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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
Text: Partial Differential Equations, by Robert McOwen, 2nd Edition, Pearson
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.