Theory of Partial Differential Equations

Name: Theory of Partial Differential Equations
Brand: Academic Press
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Lieberstein(Herausgeber*in)

Academic Press

Erschienen am 14. Mai 2014

299 Seiten

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Front Cover
Theory of Partial Differential Equations
Copyright Page
Contents
PREFACE
PART I: AN OUTLINE
Chapter 1. The Theory of Characteristics, Classification, and the Wave Equation in E2
1. D' Alembert Solution of the Cauchy Problem for the Homogeneous Wave Equation in E2
2. Nomenclature
3. Theory of Characteristics and Type Classification for Equations in E2
4. Considerations Special to Nonlinear Cases
5. Compatibility Relations and the Finite-Difference Method of Characteristics
6. Systems Larger Than Two by Two
7. Flow and Transmission Line Equations
Chapter 2. Various Boundary-Value Problems for the Homogeneous Wave Equation in E2
1. The Cauchy or Initial-Value Problem
2. The Characteristic Boundary-Value Problem
3. The Mixed Boundary-Value Problem
4. The Goursat Problem
5. The Vibrating String Problem
6. Uniqueness of the Vibrating String Problem
7. The Dirichlet Problem for the Wave Equation?
Chapter 3. Various Boundary-Value Problems for the Laplace Equation in E2
1. The Dirichlet Problem
2. Relation to Analytic Functions of a Complex Variable
3. Solution of the Dirichlet Problem on a Circle
4. Uniqueness for Regular Solutions of the Dirichlet and Neumann Problem on a Rectangle
5. Approximation Methods for the Dirichlet Problem in E2
6. The Cauchy Problem for the Laplace Equation
Chapter 4. Various Boundary-Value Problems for Simple Equations of Parabolic Type
1. The Slab Problem
2. An Alternative Proof of Uniqueness
3. Solution by Separation of Variables
4. Instability for Negative Times
5. Cauchy Problem on the Infinite Line
6. Unique Continuation
7. Poiseuille Flow
8. Mean-Square Asymptotic Uniqueness
9. Solution of a Dirichlet Problem for an Equation of Parabolic Type
Chapter 5. Expectations for Well-Posed Problems
1. Sense of Hadamard
2. Expectations
3. Boundary-Value Problems for Equations of Elliptic-Parabolic Type
4. Existence as the Limit of Regular Solutions
5. The Impulse Problem as a Prototype of a Solution in Terms of Distributions
6. The Green Identities
7. The Generalized Green Identity
8. Lp-Weak Solutions
9. Prospectus
10. The Tricomi Problem
PART II: SOME CLASSICAL RESULTS FOR NONLINEAR EQUATIONS IN TWO INDEPENDENT VARIABLES
Chapter 6. Existence and Uniqueness Considerations for the Nonhomogeneous Wave Equation in E2
1. Notation
2. Existence for the Characteristic Problem
3. Comments on Continuous Dependence and Error Bounds
4. An Example Where the Theorem as Stated Does Not Apply
5. A Theorem Using the Lipschitz Condition on a Bounded Region in E5
6. Existence Theorem for the Cauchy Problem of the Nonhomogeneous (Nonlinear) Wave Equation in E2
Chapter 7. The Riemann Method
1. Three Forms of the Generalized Green Identity
2. Riemann's Function
3. An Integral Representation of the Solution of the Characteristic Boundary-Value Problem
4. Determination of the Riemann Function for a Class of Self-Adjoint Cases
5. An Integral Representation of the Solution of the Cauchy Problem
Chapter 8. Classical Transmission Line Theory
1. The Transmission Line Equations
2. The Kelvin r-c Line
3. Pure I-c Line
4. Heaviside's r-c-l-g Distortion-Free Balanced Line
5. Contribution of Du Boise-Reymond and Picard to the Heaviside Position
6. Realization
7. Neurons
Chapter 9. The Cauchy-Kovalevski Theorem
1. Preliminaries
Multiple Series
2. Theorem Statement and Comments
3. Simplification and Restatement
4. Uniqueness
5. The first Majorant Problem
6. An Ordinary Differential Equation Problem
7. Remarks and Interpretations
PART III: SOME CLASSICAL RESULTS FOR THE LAPLACE AND WAVE EQUATIONS IN HIGHER-DIMENSIONAL SPACE
Chapter 10. A Sketch of Potential Theory
1. Uniqueness of the Dirichlet Problem Using the Divergence Theorem
2. The Third Green Identity in E3
3. Uses of the Third Identity and Its Derivation for En, n ? 3
4. The Green Function
5. Representation Theorems Using the Green Function
6. Variational Methods
7. Description of Torsional Rigidity
8. Description of Electrostatic Capacitance, Polarization, and Virtual Mass
9. The Dirichlet Integral as a Quadratic Functional
10. Dirichlet and Thompson Principles for Some Physical Entities
11. Eigenvalues as Quadratic Functionals
Chapter 11. Solution of the Cauchy Problem for the Wave Equation in Terms of Retarded Potentials
1. Introduction
2. Kirchhoff's Formula
3. Solution of the Cauchy Problem
4. The Solution in Mean-Value Form
5. Verification of the Solution of the Homogeneous Wave Equation
6. Verification of the Solution to the Homogeneous Boundary-Value Problem
7. The Hadamard Method of Descent
8. The Huyghens Principle
PART IV: BOUNDARY-VALUE PROBLEMS FOR EQUATIONS OF ELLIPTIC-PARABOLIC TYPE
Chapter 12. A Priori Inequalities
1. Some Preliminaries
2. A Property of Semidefinite Quadratic Forms
3. The Generalized Green Identity Using ? = (u2 + d)P/2
4. A First Maximum Principle
5. A Second Maximum Principle
Chapter 13. Uniqueness of Regular Solutions and Error Bounds in Numerical Approximation
1. A Combined Maximum Principle
2. Uniqueness of Regular Solutions
3. Error Bounds in Maximum Norm
4. Error Bounds in Lp-Norm
5. Computable Bounds for the L2-Norm of an Error Function
Chapter 14. Some Functional Analysis
1. General Preliminaries
2. The Hahn-Banach Theorem, Sublinear Case
3. Normed Spaces and Continuous Linear Operators
4. Banach Spaces
5. The Hahn-Banach Theorem for Normed Spaces
6. Factor Spaces
7. Statement (Only) of the Closed Graph Theorem
Chapter 15. Existence of Lp-Weak Solutions
1. A First Form of the Abstract Existence Principle
2. Function Spaces Lp and Lp/(p- 1)
Riesz Representation
3. A Reformulation of the Abstract Existence Principle
4. Application of the Reformulated Principle to Lp-Weak Existence
5. Uniqueness of Lp-Weak Solutions
6. Prospectus
NOTES
REFERENCES
INDEX

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