- Prerequisites: Math 126, 307, 308
- Text: Linear Analysis by Boyce (Custom 10th ed.);
- Description of course:
Math 309 serves as the culmination of the Math 307-8-9 program in linear analysis. It combines analysis tools and ideas (such as series expansions, complex numbers and exponentials, and differential equations) from Math 307 with comparabale content (eigenvalues, and difference equations, orthogonal bases, projections and best approximations) from Math 308. These tools are applied to the solution and qualitative study of linear systems of ordinary differential equations, and to the analysis of classical partial differential equations (heat, wave, Laplace). In the case of partial differential equations, separation of variables is used to obtain boundary value problems which are then solved and are used to generate the Fourier series solutions of the original partial differential equation.
- Questions: Please contact Brooke Miller (C-36F, 3-6830) if you have questions or suggestions concerning the syllabus.
SYLLABUS FOR 25 LECTURES
- 1. Systems of First Order Linear Equations (10 lectures):
- §7.1-§7.3: Review of linear algebra (complex linear algebra, spaces of vector-valued functions, matrix differential equations.) (3 lectures)
Students probably did NOT see the use of complex arithmetic for Gaussian elimination and finding null spaces and eigenvalues of matrices in Math 308. Teach this material assuming no previous knowledge of it (either with these sections, or when needed later in the chapter).
- §7.4-§7.8: Solving homogeneous linear systems, complex eigenvalues and diagonalization. (5 lectures)
Note that complex eigenvalues and diagonalization are not part of the required material in Math 308 and so they cannot be assumed here. The coverage of Boyce and DiPrima must be amplified. Additional examples and exercises can be easily constructed, or extracted from the current Math 308 text. We suggest only minimal emphasis on 7.7 (especially defective matrices).
- §7.9: Inhomogeneous equations. (1.5 lectures)
Emphasize connections with the method of variation of parameters for second order equations.
- Optional: Matrix exponentials via similarity (This topic is not covered in the present text; see the Strang's Linear Algebra for ideas).
- 2. Nonlinear Differential Equations and Stability (1+ lectures):
- §9.1: Introduction and the phase plane for linear systems (1+ lectures)
- Optional: §9.2: Stability for autonomous systems (?? lectures)
- 3. Partial Differential Equations and Fourier Series (14)
- §10.1: Introduction to separation of variables and boundary value problems via the heat equation (2 lectures, Appendix A optional)
- §10.2-§10.4: Fourier series as orthogonal expansions, even and odd extensions, pointwise convergence (6 lectures)
- §10.5-§10.6: More heat equation (2 lectures)
- §10.7 The wave equation (2 lectures)
- §10.8 Laplace's equation (2 lectures)