Partial Differential Equations

Transport, heat, wave and Laplace equations, Fourier methods, energy and maximum principles, weak derivatives, Sobolev spaces, well-posedness and scaling.

7 ሴኮንዶችunit-format 19 ማስተማሪያ 121 ጥያቄዎች 180 ደቂቃ

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.

ከ: Multivariable-Calculus, Differential-Equations, Linear-Algebra, Analysis

የኮርሱ

Classification and first-order equations
3 ማስተማሪያ

What a PDE is, and how to classify one First-order linear equations and characteristics Nonlinear first-order equations and shocks

የምርመራውን ፋይል አስቀምጥ
The heat equation and Fourier series
4 ማስተማሪያ

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

የምርመራውን ፋይል አስቀምጥ
The wave equation
2 ማስተማሪያ

The wave equation and d'Alembert's formula The vibrating string: modes and energy

የምርመራውን ፋይል አስቀምጥ
Laplace's equation, maximum principles and Green's functions
3 ማስተማሪያ

Laplace's equation and harmonic functions Maximum principles and uniqueness Green's functions and the fundamental solution

የምርመራውን ፋይል አስቀምጥ
Fourier transforms and energy methods
2 ማስተማሪያ

The Fourier transform and the heat kernel Energy methods: uniqueness, stability and decay

የምርመራውን ፋይል አስቀምጥ
Distributions and Sobolev spaces
2 ማስተማሪያ

Distributions and weak derivatives Sobolev spaces and embeddings

የምርመራውን ፋይል አስቀምጥ
Well-posedness, estimates and scaling
3 ማስተማሪያ

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

የምርመራውን ፋይል አስቀምጥ

የመጨረሻው ምርመራ

ጥያቄዎች ከየአካባቢው ጥያቄዎች, በጣም አስቸጋሪዎቹ መጨረሻ. የተጻፉ መልሶች በአልጀርባ ስርዓት ምልክት; ማስረጃዎች እና ፎቶግራፍ ስራዎች በሞዴል ይነበባሉ; ከዚያ ሰው ሁሉንም ጽሑፍ ውጤትዎ ከመለቀቁ በፊት ይመልከቱ. ተቀበሉ እና የራሱን አገናኝ ያለውን ማስረጃ ያገኛሉ. ተቀባይ: 70%.

መዝገብ ቤትን አስገባ