- 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.
Note: This syllabus will sometimes refer to LAA ( Linear Algebra with Applications, by Holt, the textbook for Math 308.
- 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)
Emphasize applications of complex arithmetic (from Math 307) to Gaussian elimination and null spaces of complex matrices as on pp. 263-4 of JRA, applied to eigenspaces for complex eigenvalues. Notice that there is much NEW material in these sections.
- §7.4-§7.8: Solving homogeneous linear systems, complex eigenvalues and diagonalization. (5 lectures)
Note that sections 3.6 and 3.7 in JRA 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 LAA. 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)