Difference between revisions of "MAT 5673"

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==Course description==
 +
Partial differential equations arise in many different areas as one tries
 +
to describe the behavior of a system ruled by some law. Typically, this has to do with
 +
some physical process such as heat diffusion in a material, vibrations of a bridge, circulation
 +
of fluids, the behavior of microscopic particles or the evolution of the universe as a whole.
 +
Modeling by means of partial differential equations has been successful in other disciplines
 +
as well, like in the case of the Black-Scholes equation for stock options pricing and the
 +
Hodgkin–Huxley equations for firing patterns of a neuron. Partial differential equations are
 +
an important tool in Applied Math and Pure Math. This course gives an introduction to PDE's in the setting of two independent variables.
 +
 +
'''Textbooks:
 +
'''
 +
* P. Olver: Introduction to Partial Differential Equations (Undergraduate Texts in Mathematics) 1st ed. 2014, Corr. 3rd printing 2016
 +
* L.C. Evans: Partial Differential Equations: Second Edition (Graduate Studies in Mathematics) 2nd Edition
 +
 
==Topics List==
 
==Topics List==
 
{| class="wikitable sortable"
 
{| class="wikitable sortable"
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*  
 
*  
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||
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Introduction and classification of PDE, Calculus review
 +
||
 +
Multivariable Calculus, Chain Rule
 +
||
 +
* Definition of a PDE as a relation between partial derivatives of an unknown function. Classification of PDE according to order - linear/nonlinear/quasilinear
 +
|-
 +
|Week 2
 +
||
 +
*
 +
||
 +
Applied examples of PDE
 +
||
 +
Multivariable Calculus, Chain Rule
 +
||
 +
* Origin and background of common PDE's: heat equation, wave equation, transport equation, etc.
 +
|-
 +
|Week 3
 +
||
 +
*
 +
||
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The method of characteristics for first-order quasilinear equations
 +
||
 +
Multivariable Calculus, Chain Rule
 +
||
 +
* Solving quasilinear first-order equations using the method of characteristics
 +
|-
 +
|Week 4
 +
||
 +
*
 +
||
 +
The method of characteristics for first-order fully nonlinear equations
 +
||
 +
Multivariable Calculus, Chain Rule
 +
||
 +
* Solving fully nonlinear first-order equations (e.g. the Eikonal equation) using the method of characteristics
 +
|-
 +
|Week 5
 +
||
 +
*
 +
||
 +
Heat and wave equation on the whole real line
 +
||
 +
Differentiation of integrals with respect to a parameter, integration by parts
 +
||
 +
* Fundamental solution of the heat equation, D'Alembert's formula for the wave equation
 +
|-
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|Week 6
 +
||
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*
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Initial-boundary value problem for heat and wave equation I
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||
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Partial derivatives, chain rule
 +
||
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* Separation of variables method for heat and wave equation
 +
|-
 +
|Week 7
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||
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*
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||
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Initial-boundary value problem for heat and wave equation II, introduction to Fourier series
 +
||
 +
Partial derivatives, chain rule
 +
||
 +
* Forming more general solutions out of infinite superposition of basic solutions
 +
|-
 +
|Week 8
 
||
 
||
 
*  
 
*  
 +
||
 +
Introduction to Fourier series
 +
||
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Infinite series
 +
||
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* Orthonormal systems of functions, spectral method for the wave and heat equation
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|-
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|Week 9
 
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*  
 
*  
 
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−
* Learning outcome
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Schroedinger equation
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Complex numbers
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* Basic properties of Schroedinger equation, particle in a potential well 
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|-
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|Week 10
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||
 +
 
 +
||
 +
Qualitative properties of PDE's
 +
||
 +
Differentiation of integrals with respect to parameter
 +
||
 +
* Uniqueness of solutions, finite and infinite propagation speed for wave and heat equation
 +
|-
 +
|Week 11
 +
||
 +
*
 +
||
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Introduction to numerical methods for PDE (optional)
 +
||
 +
Derivatives, Calculus, Matrices, Linear Algebra
 +
||
 +
* Basic finite difference schemes for first-order quasilinear equations, CFL condition
 +
|-
 +
|Week 12
 +
||
 +
 
 +
||
 +
Introduction to the Laplace and Poisson equation
 +
||
 +
* 
 +
||
 +
* Solving the Laplace equation on the whole space and on a simple bounded region (square, disc)
 +
|-
 +
|Week 13
 +
||
 +
*
 +
||
 +
Introduction to the Calculus of Variations
 +
||
 +
Differentiation of an integral with respect to a parameter, parametric surfaces
 +
||
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* Compute the variational derivative of a functional 
 +
|-
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|Week 14
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||
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*
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||
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Review, advanced topics
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||
 +
* 
 +
||
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* 
 
|}
 
|}

Latest revision as of 07:57, 24 March 2023

Course description

Partial differential equations arise in many different areas as one tries to describe the behavior of a system ruled by some law. Typically, this has to do with some physical process such as heat diffusion in a material, vibrations of a bridge, circulation of fluids, the behavior of microscopic particles or the evolution of the universe as a whole. Modeling by means of partial differential equations has been successful in other disciplines as well, like in the case of the Black-Scholes equation for stock options pricing and the Hodgkin–Huxley equations for firing patterns of a neuron. Partial differential equations are an important tool in Applied Math and Pure Math. This course gives an introduction to PDE's in the setting of two independent variables.

Textbooks:

  • P. Olver: Introduction to Partial Differential Equations (Undergraduate Texts in Mathematics) 1st ed. 2014, Corr. 3rd printing 2016
  • L.C. Evans: Partial Differential Equations: Second Edition (Graduate Studies in Mathematics) 2nd Edition

Topics List

Date Sections Topics Prerequisite Skills Student Learning Outcomes
Week 1

Introduction and classification of PDE, Calculus review

Multivariable Calculus, Chain Rule

  • Definition of a PDE as a relation between partial derivatives of an unknown function. Classification of PDE according to order - linear/nonlinear/quasilinear
Week 2

Applied examples of PDE

Multivariable Calculus, Chain Rule

  • Origin and background of common PDE's: heat equation, wave equation, transport equation, etc.
Week 3

The method of characteristics for first-order quasilinear equations

Multivariable Calculus, Chain Rule

  • Solving quasilinear first-order equations using the method of characteristics
Week 4

The method of characteristics for first-order fully nonlinear equations

Multivariable Calculus, Chain Rule

  • Solving fully nonlinear first-order equations (e.g. the Eikonal equation) using the method of characteristics
Week 5

Heat and wave equation on the whole real line

Differentiation of integrals with respect to a parameter, integration by parts

  • Fundamental solution of the heat equation, D'Alembert's formula for the wave equation
Week 6

Initial-boundary value problem for heat and wave equation I

Partial derivatives, chain rule

  • Separation of variables method for heat and wave equation
Week 7

Initial-boundary value problem for heat and wave equation II, introduction to Fourier series

Partial derivatives, chain rule

  • Forming more general solutions out of infinite superposition of basic solutions
Week 8

Introduction to Fourier series

Infinite series

  • Orthonormal systems of functions, spectral method for the wave and heat equation
Week 9

Schroedinger equation

Complex numbers

  • Basic properties of Schroedinger equation, particle in a potential well
Week 10

Qualitative properties of PDE's

Differentiation of integrals with respect to parameter

  • Uniqueness of solutions, finite and infinite propagation speed for wave and heat equation
Week 11

Introduction to numerical methods for PDE (optional)

Derivatives, Calculus, Matrices, Linear Algebra

  • Basic finite difference schemes for first-order quasilinear equations, CFL condition
Week 12

Introduction to the Laplace and Poisson equation

  • Solving the Laplace equation on the whole space and on a simple bounded region (square, disc)
Week 13

Introduction to the Calculus of Variations

Differentiation of an integral with respect to a parameter, parametric surfaces

  • Compute the variational derivative of a functional
Week 14

Review, advanced topics

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