Mathematical Analysis And Numerical Methods For Science And Technology - Volume 5, Evolution Problems I - Dautray Robert

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Résumé :
Volumes 5 and 6 examine evolution (i.e. time-dependent) problems. The methods presented are useful for physics and mechanics (Chapter XV) as well as electronics, automata theory and robotics, etc. (Chapter XVI). In Chapter XVII it is shown that for a large class of problems the solution may be expressed in terms of a family (or semi-group) of time-dependent operators. Chapter XVIII deals with variational methods which are the simplest and the most powerful methods applicable to both non-symmetric and time-dependent operators, and form a basis for the study of nonlinear problems. The linearised Navier-Stokes equations are considered in Chapter XIX with particular variational methods. Chapter XX presents the methods of computation for evolution problems, serving as a basis for the effective calculation of evolutions on computer. The final chapter presents the many important and specific properties of the operator of transport equations.

Sommaire:
["EVOLUTION PROBLEM: CAUCHY PROBLEMS IN IRn","The Ordinary Cauchy Problems in Finite Dimensional Spaces","Diffusion Equations","Wave Equations","The Cauchy Problem for the Schrödinger Equation, Introduction","The Cauchy Problem for Evolution Equations Related to Convolution Products","An Abstract Cauchy Problem","Ovsyannikov's Theorem","EVOLUTION PROBLEM: THE METHOD OF DIAGONALISATION","The Fourier Method or the Method of Diagonalisation","Variations","The Method of Diagonalisation for an Operator Having Continuous Spectrum","Examples of Application: The Diffusion Equation","The Wave Equation: Mathematical Examples and Examples of Application","The Schrödinger Equation","Application with an Operator having a Continuous Spectrum: Example Review of Chapter XV","Return to the problem of Vibrating Strings","EVOLUTION PROBLEM: THE METHOD OF THE LAPLACE TRANSFORM","Laplace Transform of Distributions","Laplace Transform of Vector-valued Distributions","Applications to First Order Evolution Problems","Evolution Problems of Second Order in t","Applications","EVOLUTION PROBLEMS: THE METHOD OF SEMIGROUPS","STUDY OF SEMIGROUPS","Definitions and Properties of Semigroups Acting in a Banach Space","The Infinitesimal Generator of a Semigroup","The Hille-Yosida Theorem","The Case of Groups of Class c° and Stone's Theorem","Differentiable Semigroups","Holomorphic Semigroups","Compact Semigroups","CAUCHY PROBLEMS AND SEMIGROUPS","Cauchy Problems","Semigroups and Diffusion Problems","Groups and Evolution Equations","Evolution Operators in Quantum Physics","The Liouville-von Neumann Equation","Trotter's Approximation Theorem","EVOLUTION PROBLEMS: VARIATIONAL METHODS","Some Elements of Functional Analysis","Galerkin Approximation of a Hilbert Space","Evolution problems of First Order in t","Problems of First order in t (Examples)","Evolution Problems of Second Order in t","Problems of Second order in t","Examples","Other Types of Equation."]