Professor of Mathematics
Dec 13 2010:
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This week: our final exam at 1:30 on Thursday in our usual classroom. Special pre-exam office hours: Tuesday, 11-1, Wednesday, 1-3, and Thursday, 11-1.
| Week | Meeting | Date | In Class | Homework |
|---|---|---|---|---|
| Week 1 | Lecture 1 | Tuesday, September 7 | Course Introduction. Initial and boundary value problems for partial differential equations as models for physical processes. | |
| Lecture 2 | Thursday, September 9 | Chapter 1. The Dirac delta "function". Basic distribution theory. Test functions. Regular and singular distributions. | ||
| Week 2 | Lecture 3 | Tuesday, September 14 | Comparison of distributions and functions: equality of distributions on open sets and approximation of distributions by smooth functions. Basic operations with distributions: (i) linear combinations, (ii) products of smooth functions and distributions, (iii) translations of distributions, (iv) dilations of distributions, (v) differentiation of distributions. | |
| Lecture 4 | Thursday, September 16 | Convergence of distributions. Distributions describing the regularization of divergent integrals. | ||
| Week 3 | Lecture 5 | Tuesday, September 21 | Review of classical Fourier series. Orthogonality of harmonics, formula for Fourier coefficients, Bessel's inequality, and smoothness of a function versus decay of its Fourier coefficients. Uniform convergence of Fourier series with rapidly decaying coefficients (to something...). | |
| Lecture 6 | Thursday, September 23 | Classical Fourier series, continued. Pointwise and uniform convergence of the Fourier series of a continuous and piecewise smooth function to the function itself. Distributional Fourier series and the Poisson summation formula. | ||
| Week 4 | Lecture 7 | Tuesday, September 28 | Chapter 2. Introduction to differential equations in distributions. Formal adjoints. Distributional, weak, and classical solutions of linear differential equations. The simplest differential equation: u'=f. | Homework Set 1 Due |
| Lecture 8 | Thursday, September 30 | Introduction to Green's functions and fundamental solutions. Boundary value problems in one dimension. | ||
| Week 5 | Lecture 9 | Tuesday, October 5 | The classical formulation of Green's function for one-dimensional problems. Conditions for existence of Green's functions. | |
| Lecture 10 | Thursday, October 7 | Use of Green's functions to solve inhomogeneous problems. Techniques for finding Green's functions (subtracting a fundamental solution, the method of images). | ||
| Week 6 | Lecture 11 | Tuesday, October 12 | Chapter 3. The classical theory of Fourier transforms. | Homework Set 2 Due |
| Lecture 12 | Thursday, October 13 | Distributions of slow growth (AKA tempered distributions). Generalized Fourier transforms. Applications to partial differential equations. | ||
| Week 7 | Tuesday, October 19 | FALL BREAK | ||
| Lecture 13 | Thursday, October 21 | Applications continued. The Poisson and wave equations. | ||
| Week 8 | Lecture 14 | Tuesday, October 26 | Chapter 4. Basic concepts of vector spaces. Normed linear spaces. Examples. | Homework Set 3 Due |
| Lecture 15 | Thursday, October 28 | Topological notions. Convergence and completeness. Equivalence of norms. Banach spaces. MIDTERM EXAM: 6-8 PM, 4096 East Hall |
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| Week 9 | Lecture 16 | Tuesday, November 2 | Chapter 5. Introduction to fixed point equations on vector spaces. Examples: root finding, initial-value problems for ODE, boundary-value problems for PDE. | |
| Lecture 17 | Thursday, November 4 | The method of successive approximations and the Contraction Mapping Theorem. Applications. Newton's Method. Picard iteration. | ||
| Week 10 | Lecture 18 | Tuesday, November 9 | Applications of contraction mappings continued. Solution of boundary-value problems for partial differential equations by using the "wrong" Green's function. Neumann series for linear operator inverses. | Take-home Midterm Due
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| Lecture 19 | Thursday, November 11 | Chapter 7. Inner product spaces. Hilbert spaces. Orthogonal bases and generalized Fourier series. | ||
| Week 11 | Lecture 20 | Tuesday, November 16 | Examples of Hilbert spaces. Orthogonal expansions. The generalized Bessel inequality and Parseval's Theorem. | Homework Set 4 Due |
| Lecture 21 | Thursday, November 18 | The Riesz-Fischer Theorem and uniqueness of Hilbert space. The Gram-Schmidt process. Subspaces and projections. | ||
| Week 12 | Lecture 22 | Tuesday, November 23 | Proof of the Projection Theorem and best approximation by generalized Fourier series. Linear functionals and the Riesz Representation Theorem. | Homework Set 5 Due |
| Thursday, November 25 | THANKSGIVING BREAK | |||
| Week 13 | Lecture 23 | Tuesday, November 30 | Weak convergence in Hilbert space. | |
| Lecture 24 | Thursday, December 2 | Chapter 8. The algebra of bounded operators on normed spaces. The space of bounded operators as a normed linear space. | ||
| Week 14 | Lecture 25 | Tuesday, December 7 | Operator power series and examples. Adjoints and selfadjoint operators. Eigenvalue problems and Sturm-Liouville theory. Compact operators. | |
| Lecture 26 | Thursday, December 9 | Chapter 9. The Spectral Theorem for compact selfadjoint operators. Application to Sturm-Liouville problems. | Homework Set 6 Due | |
| Week 15 | Thursday, December 16 | FINAL EXAM |