The principal object of this volume is the creation of a mathematical theory of deformations for elastic anisotropic thermodynamic piezoelastic plates, beams and shells with variable thickness. The book is divided into two parts. The first part deals with problems related to the construction of refined theories (such as those of Richhof-Love, von Karman-A. Fioppl, and Reissner) and their equivalent new models (depending on arbitrary control functions). These are investigated by means of a new variational principle. Methods of reduction, containing regular processes of study of spatial problems, are also studied. Topics treated include problems of solvability, error estimations, convergence of processes in Sobolev spaces and construction of effective schemes of solutions of two-dimensional boundary value problems for systems of partial differential equations. The second part considers stable projective methods, using classical orthogonal polynomials and a new class of spline-functions as coordinate systems, and their numerical realizations for a design of one- and two- dimensional boundary value problems from the first part. These efficient methods increase the possibilities of classical finite-difference, exponential- fitted, variational-discrete and alternating-direction methods. Audience: This book will be of interest to researchers and graduate students whose work involves mechanics, analysis, numerics and computation, mathematical modelling and industrial mathematics, calculus of variations, and design engineering.

The main purpose of this work is construction of the mathematical theory of elastic plates and shells, by means of which the investigation of basic boundary value problems of the spatial theory of elasticity in the case of cylindrical do mains reduces to the study of two-dimensional boundary value problems (BVP) of comparatively simple structure. In this respect in sections 2-5 after the introductory material, methods of re duction, known in the literature as usually being based on simplifying hypotheses, are studied. Here, in contradiction to classical methods, the problems, connected with construction of refined theories of anisotropic nonhomogeneous plates with variable thickness without the assumption of any physical and geometrical re strictions, are investigated. The comparative analysis of such reduction methods was carried out, and, in particular, in section 5, the following fact was established: the error transition, occuring with substitution of a two-dimensional model for the initial problem on the class of assumed solutions is restricted from below. Further, in section 6, Vekua's method of reduction, containing regular pro cess of study of three-dimensional problem, is investigated. In this direction, the problems, connected with solvability, convergence of processes, and construction of effective algorithms of approximate solutions are studied."