Nonclassical Problems Research Articles

Numerical Algorithms for Identification of Convection Coefficient and Source in a Magnetohydrodynamics Flow

We consider three integral observations inverse problems (IP1, IP2, IP3) for reconstruction of a time-dependent convection coefficient and a source in magnetohydrodynamics (MHD) model. On the first stage, using the integral observations, we reduce the inverse problems to nonclassical direct (forward) ones. The equivalence between the inverse and direct problems is established. Then, the well-posedness of nonclassical problems is proved. Further, to overcome the difficulties arising from the nonlinear nonlocal parabolic operators, we construct a linarization algorithm in time after their difference space discretizatons. Next, on each iteration, to solve the corresponding linear systems of algebraic equations, we propose adequate fast elimination algorithms. Computational results of test examples data are discussed.

Nonclassical Problems of the Mathematical Theory of Hydrodynamic Boundary Layer

Nonclassical Problems of the Mathematical Theory of Hydrodynamic Boundary Layer

Nonclassical problems of the mathematical theory of hydrodynamic boundary layer

Nonclassical problems of the mathematical theory of hydrodynamic boundary layer

Exponential stability for porous system of thermoelasticity-type III with past history and distributed delay term

As a continuity to the study by Ouchenane et al. [On the Porous-elastic system with thermoelasticity of type III and distributed delay: Well-posedness and stability, J. Funct. Spaces (2021) 1–12], we consider a one-dimensional linear thermoelastic system of porous type with past history and distributed delay term. This model deals with dynamics of engineering structures and non-classical problems of mathematical physics. It includes new results and their proofs with a help of the energy method combined with Lyapunov functional, we discuss the stability of system for the case of equal speeds of wave propagation.

Решение обратной задачи, описывающей медленную тепловую конвекцию во вращающемся слое

The linear inverse initial-boundary value problem arising when modeling the rotational motion of a viscous heat-conducting liquid in a flat layer is solved. It is shown that the problem has two different integral identities. Based on these identities, a priori estimates of the solution in a uniform metric are obtained and its uniqueness is proved. The conditions for the input data are also determined, under which this solution goes to the stationary mode with increasing time according to the exponential law. In the final part, the existence of a unique classical solution of the inverse problem is proved. To do this, differentiating the problem by a spatial variable, we come to a direct non-classical problem with two integral conditions instead of the usual boundary conditions. The new problem is solved by the method of separation of variables, which makes it possible to find a solution in the form of rapidly converging series on a special basis.

Stability of multispan rods

Objective. In the design of complex engineering structures, the problem of stability is now becoming especially relevant. With the need to meet the conditions of strength, rigidity in calculations and design, it is imperative to ensure the stability of equilibrium as the most important requirement. Regardless of the type of strength calculations (verification, design, load), the calculation of a structure for stability, even in the simplest formulation, is reduced to the determination of critical forces. This allows us to estimate the stability margin of the structure under a given load. The complexity of determining the critical parameters of the impact increases with the complexity of the system under consideration.Method. The use of the finite difference method is a feature in solving systems of equations in problems with boundary conditions.Result. The exact values of critical forces in non-classical problems of the stability of compressed rods are determined, which is proved by the obtained curve from the monitor screen.Conclusion. The main points of the theory of calculation in the study of the stability of multi-span rods are outlined. To find the critical forces of a multi-span rod on elastic supports, a simple algorithm for solving problems is given.

An Efficient Constructive Heuristic for the Cutting Stock Problem Applied in a Foam Mattress Industry

The cutting and packing problem belongs to the combinatorial optimization problems; it covers a wide range of practical cases in industries. The present paper investigates a new real world problem needs to be solved through a daily operations of cutting foam blocks in an industrial company. The problem is considered as one of non-classical problems in the cutting and packing area. It represents a variant of the three dimensional Cutting Stock Problem. The originality of the studied problem is indicated by a specific set of constraints related to the production process and the cutting ways. A constructive heuristic was developed to provide cutting patterns in advance. All possible combinations established from the ways of cutting right rectangular prisms from foam blocks define the cutting patterns. This heuristic performs well and shows promising results in reasonable computational times to provide efficient cutting plans in order to reduce the total material loss.

The Non-Classical Problem of Thermoelastic Stability of an Elastically Restrained Orthotropic Plate of Variable Thickness

The Non-Classical Problem of Thermoelastic Stability of an Elastically Restrained Orthotropic Plate of Variable Thickness

On General List of References to the Monograph “Eight Non-Classical Problems of Fracture Mechanics”

On General List of References to the Monograph “Eight Non-Classical Problems of Fracture Mechanics”

Анализ терминов как инструмент компаративной философии

The article continues the polemics on the problems of interaction of philosophical cultures in the era of globalization, which was started at the meetings of the Round Table "Geography of Rationality". The author gives answers to critical questions, explains the methodology and principles of her work with Indian philosophical texts. A short research of the meta-term "cognitive subject" is an example of her methods. The analysis of cognitive subject aimed to justify the absence of the concepts of reason and rationality in Indian epistemic culture, the cornerstones of Western epistemic culture, since Modern times. The justification was carried out by comparing the generalized model of the cognitive subject, abstracted from the writings of empiricists and rationalists of the XVII–XVIII centuries, with the generalized models of the cognitive subject, reconstructed on the basis of authoritative writings of three variants of Indian epistemological teachings: Advaita Vedānta, Jainism and Buddhism. From the author's point of view, the absence of the concepts of reason and rationality in India leads to the non-classical problem of pluralism of epistemic cultures, and the exploration of the meta-term "cognitive subject" allows us to find, on the one hand, intersections in the contents of epistemologies in Indian philosophy and Western metaphysics of Modern times, and on the other, their incompatible contents, which are specific manifestations of pluralism of epistemic cultures. For her reconstruction of the cognitive subject models the author takes the principle of "double perspective" in combination with the methods of hermeneutical and logical analysis of philosophical terms. The principle determinates the consideration of the theoretical object from two sides: European and Indian. Having appeared in the Western epistemic culture, these methods effectively work to objectify the results of socio-humanitarian research, thanks to which they are becoming increasingly widespread among non-Western cross-cultural philosophers. When the author applies the method of logical analysis to justify the absence of the concepts of reason and rationality in India, she is guided by the rules of logical semantics and the principles of semiotics. The compared terms, Western and Indian, are considered as signs with their own meanings and senses. The senses are understood as sets of predicates important for solving the author’s task. The author of the article, taking into account the experience of famous philosophers, negatively assesses the possibility of solving the problem of unambiguously correct translation.

Application of optimal set partitioning theory to solving problems of artificial intelligence and pattern recognition

The paper substantiates the possibility of applying the mathematical theory of continuous problems of optimal partitioning of sets of n-dimensional Euclidean space, which belong to the non-classical problems of infinite-dimensional mathematical programming, to the solution of problems of artificial intelligence and pattern recognition. The problems of pattern recognition both in conditions of certainty and in conditions of uncertainty are formulated. A particular attention is paid to the application of methods of the theory of optimal partitioning for the construction of fuzzy Voronoi diagrams. Examples of constructing fuzzy Voronoi diagrams with the optimal placement of generating points are given.

Analogies Between the Classical Problem on the Oscillation of Bodies and the Nonclassical Problem on the Propagation of Plane Waves*

Analogies Between the Classical Problem on the Oscillation of Bodies and the Nonclassical Problem on the Propagation of Plane Waves*

The combinatorial optimization problems of the discrete processes

Nonclassic combinatorial optimization problems of the discrete processes are investigated. The methodology, permitting transform there problems into equivalent typical problems, which can be solved applying dynamic programming and principle of the maximum methods, is proposed.

On The Solvability of the Cauchy Problem for a Certain Class of Multidimensional Loaded Parabolic Equations

In this paper, we examine the solvability of a new class of nonclassical direct problems for multidimensional loaded parabolic equations with Cauchy data. We obtain sufficient conditions for the solvability of the problem; the proof is based on the method of weak approximation. By an example, we demonstrate the application of the theorem proved to the study of inverse problems for multidimensional parabolic equations with Cauchy data.

On the Porous-Elastic System with Thermoelasticity of Type III and Distributed Delay: Well-Posedness and Stability

The paper deals with a one-dimensional porous-elastic system with thermoelasticity of type III and distributed delay term. This model is dealing with dynamics of engineering structures and nonclassical problems of mathematical physics. We establish the well posedness of the system, and by the energy method combined with Lyapunov functions, we discuss the stability of system for both cases of equal and nonequal speeds of wave propagation.

Методы моделирования и идентификация параметров неоднородных анормальных неврологических движений многокомпонентных нейробиосистемах с когнитивными обратными связями

Разработаны основы математического моделирования и идентификации параметров неоднородных анормальных неврологических движений (АНР) в многокомпонентных нейробиосистемах с обратными когнитивными связями. На основе развитых авторами методов интегральных превращений и спектрального анализа для неоднородных сред предложен новый подход к построению гибридных моделей распространения волнового сигнала, описывающего нежелательные дрожания конечности руки пациента (Т-объекта) в результате самопроизвольного сокращения скелетных мышц. за счет когнитивных воздействий отдельной группы нейронных узлов коры головного мозга (КГМ). Разработана гибридная модель нейробиосистемы, описывающая состояние и поведение Т-объектов, а именно посегментное описание 3D-элементов траекторий АНР Т-объекта с учетом матрицы когнитивных воздействий групп нейроузлов КГМ. На основе гибридных интегральных преобразований Фурье получено высокоскоростное аналитическое векторное решение модели, описывающей элементы траекторий на каждом АНР-сегменте. Предложена новая методика вычисления гибридной спектральной функции, спектральных значений и матрицы когнитивных воздействий нейроузлов КГМ, определяющих гибридное интегральное преобразование построения решения. Сформулированы и решены новые неклассические задачи многопараметрической идентификации нейросистем с обратной связью в неоднородных средах на основе минимизации функционала-невязки между траекториями наблюдения и их модельными аналогами. Построены высокопроизводительные алгоритмы идентификации амплитудно-частотных характеристик систем обратной связи для компонентной оценки влияния когнитивных обратных связей, позволяющих распараллелировать вычисления для многоядерных компьютеров. Выполнено компьютерное моделирование и идентификацию АНР-траекторий нейросистем обратной связи.

Two-Dimensional Stressed State of an Adhesive Joint. Nonclassical Problem

We obtain an analytic solution of the problem of stressed state of a lap adhesive joint of two rectangular plates one of which is fixed along the lateral side. In the construction of solution, it is assumed that the transverse displacements of both bearing layers are equal to zero. We use the method of separation of variables and solve the model problem.

Investigation of the Fracture of a Semibounded Body Compressed Along a Near-Surface Interface Crack

We consider the problem of compression of a piecewise homogeneous half plane by forces directed along a near-surface crack located on the interface of two materials. This problem belongs to the class of nonclassical problems of fracture mechanics because, for the analyzed loading scheme, the stressstrain state realized in the body is homogeneous and the corresponding expressions for stresses and displacements in the vicinity of the crack do not contain singular components. Due to the fact that the stress intensity factors are equal to zero, for the analyzed problem, the classical Griffith–Irwin fracture criteria are inapplicable. In the indicated situation, the onset of crack propagation is associated with the local loss of stability of the equilibrium state of a part of the material located in the region adjacent to the crack. By using the approaches of the three-dimensional linearized theory of stability of deformable bodies, we propose a mathematical statement of the problem and an approach to its investigation. By using the representations of stresses and displacements in terms of complex potentials, we analyze the case where the roots of the corresponding characteristic equation are identical for each material.

Refining the Galerkin method error estimation for parabolic type problem with a boundary condition

The article considers a parabolic-type boundary value problem with a divergent principal part, when the boundary condition contains the time derivative of the required function: {ut−d/dxiai(x,t,u,∇u)+a(x,t,u,∇u)=0,a0ut+ai(x,t,u,∇u)cos(v,xi)=g(x,t,u,),(x,t)∈St,u(x,0)=u0(x),x∈ΩSuch nonclassical problems with boundary conditions containing the time derivative of the desired function arise in the study of a number of applied problems, for example, when the surface of a body, whose temperature is the same at all its points, is washed off by a well-mixed liquid, or when a homogeneous isotropic body is placed in the inductor of an induction furnace and an electro-magnetic wave falls on its surface. Such problems have been little studied, therefore, the study of problems of parabolic type, when the boundary condition contains the time derivative of the desired function, is relevant. In this paper, the definition of a generalized solution of the considered problem in the space H˜1,1(QT) is given. This problem is solved by the approximate Bubnov-Galerkin method. The coordinate system is chosen from the spaceH1(Ω). To determine the coefficients of the approximate solution, the parabolic problem is reduced to a system of ordinary differential equations. The aim of the study is to obtain conditions under which the estimate of the error of the approximate solution in the normH1(Ω) has orderO(hk−1) The paper first explores the auxiliary elliptic problem. When the condition of the ellipticity of the problem is satisfied, inequalities are proposed for the difference of the generalized solution of the considered parabolic problem with a divergent principal part, when the boundary condition contains the time derivative of the desired function and the solution of the auxiliary elliptic problem. Using these estimates, as well as under additional conditions for the coefficients and the function included in the problem under consideration, estimates of the error of the approximate solution of the Bubnov-Galerkin method in the normH1(Ω) of orderO(hk−1) for the considered nonclassical parabolic problem with divergent principal part, when the boundary condition contains the time derivative of the desired function.

The nonclassical problem of the compressed bar stability

The non-classical problem of the stability loss of a straight vertical bar with elastic flexible elements at the ends is considered. The mathematical model for the study of stability loss consists of a linearized differential equation for buckling of a constant cross section bar, supplemented by boundary conditions. Critical force is determined by analytical and numerical methods. Two versions of computational algorithms are presented. For one design scheme of the rod in the Matlab computing environment, the critical forces are calculated using two methods. The found critical forces differ insignificantly, which confirms the reliability and efficiency of the proposed algorithms.