Data models with effectively separable algorithmic representations are considered. It is established that any such model has an enrichment, which is the unique model constructed from constants for a suitable proposal of first-order logic

We consider the problem of minimizing a nonlinear functional on a closed set in a Hilbert space. The functional to be minimized and the admissible set may be specified with errors. It is established that a necessary and sufficient condition for the existence of regularization procedures with an accuracy estimate uniform across different classes of functionals and admissible sets is the uniform well-posedness of these classes of minimization problems. A necessary and sufficient condition for the existence of a regularizing operator that does not use information about the error level of the input data is obtained. The proofs partially rely on the variational principles of Ekeland and Borwein–Preiss. Similar results were previously known for regularization procedures for ill-posed inverse problems, as well as for unconstrained optimization problems.

This article aims at the construction and analysis of a computational method for a system of two-parameter singularly perturbed second-order nonlinear differential equations with prescribed boundary conditions modeling reaction-convection-diffusion processes. A fitted mesh method consists of a classical finite difference scheme together with a Shishkin mesh constructed to solve the system. The fitted mesh method is proved to be convergent essentially first-order uniformly with respect to the perturbation parameters. An algorithm using the continuation method is designed to compute the numerical approximations. Numerical experiments support the theoretical results. Since there is no literature on systems of two-parameter singularly perturbed nonlinear differential equations, the present study reveals the characteristics of such systems and contributes to their numerical solution.

Many problems in science and engineering are naturally reduced to singular integral equations. Moreover, planar problems are reduced to one-dimensional singular integral equations. In the present paper, we develop an optimal algorithm for the approximate solution of one-dimensional singular integral equations with the Cauchy kernel. Here, we focus on finding the analytical form of the coefficients of the optimal quadrature formula. We apply these coefficients to an approximate solution of the Fredholm singular integral equation of the first kind. Thus, we demonstrate the possibility of solving singular integral equations with higher accuracy using the optimal quadrature formula.

In this paper, the Cauchy problem for the stationary and nonstationary nonlocal incompressible abstract Stokes equation is considered. The equation involves a convolution term and an abstract operator in a Banach space E. Existence, uniqueness, and coercive estimates are derived in L p spaces. Different classes of Stokes equations can be obtained by choosing the space E and the linear operator A, which occur in a wide variety of physical systems. As an application of the obtained results, the existence, uniqueness, and L p -maximal regularity properties of solutions to initial value problems for nonlocal degenerate Stokes equations and nonlocal Stokes equations with discontinuous coefficients are established.

A refined transformational mathematical model is proposed to describe the deformation process of a rod-strip having fixed and non-fixed sections along its length. It is assumed that the rod in the fixed section is connected to a support element, which has displacement components prescribed (known) at the points of connection with the rod, which makes it possible, in particular, to simulate the process of kinematic loading of the rod during tensile and compression tests. To describe the process of deformation of the non-fixed section of the rod, tangential displacements are approximated by a third-degree polynomial along the transverse coordinate, and deflection by a second-degree polynomial. In the fixed section, the approximations of the displacements that were assumed for the non-fixed section are transformed into other approximation functions along the transverse coordinate due to their compliance with the kinematic conditions of the two-sided connection with a support element with prescribed displacements. The conditions for the kinematic coupling of the fixed and non-fixed parts of the rod are formulated, taking them into account using the D’Alembert–Lagrange variational principle, the equations of equilibrium and motion of the marked parts, their corresponding boundary conditions, as well as the force conditions for coupling the fixed and non-fixed sections of the rod are obtained.

This paper studies the nonhomogeneous Hilbert boundary value problem in the half-plane with a finite index for a single generalized Cauchy–Riemann equation with a strong singularity in the coefficient. A formula for the solution of this equation is derived, and the solvability of the Hilbert problem for analytic functions with an infinite index and two vortex points of power and logarithmic orders is investigated. Based on this, the solvability of the Hilbert boundary value problem for generalized analytic functions is studied.