Contents

• 1. Introduction
  • 1.1 Basic Concepts
    • 1.1.1 The prevalence of partial differential equations
    • 1.1.2 Definitions
    • 1.1.3 Typical boundary conditions
  • 1.2 The Standard Examples
    • 1.2.1 The Laplace equation
      • 1.2.1.1 Solution stanexl-a
      • 1.2.1.2 Solution stanexl-b
      • 1.2.1.3 Solution stanexl-b1
      • 1.2.1.4 Solution stanexl-b2
      • 1.2.1.5 Solution stanexl-b3
      • 1.2.1.6 Solution stanexl-b5
      • 1.2.1.7 Solution stanexl-c
      • 1.2.1.8 Solution stanexl-d
      • 1.2.1.9 Solution stanexl-e
    • 1.2.2 The heat equation
      • 1.2.2.1 Solution stanexh-a
      • 1.2.2.2 Solution stanexh-b
    • 1.2.3 The wave equation
      • 1.2.3.1 Solution stanexw-a
      • 1.2.3.2 Solution stanexw-b
      • 1.2.3.3 Solution stanexw-c
      • 1.2.3.4 Solution stanexw-e
      • 1.2.3.5 Solution stanexw-f
  • 1.3 Classification
    • 1.3.1 Introduction
    • 1.3.2 Scalar second order equations
      • 1.3.2.1 Solution clasnd-a
  • 1.4 Changes of Coordinates
    • 1.4.1 Introduction
    • 1.4.2 The formulae for coordinate transformations
    • 1.4.3 Rotation of coordinates
      • 1.4.3.1 Solution rotcoor-a
    • 1.4.4 Explanation of the classification
      • 1.4.4.1 Solution expclass-a
  • 1.5 Two-Dimensional Coordinate Transforms
    • 1.5.1 Characteristic Coordinates
    • 1.5.2 Parabolic equations in two dimensions
    • 1.5.3 Elliptic equations in two dimensions
      • 1.5.3.1 Solution 2dcanel-a
  • 1.6 Properly Posedness
    • 1.6.1 The conditions for properly posedness
      • 1.6.1.1 Solution ppc-a
      • 1.6.1.2 Solution ppc-b
      • 1.6.1.3 Solution ppc-c
    • 1.6.2 An improperly posed parabolic problem
    • 1.6.3 An improperly posed elliptic problem
      • 1.6.3.1 Solution ppe-a
      • 1.6.3.2 Solution ppe-b
      • 1.6.3.3 Solution ppe-c
      • 1.6.3.4 Solution ppe-d
      • 1.6.3.5 Solution ppe-e
      • 1.6.3.6 Solution ppe-f
    • 1.6.4 Improperly posed hyperbolic problems
      • 1.6.4.1 Solution pph-a
      • 1.6.4.2 Solution pph-b
  • 1.7 Energy methods
    • 1.7.1 The Poisson equation
      • 1.7.1.1 Solution emp-a
      • 1.7.1.2 Solution emp-b
    • 1.7.2 The heat equation
    • 1.7.3 The wave equation
  • 1.8 Variational methods [None]
  
  
• 2. Green’s Functions
  • 2.1 Introduction
    • 2.1.1 The one-dimensional Poisson equation
      • 2.1.1.1 Solution gf1d-a
      • 2.1.1.2 Solution gf1d-b
    • 2.1.2 More on delta and Green’s functions
  • 2.2 The Poisson equation in infinite space
    • 2.2.1 Overview
    • 2.2.2 Loose derivation
      • 2.2.2.1 Solution pninfl-a
    • 2.2.3 Rigorous derivation
  • 2.3 The Poisson or Laplace equation in a finite region
    • 2.3.1 Overview
    • 2.3.2 Intro to the solution procedure
    • 2.3.3 Derivation of the integral solution
      • 2.3.3.1 Solution pnfd-a
    • 2.3.4 Boundary integral (panel) methods
    • 2.3.5 Poisson’s integral formulae
    • 2.3.6 Derivation
      • 2.3.6.1 Solution pnifd-a
      • 2.3.6.2 Solution pnifd-b
    • 2.3.7 The integral formula for the Neumann problem
    • 2.3.8 Smoothness of the solution
  
  
• 3. First Order Equations
  • 3.1 Classification and characteristics
  • 3.2 Numerical solution
  • 3.3 Analytical solution
  • 3.4 Application of the boundary or initial condition
  • 3.5 The inviscid Burgers’ equation
    • 3.5.1 Wave steepening
    • 3.5.2 Shocks
    • 3.5.3 Conservation laws
    • 3.5.4 Shock relation
    • 3.5.5 The entropy condition
  • 3.6 First order equations in more dimensions
  • 3.7 Systems of First Order Equations (None)
  
  
• 4. D'Alembert Solution of the Wave equation
  • 4.1 Introduction
  • 4.2 Extension to finite regions
    • 4.2.1 The physical problem
    • 4.2.2 The mathematical problem
    • 4.2.3 Dealing with the boundary conditions
    • 4.2.4 The final solution
  
  
• 5. Separation of Variables
  • 5.1 A simple example
    • 5.1.1 The physical problem
    • 5.1.2 The mathematical problem
    • 5.1.3 Outline of the procedure
    • 5.1.4 Step 1: Find the eigenfunctions
    • 5.1.5 Should we solve the other equation?
    • 5.1.6 Step 2: Solve the problem
  • 5.2 Comparison with D'Alembert
  • 5.3 Understanding the Procedure
    • 5.3.1 An ordinary differential equation as a model
    • 5.3.2 Vectors versus functions
    • 5.3.3 The inner product
    • 5.3.4 Matrices versus operators
    • 5.3.5 Some limitations
  • 5.4 An Example with Periodic Boundary Conditions
    • 5.4.1 The physical problem
    • 5.4.2 The mathematical problem
    • 5.4.3 Outline of the procedure
    • 5.4.4 Step 1: Find the eigenfunctions
    • 5.4.5 Step 2: Solve the problem
    • 5.4.6 Summary of the solution
  • 5.5 Finding the Green's function
  • 5.6 Inhomogeneous boundary conditions
    • 5.6.1 The physical problem
    • 5.6.2 The mathematical problem
    • 5.6.3 Outline of the procedure
    • 5.6.4 Step 0: Fix the boundary conditions
    • 5.6.5 Step 1: Find the eigenfunctions
    • 5.6.6 Step 2: Solve the problem
    • 5.6.7 Summary of the solution
  • 5.7 Finding the Green's functions
  • 5.8 An alternate procedure
    • 5.8.1 The physical problem
    • 5.8.2 The mathematical problem
    • 5.8.3 Step 0: Fix the boundary conditions
    • 5.8.4 Step 1: Find the eigenfunctions
    • 5.8.5 Step 2: Solve the problem
    • 5.8.6 Summary of the solution
  • 5.9 A Summary of Separation of Variables
    • 5.9.1 The form of the solution
    • 5.9.2 Limitations of the method
    • 5.9.3 The procedure
    • 5.9.4 More general eigenvalue problems
  • 5.10 More general eigenfunctions
    • 5.10.1 The physical problem
    • 5.10.2 The mathematical problem
    • 5.10.3 Step 0: Fix the boundary conditions
    • 5.10.4 Step 1: Find the eigenfunctions
    • 5.10.5 Step 2: Solve the problem
    • 5.10.6 Summary of the solution
    • 5.10.7 An alternative procedure
  • 5.11 A Problem in Three Independent Variables
    • 5.11.1 The physical problem
    • 5.11.2 The mathematical problem
    • 5.11.3 Step 1: Find the eigenfunctions
    • 5.11.4 Step 2: Solve the problem
    • 5.11.5 Summary of the solution
  
  
• 6. Fourier Transforms [None]
• 7. Laplace Transforms
  • 7.1 Overview of the Procedure
    • 7.1.1 Typical procedure
    • 7.1.2 About the coordinate to be transformed
  • 7.2 A parabolic example
    • 7.2.1 The physical problem
    • 7.2.2 The mathematical problem
    • 7.2.3 Transform the problem
    • 7.2.4 Solve the transformed problem
    • 7.2.5 Transform back
  • 7.3 A hyperbolic example
    • 7.3.1 The physical problem
    • 7.3.2 The mathematical problem
    • 7.3.3 Transform the problem
    • 7.3.4 Solve the transformed problem
    • 7.3.5 Transform back
    • 7.3.6 An alternate procedure
  
  
• Bibliography