Math 220B
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| Below (LN) denotes lecture notes, (S) denotes Strauss, and (E) denotes Evans. | |||
| Date | Topics | References | Event |
|---|---|---|---|
| Week 1 | |||
| 6-23 | No class. | ||
| 6-24 | Introduction; Heat Equation: Derivation; Heat Equation an an Interval; Separation of Variables | (LN) Introduction, Heat Equation; (S) 1.3, Chap. 4; (E) 2.3 | |
| 6-25 | Separation of Variables; Fourier Series; Symmetric Boundary Conditions | (LN) Heat Equation; (S) 5.1-5.3 | |
| 6-26 | Orthogonality; Complex Form of Full Fourier Series | (LN) Heat Equation; (S) 12.3; (E) 4.3.1 | |
| Week 2 | |||
| 6-30 | Fourier Transform: Motivation and Definitions; Properties of Fourier Transforms | (LN) Heat Equation; (S) 12.3; (E) 4.3.1 | |
| 7-1 | Fourier Transforms; Heat Equation in Rn | (LN) Heat Equation; (S) 2.4, 9.4;(E) 2.1.3.a-b | |
| 7-2 | Fundamental Solution of the Heat Equation; Distributions | (LN) Heat Equation; (S) 12.1, 2.4; (E) 2.1.3.a-b | |
| 7-3 | Distributions; Invariance Properties of the Heat Equation. | (LN) Heat Equation; (S) 12.1, 2.4 | Homework 1 due |
| Week 3 | |||
| 7-7 | Inhomogeneous Heat Equation | (LN) Heat Equation; (S) 3.3; (E) 2.3.1.c | |
| 7-8 | Maximum Principle. | (LN) Heat Equation; (S) 2.3; (E) 2.3.3.a | |
| 7-9 | Uniqueness. Energy Methods. | (LN) Heat Equation; (S) 2.3; (E) 2.3.3.a | |
| 7-10 | Laplace's Equation: Fundamental Solution. | (LN) Laplace's Equation; (S) 6.1; (E) 2.2.1 | Homework 2 due |
| Week 4 | |||
| 7-14 | Solution of Poissson's Equation. | (LN) Laplace's Equation; (E) 2.2.1 | |
| 7-15 | Properties of Harmonic Functions: Mean-Value Property, Maximum Principle. | (LN) Laplace's Equation; (S) 6.1 | |
| 7-16 | Properties of Harmonic Functions: Smoothness, Liouville's Theorem. | (LN) Laplace's Equation; (E) 2.2.2-2.2.3 | |
| 7-17 | Laplace's Equation on a Rectangle. Laplace's Equation on a Disk. | (LN) Laplace's Equation; (S) 6.2, 6.3 | Homework 3 due |
| Week 5 | |||
| 7-21 | Poisson's Formula. | (LN) Laplace's Equation; (S) 6.3 | |
| 7-22 | Green's Functions: Definitions, Representation Formula for solutions of Poisson's equation using Green's functions. | (LN) Green's Functions; (S) 7.1-7.3; (E) 2.2.4.a | Homework 4 due |
| 7-23 | Green's Functions continued. | (LN) Green's Functions; (S) 7.1-7.3; (E) 2.2.4.a | |
| 7-24 | Review/Catch-up | Midterm: 7:00-9:00 | |
| Week 6 | |||
| 7-28 | Green's Function on a Ball; Green's Function on a Half-Space. | (LN) Green's Functions; (S) 7.4; (E) 2.2.4.b, 2.2.4.c | |
| 7-29 | Potential Theory: Definitions and Preliminaries. | (LN) Potential Theory | |
| 7-30 | Gauss' Lemma | (LN) Potential Theory | |
| 7-31 | Solutions of Interior/Exterior Dirichlet and Neumann Problems. | (LN) Potential Theory | Homework 5 due |
| Week 7 | |||
| 8-4 | General Eigenvalue Problems. Eigenvalues as Minima of the Potential Energy. | (LN) Eigenvalues (S) 11.1; (E) 6.5.1 | |
| 8-5 | Minimization Principle continued. Rayleigh-Ritz Approximation. | (LN) Eigenvalues; (S) 11.2. | |
| 8-6 | Minimax Principle; Asymptotics of Eigenvalues | (LN) Eigenvalues; (S) 11.2, 11.6; (E) 6.5.1 | |
| 8-7 | Completeness of Eigenfunctions. | (LN) Eigenvalues; (S) 11.3. | Homework 6 due |
| Week 8 | |||
| 8-11 | Calculus of Variations; Dirichlet's Principle | (S) 14.3, 7.1; (E) 2.2.5.b, 8.1.1, 8.1.2 | |
| 8-12 | Calculus of Variations. | (S) 14.3, 7.1; (E) 2.2.5.b, 8.1.1, 8.1.2 | |
| 8-13 | Review/Catch-up | ||
| 8-14 | Review/Catch-up | Homework 7 due | |
| 8-16 | Final Exam. 12:15pm-3:15pm. Location to be announced. |
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