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 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.

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