Àwọn Ààyè-iṣẹ́ · Graduate

Partial Differential Equations

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

7 Ìṣàmúlò-ètò 19 Àwọn ìwé-ìwé 121 Àwọn àtòjọ-ètò 180 Àwọn àkóónú àtòjọ-ẹ̀yàn

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.

Tí a bá ṣẹ̀dà: Multivariable-Calculus, Differential-Equations, Linear-Algebra, Analysis

Àwọn Ìjádè

Classification and first-order equations
3 Àwọn ìwé-ìwé

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

Fi wọlé láti mú ìṣàmúlò-ètò náà
The heat equation and Fourier series
4 Àwọn ìwé-ìwé

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

Fi wọlé láti mú ìṣàmúlò-ètò náà
The wave equation
2 Àwọn ìwé-ìwé

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

Fi wọlé láti mú ìṣàmúlò-ètò náà
Laplace's equation, maximum principles and Green's functions
3 Àwọn ìwé-ìwé

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

Fi wọlé láti mú ìṣàmúlò-ètò náà
Fourier transforms and energy methods
2 Àwọn ìwé-ìwé

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

Fi wọlé láti mú ìṣàmúlò-ètò náà
Distributions and Sobolev spaces
2 Àwọn ìwé-ìwé

Distributions and weak derivatives Sobolev spaces and embeddings

Fi wọlé láti mú ìṣàmúlò-ètò náà
Well-posedness, estimates and scaling
3 Àwọn ìwé-ìwé

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

Fi wọlé láti mú ìṣàmúlò-ètò náà

Àwọn ìwé-ìwé ìparí

Questions from every unit, the hardest ones last. Typed answers are marked by the algebra system; proofs and photographed working are read by a model; then a person reviews the whole paper before your result is released. Pass and you get a certificate with its own link. Àwọn àmì-ìwé ìṣàmúlò-ètò: 70%.

Fi wọlé

Àwọn ìṣàmúlò-ètò

$39

Àwọn ààyè-iṣẹ́ gbogbo, àwọn ìwé-ìwé ìparí nípa ìṣàfarawé, àti àwọn ìṣàmúlò-ètò 20 tí o ń ṣiṣẹ́, fún ìṣàmúlò-ètò kan.

Àwọn ìwé-ìwé àìpẹ̀

$15

A single sitting of one course's final, marked and reviewed. For someone who already knows the material and wants it tested.

Àwọn Ìjánu-ìsún

$19/Àwọn àwọn ààyè-iṣẹ́

Àwọn àwọn ìṣàmúlò-ètò àti àwọn ìṣàmúlò-ètò gbogbo nínú àwọn ìwé-ìwé (ìwé-ìwé meji nínú àwọn ìwé-ìwé kan nínú oṣù) àti àwọn ìṣàmúlò-ètò iṣẹ́ 100 nínú oṣù. Gbàyàn láti inú àwọn

Àwọn ìṣàmúlò-ètò ìṣàmúlò-ètò

Àwọn ìṣàmúlò-ètò mẹ́tà: fi àwòrán tí a tí n ṣiṣẹ́ pamọ́, tàbí tẹ̀ rẹ̀, àti wò ibi tí a tì fi ṣẹ́.

Àwọn ìwé-ìwé tí a ṣì fún ìrànwọ́. Àwọn ìwé-ìwé àti àwọn ìwé-ìwé ìṣàmúlò-ètò tí a ṣì ní bayii.

Ṣayẹwo àwọn ààyè-iṣẹ́ rẹ̀

Partial Differential Equations

Àwọn Ìjádè

Classification and first-order equations

The heat equation and Fourier series

The wave equation

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

Fourier transforms and energy methods

Distributions and Sobolev spaces

Well-posedness, estimates and scaling

Àwọn ìwé-ìwé ìparí

Àwọn ìṣàmúlò-ètò

Àwọn ìṣàmúlò-ètò

Àwọn ìwé-ìwé àìpẹ̀

Àwọn Ìjánu-ìsún

Àwọn ìṣàmúlò-ètò ìṣàmúlò-ètò