S.K. Godunov – Författare

This work presents a systematization of different models in mathematical physics, a study of the structure of conservation laws, thermodynamical identities, and connection with criteria for well-posedness of the corresponding mathematical problems. The theory presented in this book stems from research carried out by the authors concerning the formulations of differential equations describing explosive deformations of metals. In such processes, elasticity equations are used in some zones, whereas hydrodynamics equations are stated in other zones. Plastic deformations appear in transition zones, which leads to residual stresses. The suggested model contains some relaxation terms which simulate these plastic deformations. Certain laws of thermodynamics are used in order to describe and study differential equations simulating the physical processes. This leads to the special formulation of differential equations using generalized thermodynamical potentials. The structure of conservation laws and new ideas and methods of constructing mathematical models are presented.The book presents an approach to the formalization of equations of continuum mechanics, in particular, relationships between the structure of thermodynamical conservation laws and representations of the rotation group. It describes the theory developed by Godunov together with his former student Evgenii Romenskii, which presents a systematization of different models of elastic media and related classification of hyperbolic equations.

This volume deals with the theory of algorithms for solving systems of linear algebraic equations having a non-full-rank matrix of co-efficients. This involves a range of problems, such as the bidiagonalization of matrices, the computation of singular values and eigenvalues, procedures for the deflation of singular values, etc. The algorithms which are discussed in this book lead to computer programs which guarantee the accuracy of the computations, leading to unambiguous solutions. Some of the algorithms and techniques described are new; for example, the bounds which include underflow effects. Also discussed is an approach for computing reliable eigenvectors from Sturm sequences of a symmetric tridiagonal matrix, and a procedure for characterizing unitary transformations which maintain Hessenberg form. The work should be of interest to researchers whose work involves numerical methods of linear algebra.

This book discusses fundamental ideas of linear algebra. The author presents the spectral theory of nonselfadjoint matrix operators and matrix pencils in a finite dimensional Euclidean space. Statements of computational problems and brief descriptions of numerical algorithms, some of them nontraditional, are given. Proved in detail are classical problems that are not usually found in standard university courses. In particular, the material shows the role of delicate estimates for the resolvent of an operator and underscores the need for the study and use of such estimates in numerical analysis.

Elements of Continuum Mechanics and Conservation Laws presents a systematization of different models in mathematical physics, a study of the structure of conservation laws, thermodynamical identities, and connection with criteria for well-posedness of the corresponding mathematical problems. The theory presented in this book stems from research carried out by the authors concerning the formulations of differential equations describing explosive deformations of metals. In such processes, elasticity equations are used in some zones, whereas hydrodynamics equations are stated in other zones. Plastic deformations appear in transition zones, which leads to residual stresses. The suggested model contains some relaxation terms which simulate these plastic deformations. Certain laws of thermodynamics are used in order to describe and study differential equations simulating the physical processes. This leads to the special formulation of differential equations using generalized thermodynamical potentials.