Nonautonomous Periodic Boundary-Value Problems in a Special Critical Case

We study the problem of finding conditions for the existence of solutions of weakly nonlinear periodic boundary-value problems for systems of ordinary differential equations and the construction of these solutions. We consider the special critical case where the equation for generating amplitudes of a weakly nonlinear periodic boundary-value problem reduces to the identity. We construct a new classification of critical cases and an iteration algorithm for the construction of solutions of weakly nonlinear periodic boundary-value problems in a special critical case.     

Solving initial–boundary-value creep problems

The Galerkin–Bubnov method with global approximations is used to find approximate solutions to initial–boundary-value creep problems. It is shown that this approach allows obtaining solutions available in the literature. The features of how the solutions of initial–boundary-value problems for oneand three-dimensional models are found are analyzed. The approximate solutions found by the Galerkin–Bubnov method with global approximations is shown to be invariant to the form of the equations of the initial–boundary-value problem. It is established that solutions of initial–boundary-value creep problems can be classified according to the form of operators in the mathematical problem formulation     

On a difficulty in the formulation of initial and boundary conditions for eigenfunction expansion solutions for the start-up of fluid flow

Most mathematics and engineering textbooks describe the process of “subtracting off” the steady state of a linear parabolic partial differential equation as a technique for obtaining a boundary-value problem with homogeneous boundary conditions that can be solved by separation of variables (i.e., eigenfunction expansions). While this method produces the correct solution for the start-up of the flow of, e.g., a Newtonian fluid between parallel plates, it can lead to erroneous solutions to the corresponding problem for a class of non-Newtonian fluids. We show that the reason for this is the non-rigorous enforcement of the start-up condition in the textbook approach, which leads to a violation of the principle of causality. Nevertheless, these boundary-value problems can be solved correctly using eigenfunction expansions, and we present the formulation that makes this possible (in essence, an application of Duhamel's principle). The solutions obtained by this new approach are shown to agree identically with those obtained by using the Laplace transform in time only, a technique that enforces the proper start-up condition implicitly (hence, the same error cannot be committed).     

Numerical-analytic method for the investigation of boundary-value problems for semilinear systems of differential equations

We propose a new numerical-analytic algorithm for the investigation of boundary-value problems for semilinear systems of differential equations.     

On a Discrete Fourier Series Solution to Static Problem for Conical Shells of Circumferentially Varying Thickness

An approach is developed to solve the two-dimensional boundary-value problems of the stress-strain state of conical shells with circumferentially varying thickness. The approach employs discrete Fourier series to separate variables and make the problem one-dimensional. The one-dimensional boundary-value problem is solved by the stable discrete-orthogonalization method. The results obtained are presented as plots and tables __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 9, pp. 26–37, September 2005.     

Application of Topological Methods to Quasilinear Parabolic Boundary-Value Problems

We apply a topological approach to the investigation of quasilinear parabolic boundary-value problems. The class of problems under investigation is reduced to an operator equation with an operator satisfying condition (S)₊. We establish theorems on solvability and give an example of the application of the approach indicated to the case of second-order parabolic equations.     

Asymptotic analysis of perforated plates and membranes. Part 1: Static problems for small holes

Static problems for the elastic plates and rods periodically perforated by small holes of different shapes are solved using the asymptotic approach based on the combination of the asymptotic technique and the multi-scale homogenization method. Using the asymptotic homogenization method the original boundary-value problem is reduced to the combination of two types of problems. First one is a recurrent system of unit cell problems with the conditions of periodic continuation. And the second problem is a homogenized boundary-value problem for the entire domain, characterized by the constant effective coefficients obtained from the solution of the unit cell problems. The combination of the perturbation method and the technique of successive approximations is applied for the solution of the unit cell problems. Taking into the account small size of holes the method of perturbation of the shape of the boundary and the Schwarz alternating method are used. The problems of torsion of a rod with perforated cross-section; deflection of the perforated membrane loaded by a normal load; and bending of perforated plates with circular and square holes are considered consecutively. The error of the applied asymptotic techniques is estimated and the high accuracy of the obtained solutions is demonstrated.     

Boundary-value problems for linear equations with a generalized invertible operator in a Banach space with basis

We consider linear boundary-value problems for operator equations with generalized invertible operators in Banach spaces that have bases. Using the technique of generalized inverse operators applied to generalized invertible operators in Banach spaces, we establish conditions for the solvability of linear boundary-value problems for these operator equations and obtain formulas for the representation of their solutions. We consider special cases of these boundary-value problems, namely, so-called n- and d-normally solvable boundary-value problems as well as normally solvable problems for Noetherian operator equations.     

A numerical analytic method for solving boundary-value problems of linear anisotropic viscoelasticity

The paper proposes an approach to solving boundary-value problems for linear viscoelastic orthotropic bodies based on the method of operator continued fractions and the boundary-element method. A problem-solving algorithm and a procedure to estimate the possible error are outlined. The solution for a viscoelastic orthotropic plate with a rigid circular inclusion under uniaxial tension is obtained as an example and compared with available ones     

On singular perturbation boundary-value problem of coupling type system of convection-diffusion equations

In this paper we consider the singular perturbation boundary-value problem of thefollowing coupling type system of convection-diffusion equationsWe advance two methods:the first one is the initial value solving method,by which theoriginal boundary-value problem is changed into a series of unperturbed initial-valueproblems of the first order ordinary differential equation or system so that an asymptoticexpansion is obtained;the second one is the boundary-value solving method,by which theoriginal problem is changed into a few boundary-value problems having no phenomenon ofboundary-layer so that the exact solution can be obtained and any classical numericalmethods can be used to obtain the numerical solution of consismethods can be used to obtainthe numerical solution of consistant high accuracy with respect to the perturbationparameterε     

The Solution Asymptotics of Classical and Nonclassical, Static and Dynamic Boundary-Value Problems for Thin Bodies

An asymptotic method for solution of classical and nonclassical boundary-value problems of the theory of elasticity for thin bodies (beams, rods, plates, and shells) is expounded. Studies on the asymptotic theory of thin bodies are reviewed. Asymptotic results are compared with those obtained by other applied theories. The asymptotic approach has been found out to be related to Saint Venant's principle. The correctness of this principle is mathematically proved for one class of problems. A fundamentally new asymptotics in the components of the stress tensor and the displacement vector is revealed in considering new classes of problems. On their basis, the applicability domains are outlined for various models of understructures. Solutions are obtained to certain classes of dynamic problems for thin bodies, particularly, those simulating seismic effects. The resonance conditions are established and ways of preventing them are pointed out.     

Asymptotic analysis of boundary-value problems in thin perforated domains with rapidly varying thickness

For a second-order symmetric uniformly elliptic differential operator with rapidly oscillating coefficients, we study the asymptotic behavior of solutions of a mixed inhomogeneous boundary-value problem and a spectral Neumann problem in a thin perforated domain with rapidly varying thickness. We obtain asymptotic estimates for the differences between solutions of the original problems and the corresponding homogenized problems. These results were announced in Dopovidi Akademii Nauk Ukrainy, No. 10, 15–19 (1991). The new results obtained in the present paper are related to the construction of an asymptotic expansion of a solution of a mixed homogeneous boundary-value problem under additional assumptions of symmetry for the coefficients of the operator and for the thin perforated domain.     

The stress state of a laminated nonlinearly elastic thick-walled spherical shell

This paper deals with spatial axisymmetric boundary-value problems of the physically nonlinear theory of elasticity for piecewise-homogeneous spherical bodies. The passage to dimensionless characteristics of the stress-strain state allows us to extract a physical dimensionless small parameter in the nonlinear state equations. The solution of nonlinear equilibrium equations and boundary-value problems is searched for in the form of series in positive degrees of the small parameter. This approach allows reducing the stated physically nonlinear boundary-value problem to a sequence of corresponding linear nonhomogeneous problems. A specific analytical solution and numerical results are obtained for a two-layer nonlinearly elastic spherical shell under bilateral pressure. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 35, No. 12, pp. 26–32, December, 1999.     

Boundary-value problems for differential equations in a Banach space

We establish a criterion for the existence of solutions of linear inhomogeneous boundary-value problems in a Banach space. We obtain conditions for the normal solvability of such problems and consider their special cases, namely, countable-dimensional boundary-value problems. Translated from Neliniini Kolyvannya, Vol. 12, No. 1, pp. 3–15, January–March, 2009.     

Solving the Stress Problem for Hollow Cylinders with Corrugated Elliptical Cross Section

The paper presents an approach based on three-dimensional elastic equations to solve boundary-value stress problems for hollow cylinders with corrugated elliptical cross section. Discrete Fourier series are used to make the problem one-dimensional and then to solve it by the stable discrete orthogonalization method. Solutions for cylinders of different thicknesses are presented     

Applications of wavelet galerkin fem to bending of beam and plate structures

I.IntroductionItilasbeenl'oundthatthewavelettheoryisapowerfLllmathematicaltooldevelopedillrecentyears.Asanewmathematicaltool,ithasbeenextensivelyappliedintheanalysisofsignalprocess,parttenrecognition,functionapproximation,andsolvingdifferentialequation(s),etc..Sinceasmallsignalinasighalprocesscanbecapturedbythewavelettheory,itsapplicationshavebeenpaidmuchattentionbothintheoryandinengneeringf'~'].Recently,thewavelettheory11asbeengeneralizedtofindanumericalsolutionofadifferentialequation.Forex…     

Application of Discrete Fourier Series in the Stress Analysis of Cylindrical Shells of Variable Thickness with Arbitrary End Conditions

An approach is proposed to solve boundary-value stress—strain problems for cylindrical shells with thickness varying in two coordinate directions. The approach employs discrete Fourier series to separate circumferential variables. This makes it possible to reduce the problem to a one-dimensional one, which can be solved by the stable discrete-orthogonalization method. Examples are given __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 6, pp. 85–94, June 2005.     

Numerical-Analytic Solution of Boundary-Value Stress-Strain Problems for Rotating Anisotropic Cylinders

A numerical-analytical approach is proposed to solve boundary-value stress-strain problems for hollow inhomogeneous cylinders under centrifugal loading. Their elastic characteristics vary in both radial and circumferential directions. The governing system of ordinary differential equations is derived using Fourier series for stresses and displacements. It is solved by the discrete-orthogonalization method. Solutions to specific problems are exemplified __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 8, pp. 82–87, August 2005.     

Weakly perturbed boundary-value problems for differential equations in a Banach space

We consider boundary-value problems for a system of ordinary differential equations with a small parameter ε in the equations and boundary conditions. We establish conditions for the bifurcation of solutions of a weakly perturbed linear boundary-value problem in a Banach space.     

Construction of Basis Functions and Their Application to Boundary-Value Problems of Mechanics of Continuous Media

A unified method for constructing basis (eigen) functions is proposed to solve problems of mechanics of continuous media, problems of cubature and quadrature, and problems of approximation of hypersurfaces. Numerical-analytical methods are described, which allow obtaining approximate solutions of internal and external boundary-value problems of mechanics of continuous media of a certain class (both linear and nonlinear). The method is based on decomposition of the sought solutions of the considered partial differential equations into series in basis functions. An algorithm is presented for linearization of partial differential equations and reduction of nonlinear boundary-value problems, which are reduced to systems of linear algebraic equations with respect to unknown coefficients without using traditional methods of linearization.