## WelcomeHello! This is the site for the Iowa State applied math qualifying exam study group. It is still under construction. ## SyllabusAPPLIED 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*- and^{ k}*L*-spaces, contraction mapping theorem.^{ p} - 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.
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