Methods of Mathematical Physics II PDE R.Courant
and D.Hilbert
I Introductory Remarks
- General Information about the Variety of Solutions
- Systems of Differential Equations
- Methods of Integration for Special Differential Equations
- Geometric Interpretation of a First Order Partial Differential Equations
in Two Independent Variables The Complete Integral
- Theory of Linear and Quasi-Linear Differential Equations of
First Order
- The Legendre Transformation
- The Existence Theorem of Cauchy and Kowalewsky
II General Theory of Partial Differential Equations of First
Order
- Geometric Theory of Quasi-Linear Differential Equations in Two Independent
Variables
- Quasi-Linear Differential Equations in n Independent Variables
- General Differential Equations in Two Independent Variables
- The Complete Integral
- Focal Curves and the Monge Equation
- Examples
- General Differential Equations in n Independent Variables
- Compete Integral and Hamiton-Jacobi Theory
- Hamilton-Jacobi Theory and the Variations
- Canonical Transformations and Appications
III Differential Equations of Higher Order
- Normal Forms for Linear and Quasi-Linear Differentail Operators
of Second Order in Two Independent Variables
- Classification in General and Characteristics
- Linear Differential Equations with Constant coefficients
- Initial Value Problems Radiation Problems for the Wave
Equation
- Solution of Initial Value Problems by Fourier Integrals
- Typical Problems in Differential Equations of Mathematical Physics
IV Potential Theory and Elliptic Differential Equations
- Basic Notions
- Poisson's Integral and Applications
- The Mean Value Theorem and Applications
- The Boundary Value Problems
- The Reduced Wave Equation Scattering
- Boundary Value Problems for More General Elliptic Differential Equations
- A Priori Estimates of Schauder and Their Applications
- Solution of the Beltrami Equations
- The Boundary Value Problem for a Special Quasi-Linear Equation
- Solution of Elliptic Differential Equations by Means of Integral
Equations
V Hyperbolic Differential Equations in Two Independent Variables
- Characteristics for Differential Equations Mainly of Second Order
- Characteristics Normal Forms for Hyperbolic Systems of First Order
- Applications to Dynamics of Compressible Fluids
- Uniqueness Domain of Dependence
- Riemann's Representation of Solutions
- Solutions of Hyperbolic Linear and Semilinear Initial Value Problems
by Iteration
- Cauchy's Problem for Quasi-Linear Systems
- Cauchy's Problem for Single Hyperbolic Differential Equations of
Higher Order
- Discontinuities of Soutions Shocks
VI Hyperbolic Differential Equations in More Than Two Independent
Variables
Part I Uniqueness,Costruction,and Geometry
of Solutions
- Differential Equations of Second Order Goemetry of Characteristics
- Second Order Equations The Role of Characteristics
- Geometry of Characteristics for Higher Order Operators
- Propagation of Discontinuities and Cauchy's Problem
- Oscillatory Initial Values ...
- Examples of Uniqueness Theorems and Domain of Dependence for Initial
Value Problems
- Domains of Dependence for Hyperblolic Problems
- Energy Integrals and ...
- Energy Estimates for Equations of Higher Order
- The Existence Theorem
Part II Representation of Solutions
- Introduction
- Equations of Second Order with Constant Coefficients
- Method of Spherical Means
- Method of Plane Mean Values
- The Solution Cauchy's Problem as Linear Functions of Data Fundamental
Solutions
- Ultrahyperbolic Differential Equations and General Differential
Equations of Second Order with Constant Coefficients
- Initial Value Problems for Non-Space-Like Initial Manifolds
- Remarks about...