Applied Analysis

The purpose of this course is to learn a variety of mathematical methods for deriving useful approximate solutions of the differential equations and integrals found in the Mathematical Sciences. The course will be structured as follows: 1. Existence and uniqueness results for ordinary differential equations: The Lipschitz condition and Picard’s theorem. Comparison theorems. 2. Integral Equations: The Volterra integral equation and initial value problems, the Fredholm integral equation and boundary value problems. 3. Sturm-Liouville Theory: The adjoint differential operator, the Sturm-Liouville problem, basic properties of a Sturm-Liouville eigenvalue problem, unboundedness of the eigenvalues, completeness in the appropriate sense of the set of eigenfunctions 4. Theory of Infinite-dimensional vector spaces: Inner product spaces, complete metric spaces, Hilbert spaces, square summable series and square integrable functions