Methods of Mathematical Physics II PDE     R.Courant and D.Hilbert

img1.gif I Introductory Remarks

  1. General Information about the Variety of Solutions
  2. Systems of Differential Equations
  3. Methods of Integration for Special Differential Equations
  4. Geometric Interpretation of a First Order Partial Differential Equations in Two Independent Variables  The Complete Integral
  5. Theory of Linear and Quasi-Linear Differential Equations  of First Order
  6. The Legendre Transformation
  7. The Existence Theorem of Cauchy and Kowalewsky

II General Theory of Partial Differential Equations of First Order

  1. Geometric Theory of Quasi-Linear Differential Equations in Two Independent Variables
  2. Quasi-Linear Differential Equations in n Independent Variables
  3. General Differential Equations in Two Independent Variables
  4. The Complete Integral
  5. Focal Curves and the Monge Equation
  6. Examples
  7. General Differential Equations in n Independent Variables
  8. Compete Integral and Hamiton-Jacobi Theory
  9. Hamilton-Jacobi Theory and the Variations
  10. Canonical Transformations and Appications

III Differential Equations of Higher Order

  1. Normal Forms for Linear and Quasi-Linear Differentail Operators of Second Order in Two Independent Variables
  2. Classification in General and Characteristics
  3. Linear Differential Equations with Constant coefficients
  4. Initial Value Problems   Radiation Problems for the Wave Equation
  5. Solution of Initial Value Problems by Fourier Integrals
  6. Typical Problems in Differential Equations of Mathematical Physics

IV Potential Theory and Elliptic Differential Equations

  1. Basic Notions
  2. Poisson's Integral and Applications
  3. The Mean Value Theorem and Applications
  4. The Boundary Value Problems
  5. The Reduced Wave Equation  Scattering
  6. Boundary Value Problems for More General Elliptic Differential Equations
  7. A Priori Estimates of Schauder and Their Applications
  8. Solution of the Beltrami Equations
  9. The Boundary Value Problem for a Special Quasi-Linear Equation
  10. Solution of Elliptic Differential Equations by Means of Integral Equations

V Hyperbolic Differential Equations in Two Independent Variables

  1. Characteristics for Differential Equations Mainly of Second Order
  2. Characteristics Normal Forms for Hyperbolic Systems of First Order
  3. Applications to Dynamics of Compressible Fluids
  4. Uniqueness  Domain of Dependence
  5. Riemann's Representation of Solutions
  6. Solutions of Hyperbolic Linear and Semilinear Initial Value Problems by Iteration
  7. Cauchy's Problem for Quasi-Linear Systems
  8. Cauchy's Problem for Single Hyperbolic Differential Equations of Higher Order
  9. Discontinuities of Soutions   Shocks

VI Hyperbolic Differential Equations in More Than Two Independent Variables

    Part I Uniqueness,Costruction,and Geometry of Solutions

  1. Differential Equations of Second Order  Goemetry of Characteristics
  2. Second Order Equations  The Role of Characteristics
  3. Geometry of Characteristics for Higher Order Operators
  4. Propagation of Discontinuities and Cauchy's Problem
  5. Oscillatory Initial Values ...
  6. Examples of Uniqueness Theorems and Domain of Dependence for Initial Value Problems
  7. Domains of Dependence for Hyperblolic Problems
  8. Energy Integrals and ...
  9. Energy Estimates for Equations of Higher Order
  10. The Existence Theorem

    Part II Representation of Solutions

  1.  Introduction
  2. Equations of Second Order with Constant Coefficients
  3. Method of Spherical Means
  4. Method of Plane Mean Values
  5. The Solution Cauchy's Problem as Linear Functions of Data Fundamental Solutions
  6. Ultrahyperbolic Differential Equations and General Differential Equations of Second Order with Constant Coefficients
  7. Initial Value Problems for Non-Space-Like Initial Manifolds
  8. Remarks about...