Hello! This is the site for the Iowa State applied math qualifying exam study group. It is still under construction.
APPLIED MATHEMATICS SYLLABUS
- Explicit solution methods for PDEs: separation of variables, characteristics, d'Alembert's formula.
- Function spaces: elementary theory of abstract Banach and Hilbert spaces, C k- and L p-spaces, contraction mapping theorem.
- Theory of distributions: test functions, calculus of distributions, tempered distributions, Sobolev spaces.
- Fourier analysis: Fourier series and Fourier transform in classical and distributional settings, convolution.
- Differential equations with distributions: fundamental solutions of differential operators, Green's functions for boundary value problems.
- Linear operators: bounded and unbounded linear operators on Banach spaces, adjoint operators, closed operators, self-adjoint and symmetric operators.
- Spectral theory: resolvent and spectrum of a linear operator, Fredholm alternative.
- Compact operators: spectral theory for compact and compact, self-adjoint operators in a Hilbert space, Hilbert-Schmidt operators, application to integral equations, Green's function and eigenfunctions of the Laplacian.
- Variational methods: variational characterization of eigenvalues, including Rayleigh-Ritz and Courant-Weyl principles, application to Sturm-Liouville theory, Euler-Lagrange equations in the Calculus of Variations, Dirichlet principle.
- Weak solutions of PDEs: weak formulation of boundary value problems, variational methods, Lax-Milgram Lemma.
Recently used textbooks
J. K. Hunter and B. Nachtergaele, Applied Analysis, World Scientific, 2001 (If you don't own this one it is freely available at the author's website)
I. Stakgold, Green's Functions and Boundary Value Problems, Wiley-Interscience, 1997
See the schedule in this folder:
Here is a spreadsheet with problems put into categories based on the syllabus. Feel free to update this at will. You should all have access to make updates in the Google document.