AMS 502, Differential Equations and Boundary Value Problems II 
Analytic solution techniques for, and properties of solutions of, partial differential equations, with concentration on second order PDEs. Techniques covered include: method of characteristics, separation of variables, eigenfunction expansions, spherical means, Green's functions and fundamental solutions, and Fourier transforms. Solution properties include: energy conservation, dispersion, dissipation, existence and uniqueness, maximum and mean value principles. 
Prerequisite: AMS 501 
3 credits, ABCF grading 

This course is offered in the spring semesters only

Text for Spring 2025:
 "Applied Partial Differential Equations" by David J. Logan; 2015, Springer Publications, ISBN: 978-3-319-30769-5

Learning Outcomes:

1) Demonstrate mastery of basic concepts and notations:
        * Domain, boundary, closure, compact support;
        * Divergence theorem;
        * PDE from physics and engineering problems.

2) Demonstrate mastery of first order equations: 
        * Method of characteristics;
        * Semilinear and quasilinear equations, parametric solution;
        * Conservation law and weak solution, jump conditions.

3) Demonstrate mastery of the classification of second order linear PDE:
        * Classification based on characteristics;
        * Canonical form of hyperbolic, parabolic and elliptic equations;
        * System of equations;
        * Adjoint, distribution and weak solutions.

4) Demonstrate mastery of hyperbolic equations:
        * D'Alembert solution, domain of dependence and range of influence;
        * Separation of variable method, nonhomogeneous equation;
        * Spherical mean and wave equation in higher dimensions;
        * Huygens principle, solution in two and three dimensions;
        * Energy method.

5) Demonstrate mastery of elliptic equations:
        * Poisson and Laplace equations, separation of variables;
        * Green's identity, maximum principle;
        * Fundamental solution and Poisson kernel;
        * Dirichlet problem and solutions in integral form.

6) Demonstrate mastery of parabolic equations:
        * Heat equation in one dimension, separation of variables;
        * Fourier transform method;
        * Fundamental solution and solutions in integral form;
        * Regularity and similarity;
        * Applications in fluid physics, thermodynamics and finance.