On Some Features of the Numerical Solving of Coefficient Inverse Problems for an Equation of the Reaction–Diffusion–Advection-Type with Data on the Position of a Reaction Front

The work continues a series of articles devoted to the peculiarities of solving coefficient inverse problems for nonlinear singularly perturbed equations of the reaction–diffusion–advection-type with data on the position of the reaction front. In this paper, we place the emphasis on some problems of the numerical solving process. One of the approaches to solving inverse problems of the class under consideration is the use of methods of asymptotic analysis. These methods, under certain conditions, make it possible to construct the so-called reduced formulation of the inverse problem. Usually, a differential equation in this formulation has a lower dimension/order with respect to the differential equation, which is included in the full statement of the inverse problem. In this paper, we consider an example that leads to a reduced formulation of the problem, the solving of which is no less a time-consuming procedure in comparison with the numerical solving of the problem in the full statement. In particular, to obtain an approximate numerical solution, one has to use the methods of the numerical diagnostics of the solution’s blow-up. Thus, it is demonstrated that the possibility of constructing a reduced formulation of the inverse problem does not guarantee its more efficient solving. Moreover, the possibility of constructing a reduced formulation of the problem does not guarantee the existence of an approximate solution that is qualitatively comparable to the true one. In previous works of the authors, it was shown that an acceptable approximate solution can be obtained only for sufficiently small values of the singular parameter included in the full statement of the problem. However, the question of how to proceed if the singular parameter is not small enough remains open. The work also gives an answer to this question.

Nonlinear Singularly Perturbed Integro-Differential Equations and Regularization Method

The paper considers a nonlinear integro-differential system of singularly perturbed equations. We discuss the question of the spectrum of its operator, which does not coincide with the spectrum of its limit operator and includes an additionally identically zero point. In the case of linear systems, this difference does not play a special role, since the regularization and construction of the space of solutions of the corresponding iterative problems are realized at nonzero points of the spectrum. In the case of nonlinear problems, the identically zero point of the spectrum plays an essential role in the construction of the solution space in the resonance and nonresonance cases (see below); therefore, in most works using the regularization method in nonlinear problems, only the nonresonance case is usually considered. In the paper, for the classical integrodifferential system, regularization (according to Lomov) is carried out and the corresponding algorithm for constructing asymptotic solutions taking into account the zero point of the spectrum is developed.

Axiomatic Approach in the Analytic Theory of Singular Perturbations

Introduced by S.A. Lomov, the concept of a pseudoanalytic (pseudoholomorphic) solution laid the foundation for the development of the singular perturbation analytical theory. In order for this concept to work in case of linear problems, an apparatus for the theory of exponential type vector spaces was developed. When considering nonlinear singularly perturbed problems, an algebraic approach is currently used. This approval is based on the properties of algebra homomorphisms for holomorphic functions with various numbers of variables, as a result of which it is possible to obtain pseudoholomorphic solutions. In this paper, formally singularly perturbed equations are considered in topological algebras, which allows the authors to formulate the main concepts of the singular perturbation analytical theory from the standpoint of maximal generality.

SINGULARLY PERTURBED PROBLEMS OF THERMO-MECHANICS OF LAYERED ROAD COATINGS

The problem of singular perturbation of the equations of elastic bending of layered pavements and thermal conductivity is discussed. It is shown that the complexity of solving these equations significantly exceeds the complexity of singularly perturbed equations known in the scientific literature. First, this is due to the fact that in the problems of road construction, the mechanics of elastic bending are described by the partial differential equations, and the integration domain reaches large dimensions. The problem of thermal deformation of the pavement and distribution in its array of temperature, displacement, and stress fields is also considered. It is proved that thesolution of the problemof thermal conductivity has the appearance of a boundary effect, localized in the elements of the coating adjacent to its free surface.Also is noted that high-gradient temperature distribution deep in the soil leads to large values of normal and tangential stresses that provoke cracking and delamination of the upper layers of the coating. They are confirmed by the results of computer simulation. KEYWORDS: SINGULARLY PERTURBED PROBLEM, LAYERED ROAD COVERING, TEMPERATURE FIELD, TRANSPORT LOAD, STRESS FIELD, THERMAL STRAIN STATE

To the analysis of weak two-locus viability selection and quasi linkage equilibrium

A model of weak viability selection at two multi-allele loci with standardization of approaches through the use of perturbation theory is examined. The estimate of the quasi-equilibrium value for the linkage disequilibrium coefficient D is analyzed, and results in terms of average effects in quantitative genetics and in terms of the theory of singular perturbations in mathematics are obtained. The approximation of a discrete-time model of a random mating population with non-overlapping generations under weak selection by ordinary differential equations is considered. Weak selection is considered as a perturbation of the model without selection. The resulting model is singularly perturbed; that is, fast (D) and slow (allele frequencies) variables can be distinguished. The first approximation equation for quasi-equilibrium of D is obtained using the first terms of the Taylor series expansion of the model functions. It coincides with the corresponding part of the system of the first approximation of the asymptotic series for solving singularly perturbed equations.