TMA4305 Partielle differensialligninger 2021
Meldinger
OFFICE HOURS BEFORE THE EXAM: Tuesday 30.XI in room 1152 SB II, from 2 o'clock to 4 o'clock.
Lectures will be in English.
Reference Group Meeting 17.IX.2021 and 26.XI.2021.
Exam 2.XII.2021 Text and Solutions. Grading according to the normal scale: 0 - 43 F; 44 -57 E; 58 - 70 D; 71- 83 C; 84-96 B; 97 -110 A. The eleven problems have the same weight.
Forelesninger
B below means the book of Borthwicks.
| Uke | Dato | Ref | Hva |
|---|---|---|---|
| 34 | Intro B 1, 2.6, 3.4 | Introduction to the course, | |
| We will cover material in Ch. 2 when we need it | |||
| 25 aug | Wed: Introduction, existence and uniqueness of solutions for ODEs. 1st order quasilinear equations in two variables. | ||
| 26 aug | Thu: Examples on quasilinear equations, in particular, Burgers' equation. Derivation of the linear wave equation. Started derivation of d'Alembert's solution. | ||
| 35 | B 2.5, 3.3, 4.1–2, notat | Mer generelle kvasilineære systemer, bølgeligningen | |
| ⮕ Notat: First order quasilinear equations ( A5 for skjerm, A4 for utskrift) Oppdatert 2020-08-30 Notes on weak solutions Rankine-Hugoniot |
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| 2 sept | Torsdag: d'Alembert's solution finished. Duhamel's Principle. Energy considerations, uniqueness | ||
| 3 Sept | Friday: Darboux's equation. Kirchhoff's formula, n = 3. | ||
| 36 | B 4.3, 4.4, 4.6, 4.7 | Bølgeligningen | |
| 9 Sep | Thursday: Poisson's formula via descent from Kirchhoff's, randverdiproblemer | ||
| 10 Sep | Friday: Local energy, uniqueness, past light cone. Derivation of the Heat Eqn. | ||
| 37 | B 6.1–4 | The Heat Equation | |
| 16 Sep | Thursday: Heat Kernel, solution of the initoal value problem (the Cauchy Problem)From Fouriertransform | ||
| 17 sep | Friday: Uniqueness via energy considerations. Duhamel's principle again. (Weak) Maximum Principle. B.9.5 | ||
| 38 | B 6.4, 9.5, 9.1, 9.2, notes 2020 | Heat Eqn, the Laplace Eqn. | |
| 23 Sept | Thursday: Short Summary of Wave Eqn, Maximum Principle in unbounded domain | ||
| ⮕ Notat: Weak maximum principle for the heat equation (A5 for skjerm · A4 for utskrift) | |||
| 24 Sept | Friday Laplace Eqn. Poisson's formula, Mean Values. Two lectures from 2020 recorded | ||
| Maksimumsprinsippet for et ubegrenset område (Panopto) | |||
| Harmoniske funksjoner og middelverdiegenskapen (Panopto) | |||
| ⮕ Notat: Harmonicfunctionology (A5 for skjerm · A4 for utskrift) · Ny versjon 2020-10-08 β2: Sterkt forenklet utledning av Theorem 22 («β2» fordi jeg ikke har korrekturlest det veldig nøye). | |||
| 39 | B Ch. 7 | ||
| 30 sep | Thu: Gave formal definitions of vector spaces, inner products, and norms. Gave a brief introduction to Lebesgue spaces, incl Holder inequality. | ||
| 1 okt | Fri: Proved Minkowski inequality. Discussed completeness and orthonormal bases. Started on self-adjointess. | ||
| 40 | B 7.6, 10.1, 3.4, 10.2 | ||
| 7 okt | Thu: Completed self-adjointness for the Laplace operator. Introduced weak derivatives, and studied characteristics for quasilinear equations. | ||
| 8 okt | Fri: Studied characteristics in more details, example from traffic flow. Proved the Rankine-Hugoniot relation. | ||
| 41 | B 10.3, 10.4, 10.5; 9.4 | Sobolev spaces, Weak solutions | |
| 14 Oct. | Thursday: Sobolev spaces. The proof in 10.4 was skipped. | ||
| 15 Oct. | Friday: Sobolev's inequality. Maximum Principle 9.4 (-L now +L) |
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| 42 | B 10.6, 11.1, 11.2, 11.3 (first part) | Weak solutions, variational methods | |
| 21 Oct | Thu: Weak formulations of evolution equations (Sec. 10.6) | ||
| 22 okt | Fri: Variational methods, Laplace equation, Poissson equation, Dirichlet principle, Poincare inequality | ||
| 43 | B 11.3 (last part), 11.4, 11.5 (partial) | Poisson equation, elliptic regularity, spectral theorem | |
| 28 Oct | Thu: Coercivity and boundedness, existence of unique weak solution of the Poisson equation | ||
| 29 Oct | Fri: Elliptic regularity, existence of lowest eigenvalue for the Dirichlet Laplacian. Rellich's theorem without proof | ||
| 44 | B 11.4–11.7 | Elliptisk regularitet, egenverdier for Laplaceoperatoren, … | |
| ??.? | Onsdag: Elliptisk regularitet, kompakthet | ||
| ??.? | Fredag: Spektralteoremet for Laplace-operatoren med Dirichlet randbetingelser | ||
| ⮕ Håndskrevet notat: kompakthet (fordi det ikke er dekket så godt i Lineære metoder?) – se også avsnitt 2.8 i C. Heil: Metrics, Norms, Inner Products, and Operator Theory. |
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| 45 | B 12.1–12.4 | Theory of distributions | |
| 11 Nov. | Thursday: Distributions, Dirac's Delta | ||
| 12 Nov. | Friday: Distributions, Fundamental Solutions | ||
| 46 | B 12.5 | Distributions, Green's Functions | |
| 18 nov | Thursday: Green's function, fundamental solutions | ||
| 19 nov | Friday: Green's functions. The "confusing" B 12.6 is replaced by the Fundamental Solution of the Heat Eqn (… = delta(x,t)). Characteristics and classification of 2nd order quasilinear eqns, F. John: section 2.1. | ||
| 47 | Last Week with lectures. Repetiton, examples | ||
| 25 Nov | Thursday: Repetition | ||
| 26 Nov | Friday: Last Lecture Examples, Repetition | ||
| From week 46: Section 2.1 in Fritz John (SpringerLink; du må være innenfor NTNUs nett fysisk eller på VPN. ) |