MAT 5673
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 BlackScholes 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 

Week 2 

Applied examples of PDE 
Multivariable Calculus, Chain Rule 

Week 3 

The method of characteristics for firstorder quasilinear equations 
Multivariable Calculus, Chain Rule 

Week 4 

The method of characteristics for firstorder fully nonlinear equations 
Multivariable Calculus, Chain Rule 

Week 5 

Heat and wave equation on the whole real line 
Differentiation of integrals with respect to a parameter, integration by parts 

Week 6 

Initialboundary value problem for heat and wave equation I 
Partial derivatives, chain rule 

Week 7 

Initialboundary value problem for heat and wave equation II, introduction to Fourier series 
Partial derivatives, chain rule 

Week 8 

Introduction to Fourier series 
Infinite series 

Week 9 

Schroedinger equation 
Complex numbers 

Week 10 
Qualitative properties of PDE's 
Differentiation of integrals with respect to parameter 
 
Week 11 

Introduction to numerical methods for PDE (optional) 
Derivatives, Calculus, Matrices, Linear Algebra 

Week 12 
Introduction to the Laplace and Poisson equation 

 
Week 13 

Introduction to the Calculus of Variations 
Differentiation of an integral with respect to a parameter, parametric surfaces 

Week 14 

Review, advanced topics 

