Asymptotic Solutions Of Boundary Value Problems Research Articles

The boundary value problems for the linear singularly perturbed systems of differential-algebraic equations

The article considers the main stages of the development of the theory of asymptotic integration of boundary-value problems for linear singularly perturbed differential-algebraic systems. The need of developing constructive methods of approximate integration of boundary-value problems for differential-algebraic systems is due to the importance of their practical application in the theory of nonlinear oscillations, stability of motion, control theory, radio engineering, and biology.
 In the present paper the authors offer a review of literary sources, which consider the methods of constructing asymptotic solutions of singularly perturbed systems of differential equations with a degenerate matrix with derivatives under the condition of stability of the spectrum of the limit pencil of matrices. It is noted that the problem of constructing asymptotic solutions of boundary-value problems for systems of this type is poorly studied, and therefore relevant. In particular, the question of the conditions for the existence and uniqueness of the solutions of these problems and the development of methods for constructing their asymptotics in various cases related to the behavior of the spectrum of the limit pencil of matrices has been poorly researched.

Free vibrations of nonlocally elastic rods

Several of the Eringen’s nonlocal stress models, including two-phase and purely nonlocal integral models, along with the simplified differential model, are studied in the case of free longitudinal vibrations of a nanorod, for various types of boundary conditions. Assuming the exponential attenuation kernel in the nonlocal integral models, the integro-differential equation corresponding to the two-phase nonlocal model is reduced to a fourth-order differential equation with additional boundary conditions, taking into account nonlocal effects in the neighbourhood of the rod ends. Exact analytical and asymptotic solutions of boundary-value problems are constructed. Formulas for natural frequencies and associated modes found in the framework of the purely nonlocal model and its ‘equivalent’ differential analogue are also compared. A detailed analysis of solutions suggests that the purely nonlocal and differential models lead to ill-posed problems.

Asymptotically Confirmed Hypotheses Method for the Construction of Micropolar and Classical Theories of Elastic Thin Shells

In the present paper asymptotic solution of boundary-value problem of three-dimensional micropolar theory of elasticity with free fields of displacements and rotations is constructed in thin domain of the shell. This boundary-value problem is singularly perturbed with small geometric parameter. Internal iteration process and boundary layers are constructed, problem of their jointing is studied and boundary conditions for each of them are obtained. On the basis of the results of the internal boundary-value problem the asymptotic two-dimensional model of micropolar elastic thin shells is constructed. Further, the qualitative aspects of the asymptotic solution are accepted as hypotheses and on the basis of them general applied theory of micropolar elastic thin shells is constructed. It is shown that both the constructed general applied theory of micropolar elastic thin shells and the classical theory of elastic thin shells with consideration of transverse shear deformations are asymptotically confirmed theories.

Asymptotic integration of degenerate singularly perturbed systems of parabolic partial differential equations

Eidel’man, Butuzov, and Ivasishen. In particular, Ladyzhenskaya and Ural’tseva established sufficient conditions for the existence and uniqueness of classical and generalized solutions of main boundary-value problems in the linear case and substantiated the applicability of the Fourier method to the solution of these problems [1, pp. 295‐299]. Analogous results on the existence and uniqueness of solutions of the corresponding problems for singularly perturbed equations were obtained by Oleinik in [2]. In [3], Butuzov proposed an original method for the solution of boundary-value problems for singularly perturbed partial differential equations. According to the method of angular boundary functions developed by Butuzov, asymptotic solutions of boundary-value problems are constructed in the form of regular and boundary-layer parts. In this case, functions of the boundary-layer part are solutions of certain differential equations with constant coefficients that satisfy certain boundary conditions. In the present paper, we consider the first boundary-value problem for a degenerate, linear, singularly perturbed parabolic system. The statement of the problem is close to problems investigated by Mitropol’skii and Khoma for regularly perturbed quasilinear and nonlinear equations of the hyperbolic type [4, pp. 137‐ 225]. In the solution of the first boundary-value problem for hyperbolic equations with slowly varying coefficients, Feshchenko and Shkil’ used the Fourier method [5, pp. 226‐245]. However, the question of the convergence of the corresponding series and the possibility of their term-by-term differentiation remains open. Note that equations with slowly varying coefficients can be reduced to singularly perturbed equations by a change of the independent variable. Consider the problem

Mathematical model of micropolar elastic thin plates and their strength and stiffness characteristics

A number of hypotheses were formulated using the properties of an asymptotic solution of boundary-value problems of the three-dimensional micropolar (moment asymmetric) theory of elasticity for areas with one geometrical parameter being substantially smaller than the other two (plates and shells). A general theory of bending deformation of micropolar elastic thin plates with independent fields of displacements and rotations is constructed. In the constructed model of a micropolar elastic plate, transverse shear strains are fully taken into account. A problem of determining the stress-strain state in bending deformation of micropolar elastic thin rectangular plates is considered. The numerical analysis reveals that plates made of a micropolar elastic material have high strength and stiffness characteristics.

Asymptotic Solution of a Boundary-Value Problem for Linear Singularly-Perturbed Functional Differential Equations Arising in Optimal Control Theory

The Hamiltonian boundary-value problem, associated with a singularly-perturbed linear-quadratic optimal control problem with delay in the state variables, is considered. A formal asymptotic solution of this boundary-value problem is constructed by application of the boundary function method. The justification of this asymptotic solution is done. The asymptotic solution of the Hamiltonian boundary-value problem is constructed and justified assuming boundary-layer stabilizability and detectability.

Steady heat-conduction problem for a double cylinder

An analytical solution of the problem stated in the title is obtained for various conditions at the ends of the cylinder. The method of asymptotic solution of boundary-value problems for elliptic equations in thin domains is translated to the case of boundary conditions of the third kind.