Partial Differential Equations

Дараа нь: Multivariable-Calculus, Differential-Equations, Linear-Algebra, Analysis

Сургууль

1. Classification and first-order equations

   3 хичээл
   ∂What a PDE is, and how to classify one∂First-order linear equations and characteristics∂Nonlinear first-order equations and shocks
   Сонирхолтой асуултууд

2. The heat equation and Fourier series

   4 хичээл
   ∂The heat equation: where it comes from∂Separation of variables∂Fourier series: coefficients, convergence and Parseval∂The heat equation on an interval
   Сонирхолтой асуултууд

3. The wave equation

   2 хичээл
   ∂The wave equation and d'Alembert's formula∂The vibrating string: modes and energy
   Сонирхолтой асуултууд

4. Laplace's equation, maximum principles and Green's functions

   3 хичээл
   ∂Laplace's equation and harmonic functions∂Maximum principles and uniqueness∂Green's functions and the fundamental solution
   Сонирхолтой асуултууд

5. Fourier transforms and energy methods

   2 хичээл
   ∂The Fourier transform and the heat kernel∂Energy methods: uniqueness, stability and decay
   Сонирхолтой асуултууд

6. Distributions and Sobolev spaces

   2 хичээл
   ∂Distributions and weak derivatives∂Sobolev spaces and embeddings
   Сонирхолтой асуултууд

7. Well-posedness, estimates and scaling

   3 хичээл
   ∂Well-posedness in the sense of Hadamard∂A priori estimates, local and global existence∂Scaling and criticality: why dimension matters
   Сонирхолтой асуултууд

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