Mathematical Analysis and Numerical Methods for Science and Technology Volume 5 Evolution Problems I / by Robert Dautray, Jacques-Louis Lions.
299 G(t), and to obtain the corresponding properties of its Laplace transform (called the resolvent of - A) R(p) = (A + pl)-l , whose existence is linked with the spectrum of A. The functional space framework used will be, for simplicity, a Banach space(3). To summarise, we wish to extend definition...
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Berlin, Heidelberg :
Springer Berlin Heidelberg : Imprint: Springer,
2000.
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| Edition: | 1st ed. 2000. |
| Series: | Springer eBook Collection.
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Table of Contents:
- XIV. Evolution Problems: Cauchy Problems in IRn
- §1. The Ordinary Cauchy Problems in Finite Dimensional Spaces
- §2. Diffusion Equations
- §3. Wave Equations
- §4. The Cauchy Problem for the Schrödinger Equation, Introduction
- §5. The Cauchy Problem for Evolution Equations Related to Convolution Products
- §6. An Abstract Cauchy Problem. Ovsyannikov’s Theorem
- Review of Chapter XIV
- XV. Evolution Problems: The Method of Diagonalisation
- §1. The Fourier Method or the Method of Diagonalisation
- §2. Variations. The Method of Diagonalisation for an Operator Having Continuous Spectrum
- §3. Examples of Application: The Diffusion Equation
- §4. The Wave Equation: Mathematical Examples and Examples of Application
- §5. The Schrödinger Equation
- §6. Application with an Operator Having a Continuous Spectrum: Example
- Review of Chapter XV
- Appendix. Return to the Problem of Vibrating Strings
- XVI. Evolution Problems: The Method of the Laplace Transform
- §1. Laplace Transform of Distributions
- §2. Laplace Transform of Vector-valued Distributions
- §3. Applications to First Order Evolution Problems
- §4. Evolution Problems of Second Order in t
- §5. Applications
- Review of Chapter XVI
- XVII. Evolution Problems: The Method of Semigroups
- A. Study of Semigroups
- §1. Definitions and Properties of Semigroups Acting in a Banach Space
- §2. The Infinitesimal Generator of a Semigroup
- §3. The Hille—Yosida Theorem
- §4. The Case of Groups of Class &0 and Stone’s Theorem
- §5. Differentiable Semigroups
- §6. Holomorphic Semigroups
- §7. Compact Semigroups
- B. Cauchy Problems and Semigroups
- §1. Cauchy Problems
- §2. Asymptotic Behaviour of Solutions as t ? + ?. Conservation and Dissipation in Evolution Equations
- §3. Semigroups and Diffusion Problems
- §4. Groups and Evolution Equations
- §5. Evolution Operators in Quantum Physics. The Liouville—von Neumann Equation
- §6. Trotter’s Approximation Theorem
- Summary of Chapter XVII
- XVIII. Evolution Problems: Variational Methods
- Orientation
- §1. Some Elements of Functional Analysis
- §2. Galerkin Approximation of a Hilbert Space
- §3. Evolution Problems of First Order in t
- §4. Problems of First Order in t (Examples)
- §5. Evolution Problems of Second Order in t
- §6. Problems of Second Order in t. Examples
- §7. Other Types of Equation
- Review of Chapter XVIII
- Table of Notations
- of Volumes 1–4, 6.