Partial Differential Equations

Basics

• What is a Partial Differential Equation
  • Hyperbolic Partial Differential Equation
  • Parabolic Partial Differential Equation
  • Elliptic Partial Differential Equation
• Differential Operators and Symbols
• Fundamental Solution of Differential Equations, Green’s Function
• Multi-Index Notation
• Smoothness of Boundaries
• Outward Unit Normal Vector
• Green-Gauss Theorem, Partial Integration Formula
• Green’s Formula
• Mollifier
• Mollification
  • Convergence of Mollification
• Boundary Value Problem (BVP)
  • Dirichlet Boundary Condition
  • Neumann Boundary Condition
  • Robin Boundary Condition
• Cauchy Problem, Initial Value Problem (IVP)

Linear Partial Differential Equations

• Solution of the Standing Wave Partial Differential Equation
• Solution of the Uniform Progressive Wave Partial Differential Equation
• Solution of the Non-uniform Progressive Wave Partial Differential Equation

Transport Equation

• Transport Equation
• Initial Value Problem and Nonhomogeneous Problem

Laplace Equation

• Laplace Equation and Poisson Equation
• Invariance with Respect to Orthogonal Transformation
• Fundamental Solution of Laplace’s Equation
• Mean Value Formula
• Properties of Harmonic Functions
  • Maximum Principle
  • Smoothing Effect

Spherical Coordinates

• General Solution of the Radial Component
• General Solution of the Polar and Azimuthal Components: Spherical Harmonic Functions
• General Solution

Poisson Equation

• Fundamental Solution
• Uniqueness of the Solution to Dirichlet’s Problem

Heat Equation

• Heat Equation
• Fourier Series
• Solution of the Heat Equation
• Solution of the Initial Value Problem for the Heat Equation with Dirichlet Boundary Conditions

Wave Equation

• Wave Equation
• Solution of the Cauchy Problem for the Wave Equation
  • One-dimensional D’Alembert’s Formula
• Solution of the Initial Value Problem for the Wave Equation with Dirichlet Boundary Conditions
• Solution of the Wave Equation with Initial Conditions (0)

Helmholtz Equation

• Helmholtz Equation

Nonlinear Partial Differential Equations

Nonlinear First Order Partial Differential Equations

• Notation
• Characteristic Equations
• Solution Using Characteristic Equations
• Linearization of Boundaries

Burgers’ Equation

• Solution of the Inviscid Burgers’ Equation
• Mass Conservation Law in the Inviscid Burgers’ Equation
• Rankine-Hugoniot Condition and Entropy Condition
• Solution of the Riemann Problem for Burgers’ Equation

Hamilton-Jacobi Equation

• Lagrangian and Euler-Lagrange Equation
• Hamilton-Jacobi Equation and Hamilton’s Equations
• Derivation of Hamilton’s Equations from Variational Methods and Euler-Lagrange Equations
• Legendre Transformation
• Convex Duality of Hamiltonian and Lagrangian
• Hopf-Lax Formula
  • The Hopf-Lax Formula Satisfies the Hamilton-Jacobi Equation

References

• Lawrence C. Evans, Partial Differential Equations (2nd Edition, 2010)