Compression of an orthotropic prism

The formulation and mathematically correct solution of a three-dimensional boundary value problem are given. A numerical calculation is presented for a prism of Canadian spruce and the displacement and stress fields are determined.Presented at a scientific seminar on polymer mechanics held in the Mechanics-Mathematics Department of Moscow State University, February 7, 1968.M. V. Lomonosov Moscow State University. Translated from Mekhanika Polimerov, Vol. 4, No. 5, pp. 810–815, September–October, 1968.     

Statistical boundary value problems of polymer mechanics

Research on statistical boundary value problems with applications in the mechanics of reinforced polymers is briefly reviewed. The common features of the models and mathematical methods used in connection with polycrystalline metals and materials of the glass-reinforced plastic type are noted.Sverdlovsk Branch of the V. A. Steklov Mathematical Institute, Academy of Sciences of the USSR. Translated from Mekhanika Polimerov, No. 1, pp. 63–69, January–February, 1969.     

The coupling problem in polymer mechanics

The present status of the coupling problem in polymer mechanics is discussed. The coupled problems of thermoelasticity and thermoviscoelasticity are formulated and methods of solving them are examined.Presented at the 11th Scientific Conference on Thermal Stresses in Structural Elements, Kanev, June 2, 1970.Lomonosov Moscow State University. Translated from Mekhanika Polimerov, No. 1, pp. 59–65, January–February, 1971.     

The Dependence of Solution Uniqueness Classes of Boundary Value Problems for General Parabolic Systems on the Geometry of an Unbounded Domain

General boundary value problems are considered for general parabolic (in the Douglas–Nirenberg–Solonnikov sense) systems. The dependence of solution uniqueness classes of these problems on the geometry of a nonbounded domain is established.     

Several existence theorems of nonlinear m-point boundary value problem for p-Laplacian dynamic equations on time scales

In this paper, several existence theorems of positive solutions are established for nonlinear m-point boundary value problem for p-Laplacian dynamic equations on time scales, as an application, an example to demonstrate our results is given. The conditions we used in the paper are different from those in [H.R. Sun, W.T. Li, Positive solutions for nonlinear three-point boundary value problems on time scales, J. Math. Anal. Appl. 299 (2004) 508–524; H.R. Sun, W.T. Li, Positive solutions for nonlinear m-point boundary value problems on time scales, Acta Math. Sinica 49 (2006) 369–380 (in Chinese); Y. Wang, C. Hou, Existence of multiple positive solutions for one-dimensional p-Laplacian, J. Math. Anal. Appl. 315 (2006) 144–153; Y. Wang, W. Ge, Positive solutions for multipoint boundary value problems with one-dimensional p-Laplacian, Nonlinear Appl. 66 (6) (2007) 1246–1256].     

Towards computability of elliptic boundary value problems in variational formulation

We present computable versions of the Fréchet–Riesz Representation Theorem and the Lax–Milgram Theorem. The classical versions of these theorems play important roles in various problems of mathematical analysis, including boundary value problems of elliptic equations. We demonstrate how their computable versions yield computable solutions of the Neumann and Dirichlet boundary value problems for a simple non-symmetric elliptic differential equation in the one-dimensional case. For the discussion of these elementary boundary value problems, we also provide a computable version of the Theorem of Schauder, which shows that the adjoint of a computably compact operator on Hilbert spaces is computably compact again.     

Oscillation properties for the equation of vibrating beam with irregular boundary conditions

In this paper we study the oscillatory properties for the eigenfunctions of some fourth-order eigenvalue problems, where the boundary conditions are irregular in the sense of the classification of [S. Janczewski, Oscillation theorems for the differential boundary value problems of the fourth order, Ann. of Math. 29 (1928) 521–542]. In this case, we show that these oscillatory properties are different from those of the Sturm–Liouville problem.     

分数阶微分方程边值问题研究简介

分数阶微积分是一个古老而又新颖的课题,近30年来,由于在包括分形现象在内的物理、工程等诸多应用学科领域应用的拓展,激发了科研人员对分数阶微积分的巨大热情。分数阶微分方程现在已应用于分数物理学、混沌与湍流、粘弹性力学与非牛顿流体力学、高分子材料的解链、自动控制理论、化学物理、随机过程和反常扩散等许多科学领域。分数阶微分方程边值问题是非线性常微分方程理论研究中一个活跃而成果丰硕的领域。本文讨论了分数阶微分方程边值问题的一些理论,介绍了作者的著作《分数阶微分方程边值问题理论及应用》的基本内容。     

Scalar discrete nonlinear multipoint boundary value problems

In this paper we provide sufficient conditions for the existence of solutions to scalar discrete nonlinear multipoint boundary value problems. By allowing more general boundary conditions and by imposing less restrictions on the nonlinearities, we obtain results that extend previous work in the area of discrete boundary value problems [Debra L. Etheridge, Jesús Rodriguez, Periodic solutions of nonlinear discrete-time systems, Appl. Anal. 62 (1996) 119–137; Debra L. Etheridge, Jesús Rodriguez, Scalar discrete nonlinear two-point boundary value problems, J. Difference Equ. Appl. 4 (1998) 127–144].     

Boundary Hemivariational Inequalities of Hyperbolic Type and Applications

In this paper we examine two classes of nonlinear hyperbolic initial boundary value problems with nonmonotone multivalued boundary conditions characterized by the Clarke subdifferential. We prove two existence results for multidimensional hemivariational inequalities: one for the inequalities with relation between reaction and velocity and the other for the expressions containing the reaction–displacement law. The existence of weak solutions is established by using a surjectivity result for pseudomonotone operators and a priori estimates. We present also an example of dynamic viscoelastic contact problem in mechanics which illustrate the applicability of our results.Mathematics Subject Classifications (2000). 34G20, 35A15, 35L85, 35L70, 74H20     

Imposing boundary conditions in Sinc method using highest derivative approximation

Sinc approximate methods are often used to solve complex boundary value problems such as problems on unbounded domains or problems with endpoint singularities. A recent implementation of the Sinc method [Li, C. and Wu, X., Numerical solution of differential equations using Sinc method based on the interpolation of the highest derivatives, Applied Mathematical Modeling 31 (1) 2007 1–9] in which Sinc basis functions are used to approximate the highest derivative in the governing equation of the boundary value problem is evaluated for structural mechanics applications in which interlaminar stresses are desired. We suggest an alternative approach for specifying the boundary conditions, and we compare the numerical results for analysis of a laminated composite Timoshenko beam, implementing both Li and Wu’s approach and our alternative approach for applying the boundary conditions. For the Timoshenko beam problem, we obtain accurate results using both approaches, including transverse shear stress by integration of the 3D equilibrium equations of elasticity. The beam results indicate our approach is less dependent on the selection of the Sinc mesh size than Li and Wu’s SIHD. We also apply SIHD to analyze a classical laminated composite plate. For the plate example, we experience difficulty in obtaining a complete system of equations using Li and Wu’s approach. For our approach, we suggest that additional necessary information may be obtained by applying the derivatives of the boundary conditions on each edge. Using this technique, we obtain accurate results for deflection and stresses, including interlaminar stresses by integration of the 3D equilibrium equations of elasticity. Our results for both the beam and the plate problems indicate that this approach is easily implemented, has a high level of accuracy, and good convergence properties.     

Multiplicity results for Sturm–Liouville boundary value problems

Multiplicity results for Sturm–Liouville boundary value problems are obtained. Proofs are based on variational methods.     

Analysis of glass-reinforced plastic bottoms of cylindrical pressure vessels

Solutions are proposed for boundary value problems associated with spherical, ellipsoidal and optimal-design glass-reinforced plastic bottoms filament-wound as an integral part of cylindrical shells.Moscow Region. Translated from Mekhanika Polimerov, Vol. 4, No. 5, pp. 918–926, September–October, 1968.     

Fundamental theories and methods of polymer mechanics

The fundamental theories and methods of polymer mechanics are analyzed in the light of modern ideas about the structure and mechanical properties of polymers.Lomonosov Moscow State University. Translated from Mekhanika Polimerov, Vol. 5, No. 1, pp. 3–13, January–February, 1969.     

Vibrations of isotropic viscoelastic shells

It is shown that the boundary value problems for viscoelastic and elastic shells coincide. The natural and forced vibrations of shells are investigated by the averaging method.Lomonosov Moscow State University. Translated from Mekhanika Polimerov, No. 1, pp. 157–163, January–February, 1971.     

Free vibrations of a gently sloping spherical shell with elastic periphery

The problem of free vibrations of a gently sloping spherical shell, rectangular in plan and with a periphery elastic in relation to the angle of rotation, is investigated by the method of decomposition. A simple approximate formula for the value of the lowest frequency is obtained. The formula is analyzed, and a comparison with known solutions for special cases of boundary conditions is provided. In distinction to some previously examined problems of structural mechanics, the method was used for solving a system of differential equations.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 20, pp. 76–80, 1989.     

Three-Dimensional Boundary Value Problems of Elastothermodiffusion with Mixed Boundary Conditions

We investigate the basic boundary value problems of the connected theory of elastothermodiffusion for three-dimensional domains bounded by several closed surfaces when the same boundary conditions are fulfilled on every separate boundary surface, but these conditions differ on different groups of surfaces. Using the results of papers [1–8], we prove theorems on the existence and uniqueness of the classical solutions of these problems.     

Uniqueness of solutions to inverse Sturm–Liouville problems with potential using three spectra

The uniqueness of solutions to two inverse Sturm–Liouville problems using three spectra is proven, based on the uniqueness of the solution-pair to an overdetermined Goursat–Cauchy boundary value problem. We discuss the uniqueness of the potential for a Dirichlet boundary condition at an arbitrary interior node, and for a Robin boundary condition at an arbitrary interior node, whereas at the exterior nodes we have Dirichlet boundary conditions in both situations. Here we are particularly concerned with potential functions that are L²(0,a).     

On a new method of solving a class of problems of applied mechanics usingR-functions,weighted discrepancies,and least squares

We propose a new approach to solving a class of problems of applied mechanics using R-functions and boundary integral equations. We give the results of numerical analysis of the solution of second-order differential equations with constant coefficients using the example of the torsion of orthotropic rods.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 36, 1992, pp. 25–30.     

Linear Noetherian boundary value problems for differential systems with an impulse action

A solvability criterion of linear nonhomogeneous boundary value problems is obtained for systems of ordinary differential equations with impulse action in the general case when the number of boundary conditions does not coincide with the order of the differential system (Noetherian problems). The generalized Green operator of such boundary value problems is constructed. Its connection with the generalized inverse operator of the operator of the original boundary value problem is shown.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 4, pp. 564–568, April, 1992.