Application of Analogues of General Kolosov–Muskhelishvili Representations in the Theory of Elastic Mixtures

The existence and uniqueness of a solution of the first, the second and the third plane boundary value problem are considered for the basic homogeneous equations of statics in the theory of elastic mixtures. Applying the general Kolosov–Muskhelishvili representations from [1], these problems can be split and reduced to the first and the second boundary value problem for an elliptic equation which structurally coincides with the equation of statics of an isotropic elastic body.     

Explicit solution of the boundary value problem of the theory of thermoelasticity with microtemperatures for elastic circle

In this paper we solve explicitly, by means of absolutely and uniformly convergent series, the 2D boundary value problem (BVP) of statics of the linear theory of thermoelasticity with microtemperatures for an elastic circle. The uniqueness theorem of the internal BVP is proved. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Basic Mixed Problem for an Anisotropic Elastic Body

The mixed boundary value problem is considered for an anisotropic elastic body under the condition that a boundary value of the displacement vector is given on some part of the boundary and a boundary value of the generalized stress vector on the remainder. Using the potential method and the theory of singular integral equations with discontinuous coefficients, the existence of a solution of the mixed boundary value problem is proved.     

Study of the solvability of a boundary value problem for the system of nonlinear differential equations of the theory of shallow shells of the Timoshenko type

We study the solvability of a boundary value problem for a system of nonlinear second-order partial differential equations under given boundary conditions, which describes the equilibrium of elastic shallow shells with hinged edges in the framework of the Timoshenko shear model. The study method implies the reduction of the original system of equations to a single nonlinear differential equation whose solvability is proved with the use of the contraction mapping principle.     

The boundary equation method in the third initial boundary value problem of the theory of elasticity. Part 2: Methods for approximate solutions

This work considers the methods for solving approximately the four types of boundary equations arising when the third initial boundary value problem of the theory of elasticity is solved with the help of retarded elastic potentials. The convergence of these methods is proved.     

Some basic boundary value problems of the plane thermoelasticity with microtemperatures

The present paper is devoted to the two‐dimensional version of statics of the linear theory of elastic materials with inner structure whose particles, in addition to the classical displacement and temperature fields, possess microtemperatures. Some results of the classical theories of elasticity and thermoelasticity are generalized. The Green's formulas in the case under consideration are obtained, basic boundary value problems are formulated, and uniqueness theorems are proved. The fundamental matrix of solutions for the governing system of the model and the corresponding single and double layer thermoelastopotentials are constructed. Properties of the potentials are studied. Applying the potential method, for the first and second boundary value problems, we construct singular integral equations of the second kind and prove the existence theorems of solutions for the bounded and unbounded domains. This paper describes the use of the LaTeX2? mmaauth.cls class file for setting papers for Mathematical Methods in the Applied Sciences. Copyright © 2012 John Wiley & Sons, Ltd.     

非线性边界条件的非线性发展方程解的爆破

讨论一类非线性发展方程具有非线性边界条件的初边值问题.在某些假设条件下,利用抛物型方程的最大值原理和凸性方法证明了其解在有限时间内具有爆破性质.     

具非线性边界条件的非线性发展方程解的Blow up

讨论一类非线性发展方程具有非线性边界条件的初边值问题.在某些假设条件下,利用抛物型方程的最大值原理和凸性方法证明了其解在有限时间内具有爆破性质.     

Steady vibrations problem in the theory of viscoelasticity for Kelvin-Voigt materials with voids

In this paper the linear theory of viscoelasticity for Kelvin-Voigt materials with voids is considered. The uniqueness and existence theorems for internal boundary value problem (BVP) of steady vibrations are proved by means of the potential method (boundary integral method) and the theory of singular integral equations. The application of this method to the 3D BVP of the considered theory reduces this problem to 2D singular integral equation. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

复合材料力学的Hamilton体系和辛几何方法(Ⅰ)——一般原理*

首次把用于动态体系的Hamilton系统引入到静力学中,建立了与原控制方程相对应的Hami-lton方程,可以对全状态向量分离变量,求出解析解和半解析解,特别适合于求解矩形域平面问题和柱形域空间问题.本文建立了一种求解偏微分方程的新方法,并对复合材料力学中的层合板的弯曲和平面应力问题的求解做了详细说明.     

半线性拟双曲型积分微分方程的初边值问题和初值问题

本文讨论半线性拟双曲型积分微分方程的初边值和初值问题。用积分方程的理论和Ｇａｌｅｒｋｉｎ方法证明了问题的整体强解的存在性、唯一性和正则性。     

Recovery of the unknown coefficients in a two-dimensional hyperbolic equation

In this paper, an inverse boundary value problem for a two-dimensional hyperbolic equation with overdetermination conditions is studied. To investigate the solvability of the original problem, we first consider an auxiliary inverse boundary value problem and prove its equivalence to the original problem in a certain sense. We then use the Fourier method to reduce such an equivalent problem to a system of integral equations. Furthermore, we prove the existence and uniqueness theorem for the auxiliary problem by the contraction mappings principle. Based on the equivalency of these problems, the existence and uniqueness theorem for the classical solution of the original inverse problem is proved. Some discussions on the numerical solutions for this inverse problem are presented including some numerical examples.     

An approximate solution of the axisymmetric contact problem for an elastic sphere

An axisymmetric, fractionally non-linear contact problem for an elastic sphere with a priori unknown boundary of the contactarea is considered. An integral equation for determining the density of the contact pressures is constructed taking account of the shear displacements of the boundary points of the elastic body. An approximate solution, which refines the equations of Hertz' theory, is constructed in the case of a small contact area.     

The boundary equation method in the third initial boundary value problem of the theory of elasticity. Part 1. Existence theorems

This work deals with four variants of boundary equations appearing if the third main initial boundary value problem is solved with the help of elastic retarded potentials. The solvability of these equations is proved, and the smoothness of their solutions researched.     

Irregular Boundary Value Problems with Discontinuous Coefficients and the Eigenvalue Parameter

In this study, a Birkhoff-irregular boundary value problem for linear ordinary differential equations of the second order with discontinuous coefficients and the spectral parameter has been considered. Therefore, at the discontinuous point, two additional boundary conditions (called transmission conditions) have been added to the boundary conditions. The eigenvalue parameter is of the second degree in the differential equation and of the first degree in a boundary condition. The equation contains an abstract linear operator which is (usually) unbounded in the space Lq(−1, 1). Isomorphism and coerciveness with defects 1 and 2 are proved for this problem. The case of the biharmonic equation is also studied.     

弹性薄板弯曲问题的等价的边界积分方程

用非解析开拓数学方法建立平面弹性薄板弯曲问题理论中具有间接变量的等价边界积分方程,并采用变分法进行了严格的说明.以往出现的三种间接变量边界积分方程,它们都不是等价的,对此我们进行了深入的讨论.     

Die Linienmethode bei nichtlinearen parabolischen Differentialgleichungen

The method of lines for parabolic differential equations consists in the discretization of the spatial variable only. In this way the first boundary value problem for a parabolic differential equation is transformed into an initial value problem for a system of ordinary differential equations. In this paper it is proved that for the general nonlinear parabolic equation the solution of the discrete problem converges to the solution of the original problem, when the mesh size tends to zero. The principal tool in this investigation is the theory of ordinary differential inequalities and especially the concept of quasimonotonicity.     

First Boundary Value Problem of Electroelasticity for a Transversally Isotropic Plane with Curvilinear Cuts

The first boundary value problem of electroelasticity for a transversally isotropic plane with curvilinear cuts is investigated. The solvability of a system of singular integral equations is proved by using the potential method and the theory of singular integral equations.     

Differentiation of the functional in a parametric optimization problem for a coefficient of a semilinear elliptic equation

We study parametric optimization with respect to an integral criterion of the higher coefficient and the right-hand side of a second-order semilinear elliptic equation with the Dirichlet boundary condition. We obtain formulas for the first partial derivatives of the objective functional with respect to the control parameters. The total preservation (preservation for the entire set of control parameters) of the unique solvability of the boundary value problem for this equation is proved based on the theory of monotone operators.     

高阶椭圆型偏微分方程奇异摄动问题一致收敛差分格式

本文,我们讨论了一类高阶椭圆型偏微分方程奇异摄动问题。给出了连续问题解的先验估计。另外,我们还提供了一种数值求解该类问题的指数型差分格式。最后,证明了差分问题的解在能量范数意义下关于小参数一致收敛到连续问题的解。