereditary graphs.     

On the Global-in-Time Existence of a Generalized Solution to a Free-Boundary Problem

A problem with free (unknown) boundary for a one-dimensional diffusion-convection equation is considered. The unknown boundary is found from an additional condition on the free boundary. By the extension of the variables, the problem in an unknown domain is reduced to an initial boundary-value problem for a strictly parabolic equation with unknown coefficients in a known domain. These coefficients are found from an additional boundary condition that enables the construction of a nonlinear operator whose fixed points determine a solution of the original problem.

     

Action of a flat stamp on a half-plane of viscoelastic material

A plane problem concerning the action of a flat stamp on a reinforced, linearly viscoelastic medium is considered. One of the main axes of elasticity is parallel to the boundary of the half-plane. The reinforcing material is supposed to be perfectly elastic, and the matrix material linearly viscoelastic. The solution of the problem is based on the assumption that up to the moment of application of the forces, the medium is stress-free and at rest. The formulation of the problem is given, and a method of solution presented. A numerical example is offered.Mekhanika Polimerov, Vol. 2, No. 5, pp. 688–692, 1966     

Propagation of TM waves in a layer with arbitrary nonlinearity

A boundary value problem for Maxwell’s equations describing propagation of TM waves in a nonlinear dielectric layer with arbitrary nonlinearity is considered. The layer is located between two linear semi-infinite media. The problem is reduced to a nonlinear boundary eigenvalue problem for a system of second-order nonlinear ordinary differential equations. A dispersion equation for the eigenvalues of the problem (propagation constants) is derived. For a given nonlinearity function, the dispersion equation can be studied both analytically and numerically. A sufficient condition for the existence of at least one eigenvalue is formulated.     

Inverse Problem for a Mathematical Model of Adsorption Dynamics

A mixed problem is considered for a system of partial differential equations modeling the process of adsorption dynamics. An existence and uniqueness theorem is proved for this problem, and the solution properties are investigated. The inverse problem is posed, involving the determination of the system coefficient given additional information about the solution. A uniqueness theorem is proved for the solution of the inverse problem.__________Translated from Prikladnaya Matematika i Informatika, No. 16, pp. 5 – 14, 2004.     

Numerical solution of a one-dimensional problem of filtration consolidation of saline soils in a nonisothermal regime

A mathematical model of a problem of filtration consolidation of saline soils in a nonisothermal regime with allowance for the presence of salts in the liquid and solid phases is constructed. A numerical solution of the corresponding one-dimensional boundary-value problem is found by the method of finite differences. As an example, we investigate a problem of filtration consolidation of a massif of clay soil of finite thickness. Results of numerical investigations and their analysis are presented. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 51, No. 1, pp. 197–204, January–March, 2008.     

On Yudin’s scheme of finding a lower estimate for the potential energy of a system of charges on a sphere

We consider Yudin’s problem of finding a lower estimate for the minimum of the potential energy of a system of equal charges on the unit sphere $ \mathbb{S}^2 $ \mathbb{S}^2 in Euclidean space ℝ³. For this problem, we write a dual problem in the case of an arbitrary number of charges. A solution of the dual problem is given for the cases of 5, 6, 7, 8, and 12 charges. In addition, the primal problem is solved and hypothetically optimal arrangements are specified for 7 and 8 charges. It is established that Yudin’s method does not allow one to prove the optimality of these arrangements.     

A generalization of a minimax theorem of Fan via a theorem of the alternative

A minimization problem for a functional on a convex subsetC of a normed linear space is considered. Under certain hypotheses, optimality in a certain subset ofC implies the validity of first-order necessary optimality conditions for the problem inC. The result is applied to a problem in optimal periodic control of neutral functional differential equations.This work was partially supported by a grant from Deutsche Forschungsgemeinschaft and by AFOSR under Grant No. AFOSR-84-0398.     

Convergence of solutions for a network approximation of the two-dimensional Laplace equation in a domain with a system of absolutely conducting disks

A boundary value problem for the Laplace equation describing the (electric, thermal, etc.) field of a system of ideally conducting disks of radius R is considered. The solution to the problem is analyzed under the condition that the characteristic distance δ between the disks is small. It was previously proved that the original continuous problem can be approximated as δ → 0 by a finite-dimensional network problem in the sense that the effective conductivities (energies) of the continuous problem are close to those of its network model. It is shown that the potentials of the ideally conducting disks determined from the continuous problem and the network model are also close to each other as δ → 0, and the difference between the potentials is O(ε^1/4), where ε = δ/R is the characteristic relative distance between the disks.     

Control of the transition probabilities of input rates of a flow in a network

A stochastic control problem concerning the flow in a network is considered. Some of the nodes in the network are fed by an external input whose rate constitute a component of a continuous in time L-state Markov process θ. A numerical study is conducted on the dependence of the network performance on the choice of the q-matrix (the matrix of the infinitesimal characteristics of θ) of θ. Also, for the sake of completeness, the problem of an optimal q-matrix is discussed.     

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.     

Analysis of a one-dimensional free boundary flow problem

A one-dimensional free surface problem is considered. It consists in Burgers’ equation with an additional diffusion term on a moving interval. The well-posedness of the problem is investigated and existence and uniqueness results are obtained locally in time. A semi-discretization in space with a piecewise linear finite element method is considered. A priori and a posteriori error estimates are given for the semi-discretization in space. A time splitting scheme allows to obtain numerical results in agreement with the theoretical investigations.Supported by the Swiss National Science Foundation     

On a problem of optimal harvesting from a stochastic system with a jump component

In this paper, we study a problem of optimal harvesting from a stochastic system modeled by a geometric Lévy process. A verification theorem of the variational inequality type is also given and proved. The paper has been motivated by I. Elsanosi et al. [Stochastics Stochastics Rep. (2000)], where the authors considered an optimal harvesting problem with price dynamics following a stochastic differential delay equation.     

Minimizing total tardiness in a scheduling problem with a learning effect

Minimizing of total tardiness is one of the most studied topics on single machine problems. Researchers develop a number of optimizing and heuristic methods to solve this NP-hard problem. In this paper, the problem of minimizing total tardiness is examined in a learning effect situation. The concept of learning effects describes the reduction of processing times arising from process repetition. A 0–1 integer programming model is developed to solve the problem. Also, a random search, the tabu search and the simulated annealing-based methods are proposed for the problem and the solutions of the large size problems with up to 1000 jobs are found by these methods. To the best of our knowledge, no works exists on the total tardiness problem with a learning effect tackled in this paper.     

Advertising decisions for a segmented market

We consider the problem of choosing the levels of a set of advertising media in order to maximize the firm profit when the market is heterogeneous. Advertising efforts affect the demand of the different segments variably and we assume that the advertising effects on demand over time are mediated by a vector goodwill variable. A first general advertising decision problem is stated and solved in the non-linear programming framework. A preference index is then obtained for the medium selection problem when each segment demand function is linear in goodwill and each medium advertising cost function is quadratic in its level. Finally the theoretical case of disjoint advertising media is discussed.