ኮርስ · 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 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

  1. Classification and first-order equations

    3 ማስተማሪያ
    What a PDE is, and how to classify oneFirst-order linear equations and characteristicsNonlinear first-order equations and shocks
    የምርመራውን ፋይል አስቀምጥ

  2. The heat equation and Fourier series

    4 ማስተማሪያ
    The heat equation: where it comes fromSeparation of variablesFourier series: coefficients, convergence and ParsevalThe heat equation on an interval
    የምርመራውን ፋይል አስቀምጥ

  3. The wave equation

    2 ማስተማሪያ
    The wave equation and d'Alembert's formulaThe vibrating string: modes and energy
    የምርመራውን ፋይል አስቀምጥ

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

    3 ማስተማሪያ
    Laplace's equation and harmonic functionsMaximum principles and uniquenessGreen's functions and the fundamental solution
    የምርመራውን ፋይል አስቀምጥ

  5. Fourier transforms and energy methods

    2 ማስተማሪያ
    The Fourier transform and the heat kernelEnergy methods: uniqueness, stability and decay
    የምርመራውን ፋይል አስቀምጥ

  6. Distributions and Sobolev spaces

    2 ማስተማሪያ
    Distributions and weak derivativesSobolev spaces and embeddings
    የምርመራውን ፋይል አስቀምጥ

  7. Well-posedness, estimates and scaling

    3 ማስተማሪያ
    Well-posedness in the sense of HadamardA priori estimates, local and global existenceScaling 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

ዋጋዎች

ኮርሶ

$39

የእያንዳንዱ ክፍል ጥያቄ፣ የመጨረሻው ፈተና አንድ ሪታክ፣ እና 20 የራስህ ሥራን ማጣራት፣ ለአንድ ኮርስ.

አንድ የመጨረሻ ምርመራ

$15

የአንድ ኮርስ መጨረሻ አንድ ጊዜ ስብሰባ, ምልክት እና ተመልከት. ለአንድ ሰው ያውቃል ነገር ግን ያውቃል እና ይፈልጋል ፈተና.

ርዕስ

$19/ሰዓት

ሁሉንም የምርመራ እና የመጨረሻው በሁሉም ኮርስ (ሁለት መጨረሻዎች በኮርስ በወር) እና 100 የሥራ ማረጋገጫዎች በወር. ከሂሳብዎ ይቁሙ.

የስራ ምርመራዎች

$9

ዐሥር ምርመራዎች: የስራውን ፎቶ ይጫኑ ወይም ይጻፉት፣ እና ስህተቱ የት እንደነበረ ይመልከቱ።

ኮርሶች በቅርቡ ለመግዛት ይከፍታሉ. ትምህርት ቤቶችና የእያንዳንዱ ኮርሶች የመጀመሪያው ምርመራ አሁን ይከፍታሉ.

የራስህ ስራን ፈትሽ