Difference between revisions of "MAT 5673"

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Hodgkin–Huxley equations for firing patterns of a neuron. Partial differential equations are
 
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.
 
an important tool in Applied Math and Pure Math. This course gives an introduction to PDE's in the setting of two independent variables.
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'''Textbooks:
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'''
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* P. Olver: Introduction to Partial Differential Equations (Undergraduate Texts in Mathematics) 1st ed. 2014, Corr. 3rd printing 2016
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* L.C. Evans: Partial Differential Equations: Second Edition (Graduate Studies in Mathematics) 2nd Edition
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==Topics List==
 
==Topics List==
 
{| class="wikitable sortable"
 
{| class="wikitable sortable"
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|Week 10
 
|Week 10
 
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Complex numbers
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Qualitative properties of PDE's
 
Qualitative properties of PDE's
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Introduction to numerical methods for PDE (optional)
 
Introduction to numerical methods for PDE (optional)
 
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Derivatives, Calculus
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Derivatives, Calculus, Matrices, Linear Algebra
 
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* Basic finite difference schemes for first-order quasilinear equations, CFL condition  
 
* Basic finite difference schemes for first-order quasilinear equations, CFL condition  
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|Week 12
 
|Week 12
 
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Matrices, Linear Algebra
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Introduction to the Laplace and Poisson equation  
 
Introduction to the Laplace and Poisson equation  

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