In this paper we discuss a priori error estimates and superconvergence of splitting positive definite mixed finite element methods for optimal control problems governed by pseudo-hyperbolic integro-differential equations. The state variables and co-state variables are approximated by the lowest order Raviart–Thomas mixed finite element functions, and the control variable is approximated by piecewise constant functions. First, we derive a priori error estimates for the control variable, state variables, and co-state variables. Second, we obtain a superconvergence result for the control variable.

We consider the resource allocation problem and its numerical solution. The following constructions are demonstrated: 1) Walrasian price-adjustment mechanism for determining the equilibrium; 2) Decentralized role of the prices; 3) Slater's method for price restrictions (dual Lagrange multipliers); 4) A new mechanism for determining equilibrium prices, in which prices are fully controlled not by Center (Government), but by economic agents -- nodes (factories). In economic literature the convergence of the considered methods is only proved. In contrast, this paper provides an accurate analysis of the convergence rate of the described procedures for determining the equilibrium. The analysis is based on the primal-dual nature of the suggested algorithms. More precisely, in this article we propose the economic interpretation of the following numerical primal-dual methods of convex optimization: dichotomy and subgradient projection method. Numerical experiments conclude the paper.

A three-dimensional unsteady periodic flow of blood in xenogenic vascular bioprostheses is simulated by computational fluid dynamics methods. The geometry of the computational domain is based on microtomographic scanning of bioprostheses. To set a variable pressure gradient causing an unsteady flow in the prostheses, personal-specific data of the Doppler echography of the blood flow of a particular patient are used. A comparative analysis of the velocity fields in the flow areas corresponding to three real samples of bioprostheses with multiple stenoses is carried out. In the zones of stenosis and outside of them, the distribution of the near-wall shear stress, which affects the risk factors for thrombosis in the prostheses, is analyzed. An algorithm for predicting the hemodynamic effects arising in vascular bioprostheses is proposed; the algorithm is based on the numerical modeling of the blood flow in these prostheses.

We propose a method for solving three-dimensional boundary value problems for Laplace’s equation in an unbounded domain. It is based on non-overlapping decomposition of the exterior domain into two subdomains so that the initial problem is reduced to two subproblems, namely, exterior and interior boundary value problems on a sphere. To solve the exterior boundary value problem, we propose a singularity isolation method. To match the solutions on the interface between the subdomains (the sphere), we introduce a special operator equation approximated by a system of linear algebraic equations. This system is solved by iterative methods in Krylov subspaces. The performance of the method is illustrated by solving model problems.

In a Hilbert space we consider a class of conditionally well-posed inverse problems for which a Holder-type estimate of conditional stability on a closed convex bounded subset holds. We investigate the Ivanov quasi-solution method and its finite-dimensional version associated with minimization of a multi-extremal discrepancy functional over a conditional stability set or over a finite-dimensional section of this set, respectively. For these optimization problems, we prove that each of their stationary points that is located not too far from the desired solution of the original inverse problem belongs to a small neighborhood of the solution. Estimates for the diameter of this neighborhood in terms of error levels in input data are also given.

The concept of a regularizing decomposition was introduced by R. Horn and V. Sergeichuk. It means the representation of a square matrix by a direct sum of Jordan blocks with zero on the principal diagonal and a nonsingular matrix. Such a representation is attained via congruence transformations and differs from the Jordan normal form. For the reasons explained in this paper, we prefer to speak of an SR decomposition (in other words, a singular-regular decomposition) of a matrix rather than a regularizing decomposition. Accordingly, algorithms providing this decomposition are called SR algorithms.We develop a rational algorithm that considerably simplifies the SR algorithms proposed by Horn and Sergeichuk.

A new method to obtain exact solutions to the two-dimensional eikonal equation where the velocity of the medium depends on one of the spatial coordinates alone is proposed. Several examples of reducing the initial problem to one or several ordinary differential equations by substituting the solution into a suitable general form are presented. The dynamics of wave propagation is illustrated for each of the solutions thus obtained.

The optimal q-homotopy analysis method is employed in order to solve partial differential equations (PDEs) featuring a time-fractional derivative. To illustrate the simplicity and ability of the suggested approach, some specific and clear examples are given. All numerical calculations in this manuscript have been carried out withMathematica package.

In this paper, various difference schemes with oblique stencils, i.e., schemes using different space grids at different time levels, are studied. Such schemes may be useful in solving boundary value problems with moving boundaries, regular grids of a non-standard structure (for example, triangular or cellular ones), and adaptive methods. To study the stability of finite difference schemes with oblique stencils, we analyze the first differential approximation and dispersion. We study stability conditions as constraints on the geometric locations of stencil elements with respect to characteristics of the equation. We compare our results with a geometric interpretation of the stability of some classical schemes. The paper also presents generalized oblique schemes for a quasilinear equation of transport and the results of numerical experiments with these schemes.

We say that square matrices A and B of the same order quasi-commute if AB = σBA for a certain scalar σ. The classical definitions of commutation and anti-commutation are particular cases of the above definition. We give a complete description of pairs of quasi-commuting Toeplitz and Hankel matrices for σ ≠ ±1.