SYLLABUS

Math 505 Applied Mathematics II

During the course of the semester we will cover the following topics:

1. Partial Differential Equations

· An overview of partial differential equations(PDE’s) and their applications

· First order partial differential equations : linear equations, quasi-linear equations and method of characteristics

· Linear second order PDE’s: classification, canonical forms.

· Elements of Fourier analysis: Fourier series, Fourier transform

· Wave equations: the Cauchy problem and d’Alamberts formula, the initial boundary value problem, energy equality.

· The heat equations: Cauchy problem, initial boundary value problem, maximum principle

· The Laplace and Poisson equations: harmonic functions and their properties, boundary value problems, maximum principle

· Some nonlinear equations of mathematical physics: Burgers equation, Kortewg de Vriez equation and nonlinear Schrödinger equation.

2. Elements of Calculus of Variations

· Some classical variational problems.

· Variation of a functional. Euler equation.

· Canonical form of the Euler equation. Conservation laws.

Textbooks and lecture notes:

I will use the following textbooks for my lectures:

P. V. O'Neil, Beginning Partial Differential Equations, John Willey & Sons, 1995

W. A. Strauss, Partial Differential equations, , John Willey & Sons, 2007

L. C. Evans , Partial Differential equations, American Math. Soc., 1998

I.M.Gelfand and S.V.Fomin, Calculus of Variations, Dover Publications, INC,2000

Each weekend lecture notes covering the corresponding material will be placed on the course web page.

There will be several homework assignments, a midterm exam and a final exam. The homework grades will account for 30% of the total grade. The midterm examination will count for 30% of the grade and the final examination - 40% of the total grade