Vladimir A. Marchenko – författare
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2 produkter
2 produkter
1 064 kr
Skickas inom 10-15 vardagar
Homogenization is a method for modelling processes in complex structures. These processes are far too complex for analytic and numerical methods and are best described by PDEs with rapidly oscillating coefficients - a technique that has become increasingly important in the last three decades due to its multiple applications in the areas of optimization, radiophysics, filtration theory, rheology, elasticity theory, and other domains of mechanics, physics, and technology. The present monograph is a comprehensive study of homogenization problems describing various physical processes in micro-inhomogeneous media. From the technical viewpoint the work focuses on the construction of nonstandard models for media characterized by several small-scale parameters (multiscale models). A variety of techniques are used --- specifically functional analysis, the spectral theory for differential operators, the Laplace transform, and, most importantly, a new variational PDE method for studying the asymptotic behavior of solutions of stationary boundary value problems. This new method can be applied to a wide variety of problems.Key topics in this systematic exposition include asymptotic analysis, Dirichlet- and Neumann-type boundary value problems, differential equations with rapidly oscillating coefficients, homogenization, homogenized and non-local models.Along with complete proofs of all main results, numerous examples of typical structures of micro-inhomogeneous media with their corresponding homogenized models are provided. Applied mathematicians, advanced-level graduate students, physicists, engineers, and specialists in mechanics will benefit from this monograph, which may be used in the classroom or as a comprehensive reference text.
859 kr
Skickas inom 11-20 vardagar
The spectral theory of Sturm-Liouville operators is a classical domain of analysis, comprising a wide variety of problems. Besides the basic results on the structure of the spectrum and the eigenfunction expansion of regular and singular Sturm-Liouville problems, it is in this domain that one-dimensional quantum scattering theory, inverse spectral problems, and the surprising connections of the theory with nonlinear evolution equations first become related. The main goal of this book is to show what can be achieved with the aid of transformation operators in spectral theory as well as in their applications. The main methods and results in this area (many of which are credited to the author) are for the first time examined from a unified point of view. The direct and inverse problems of spectral analysis and the inverse scattering problem are solved with the help of the transformation operators in both self-adjoint and nonself-adjoint cases. The asymptotic formulae for spectral functions, trace formulae, and the exact relation (in both directions) between the smoothness of potential and the asymptotics of eigenvalues (or the lengths of gaps in the spectrum) are obtained. Also, the applications of transformation operators and their generalizations to soliton theory (i.e., solving nonlinear equations of Korteweg-de Vries type) are considered. The new Chapter 5 is devoted to the stability of the inverse problem solutions. The estimation of the accuracy with which the potential of the Sturm-Liouville operator can be restored from the scattering data or the spectral function, if they are only known on a finite interval of a spectral parameter (i.e., on a finite interval of energy), is obtained.