Nhazi · 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 units 19 Ụmụ 121 Ajụjụ 180 nkeji

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.

Na-abịa mgbe: Multivariable-Calculus, Differential-Equations, Linear-Algebra, Analysis

Nhazi

  1. Classification and first-order equations

    3 Ụmụ
    What a PDE is, and how to classify oneFirst-order linear equations and characteristicsNonlinear first-order equations and shocks
    Nbanye ka ịnabata ajụjụ ahụ

  2. The heat equation and Fourier series

    4 Ụmụ
    The heat equation: where it comes fromSeparation of variablesFourier series: coefficients, convergence and ParsevalThe heat equation on an interval
    Nbanye ka ịnabata ajụjụ ahụ

  3. The wave equation

    2 Ụmụ
    The wave equation and d'Alembert's formulaThe vibrating string: modes and energy
    Nbanye ka ịnabata ajụjụ ahụ

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

    3 Ụmụ
    Laplace's equation and harmonic functionsMaximum principles and uniquenessGreen's functions and the fundamental solution
    Nbanye ka ịnabata ajụjụ ahụ

  5. Fourier transforms and energy methods

    2 Ụmụ
    The Fourier transform and the heat kernelEnergy methods: uniqueness, stability and decay
    Nbanye ka ịnabata ajụjụ ahụ

  6. Distributions and Sobolev spaces

    2 Ụmụ
    Distributions and weak derivativesSobolev spaces and embeddings
    Nbanye ka ịnabata ajụjụ ahụ

  7. Well-posedness, estimates and scaling

    3 Ụmụ
    Well-posedness in the sense of HadamardA priori estimates, local and global existenceScaling and criticality: why dimension matters
    Nbanye ka ịnabata ajụjụ ahụ

Nnyocha nke ikpeazụ

Ajụjụ site n'ọdịnihu ọbụla, ndị dị njọ bụ ndị na-adịbeghị anya. Ajụjụ ndị a tinyere bụ ndị a na-egosipụta site n'algebra sistem; ihe ngosipụta na ọrụ foto a na-agụ site na model; mgbe ahụ onye na-enyocha akwụkwọ ahụ dum tupu ihe a na-eme gị. Ọ ga-agafe ma ị ga-enweta asambodo na njikọ ya. Nhazi: 70%.

Nbanye

Nhazi

Nhazi

$39

N'otu n'otu n'ime nnyocha, nnyocha nke ikpeazụ na-eji otu n'ime ha na-achọpụta, na 20 nnyocha nke gị onwe gị na-arụ ọrụ, maka otu nnyocha.

Otu nnyocha nke ikpeazụ

$15

Nhazi nkeonwe nke ngwụcha nke otu n'ime nkuzi, egosipụtara na nyochara. Maka onye ọbụla nke maara ihenhọrọ ahụ na-achọ ka a chọpụta ya.

Nhazi ọbụla

$19/ọnwa

Nnyocha niile na ngwụcha n'ime kọ̀ntị ọbụla (ngwụcha abụọ n'ime kọ̀ntị ọbụla n'ọnwa) na 100 ọrụ nlele n'ọnwa. Wepụ site na akaụntụ gị.

Nlekọta ọrụ

$9

Nlekọta iri: gbanye foto nke ọbụla na-arụ ọrụ, mọọbụ tinye ya, ma hụ ebe ọ ga-aga n'ezighị ezi.

N'oge na-adịghị anya, kọ̀nụ̀tụ̀ ndị ahụ ga-emepe maka ịzụta. A na-emepe agụụ na nnyocha mbụ nke kọ̀nụ̀tụ̀ ọbụla ugbua.

Nlekọta akụkụ nke ọrụ gị onwe gị