コース · Graduate
Partial Differential Equations
Transport, heat, wave and Laplace equations, Fourier methods, energy and maximum principles, weak derivatives, Sobolev spaces, well-posedness and scaling.
A graduate course in partial differential equations that starts with classification and characteristics and ends with the tools used to study the Navier-Stokes equations. You solve the four model equations by hand (characteristics, separation of variables, Fourier series and transforms, d'Alembert, Green's functions), then learn the methods that work without formulas: maximum principles, energy estimates, distributions and weak derivatives, Sobolev spaces, a priori estimates and scaling.
It assumes multivariable calculus, ordinary differential equations, linear algebra and a first course in analysis, and it prepares you for the fluid dynamics course.
後に来る: Multivariable-Calculus, Differential-Equations, Linear-Algebra, Analysis
コース
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Classification and first-order equations
3 レッスン -
The heat equation and Fourier series
4 レッスン -
The wave equation
2 レッスン -
Laplace's equation, maximum principles and Green's functions
3 レッスン -
Fourier transforms and energy methods
2 レッスン -
Distributions and Sobolev spaces
2 レッスン -
Well-posedness, estimates and scaling
3 レッスン
卒業試験
ユニットごとに問題があり、最も難しい問題が残ります。タイプした答えは代数システムでマークされ、証明と写真の作業はモデルによって読まれ、その後、結果を公開する前に人が全ての論文をレビューします。合格すると、リンク付きの証明書を受け取ります。 合格点: 70%.
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