====== TMA4305 Partielle differensialligninger 2021 ======


{{ :tma4305:2020h:20180922t101208_0044h.jpg?nolink |}}


===== 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 {{ :tma4305:2021h:17092021meeting.pdf |17.IX.2021}} and {{ :tma4305:2021h:26112021meeting.pdf |26.XI.2021}}.

Exam 2.XII.2021 {{ :tma4305:2021h:exam2021.pdf |Text}} and {{ :tma4305:2021h:eksamen.pdf |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  | @hsl(240,100%,93%): Introduction to the  course,  |
^ :::  | | :::    | We will cover material in Ch. 2 when we need it |
^ :::  |  25 aug  | ::: | @hsl(240,100%,98%): Wed: Introduction, existence and uniqueness of solutions for ODEs. 1st order quasilinear equations in two variables. |
^ :::  |  26 aug  | ::: | @hsl(240,100%,98%): 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  | @hsl(240,100%,93%): Mer generelle kvasilineære systemer, bølgeligningen |
^ :::  | |  :::    | @hsl(100,100%,98%): <color hsl(220,100%,70%)>⮕</color> **Notat:** //First order quasilinear equations// ( {{ :tma4305:2020h:quasilin.pdf |A5 for skjerm}}, {{ :tma4305:2020h:quasilin-a4.pdf |A4 for utskrift}})\\ <color hsl(0,100%,20%)>Oppdatert 2020-08-30</color> Notes on weak solutions {{ :tma4305:2021h:weaksolutions14.pdf |Rankine-Hugoniot}}|
^ :::  | 2 sept | ::: | @hsl(240,100%,98%): Torsdag: d'Alembert's solution finished. Duhamel's Principle. Energy considerations, uniqueness |
^ :::  | 3 Sept | :::  | @hsl(240,100%,98%): Friday: Darboux's equation. Kirchhoff's formula, n = 3.|
^  36  | | **B** 4.3, 4.4, 4.6, 4.7 | @hsl(240,100%,93%): Bølgeligningen |
^ :::  |  9 Sep  | ::: | @hsl(240,100%,98%): Thursday: Poisson's formula via descent from Kirchhoff's,  randverdiproblemer |
^ :::  |  10 Sep  | ::: | @hsl(240,100%,98%): Friday: Local energy, uniqueness, past light cone. Derivation of the Heat Eqn. |
^  37  | | **B**  6.1–4 | @hsl(240,100%,93%): The Heat Equation|
^ :::  |  16 Sep| ::: | @hsl(240,100%,98%): Thursday: Heat Kernel, solution of the initoal value problem (the Cauchy Problem){{ :tma4305:2021h:heat2018.pdf |From Fouriertransform}}|
^ :::  |  17 sep  | ::: | @hsl(240,100%,98%): 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  | @hsl(240,100%,93%): Heat Eqn, the Laplace Eqn. |
^ :::  |  23 Sept  | ::: | @hsl(240,100%,98%): Thursday: Short Summary of Wave Eqn, Maximum Principle in unbounded domain |
^ :::  | |  :::    | @hsl(100,100%,98%): <color hsl(220,100%,70%)>⮕</color> **Notat:** Weak maximum principle for the heat equation ({{ :tma4305:2020h:paramax.pdf |A5 for skjerm}} · {{ :tma4305:2020h:paramax-a4.pdf |A4 for utskrift}}) |
^ :::  |  24 Sept | ::: | @hsl(240,100%,98%): **Friday**  Laplace Eqn. Poisson's formula, Mean Values.  Two lectures from 2020 recorded |
^ :::  |  :::  | ::: | @hsl(240,100%,98%): [[https://ntnu.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=f41a0fcf-bb10-4ff7-b575-ac38008311b3|Maksimumsprinsippet for et ubegrenset område]] (Panopto) |
^ :::  |  :::  | ::: | @hsl(240,100%,98%): [[https://ntnu.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=7aba7703-f300-42c8-acde-ac38009610f8|Harmoniske funksjoner og middelverdiegenskapen]] (Panopto) |
^ :::  | |  :::    | @hsl(100,100%,98%): <color hsl(220,100%,70%)>⮕</color> **Notat:** Harmonicfunctionology ({{ :tma4305:2020h:laplace.pdf |A5 for skjerm}} · {{ :tma4305:2020h:laplace-a4.pdf |A4 for utskrift}}) ·  <color hsl(0,100%,20%)>**Ny versjon** 2020-10-08 β2: Sterkt forenklet utledning av Theorem 22 («β2» fordi jeg ikke har korrekturlest det veldig nøye).</color> |
^  39  | | **B** Ch. 7  |  |
^ :::  |  30 sep  | ::: | @hsl(240,100%,98%): Thu: Gave formal definitions of vector spaces, inner products, and norms. Gave a brief introduction to Lebesgue spaces, incl Holder inequality. |
^ :::  |  1 okt | ::: | @hsl(240,100%,98%): 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  | ::: | @hsl(240,100%,98%): Thu: Completed self-adjointness for the Laplace operator. Introduced weak derivatives, and studied characteristics for quasilinear equations. |
^ :::  |  8 okt | ::: | @hsl(240,100%,98%): 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 | @hsl(240,100%,93%): Sobolev spaces, Weak solutions|
^ :::  |   14 Oct.| ::: | @hsl(240,100%,98%): Thursday: Sobolev spaces. The proof in 10.4 was skipped. | 
^ :::  |   15 Oct.| ::: | @hsl(240,100%,98%):  Friday: Sobolev's inequality. Maximum Principle {{ :tma4305:2021h:15102021pde.pdf |9.4 (-L now +L)}}\\  |
^  42  | |**B** 10.6, 11.1, 11.2, 11.3 (first part) | @hsl(240,100%,93%): Weak solutions, variational methods|
^ :::  |   21 Oct  | ::: | @hsl(240,100%,98%): Thu: Weak formulations of evolution equations (Sec. 10.6) |
^ :::  |   22 okt  | ::: | @hsl(240,100%,98%):​ Fri: Variational methods, Laplace equation, Poissson equation, Dirichlet principle, Poincare inequality|
^  43  | | **B**  11.3 (last part), 11.4, 11.5 (partial)  | @hsl(240,100%,93%): Poisson equation, elliptic regularity, spectral theorem |
^ :::  |   28 Oct  | ::: | @hsl(240,100%,98%): Thu: Coercivity and boundedness, existence of unique weak solution of the Poisson equation |
^ :::  |   29 Oct    | ::: | @hsl(240,100%,98%):  Fri: Elliptic regularity, existence of lowest eigenvalue for the Dirichlet Laplacian. Rellich's theorem without proof |
^  44  | | **B**  11.4–11.7  | @hsl(240,100%,93%): Elliptisk regularitet, egenverdier for Laplaceoperatoren, … |
^ :::  |   ??.?  | ::: | @hsl(240,100%,98%):​ Onsdag: Elliptisk regularitet, kompakthet  |
^ :::  |   ??.?  | ::: | @hsl(240,100%,98%):​Fredag: Spektralteoremet for Laplace-operatoren med Dirichlet randbetingelser |
^ :::  | |  :::    | @hsl(100,100%,98%): <color hsl(220,100%,70%)>⮕</color> **Håndskrevet notat:** {{ :tma4305:2020h:kompakthet.pdf |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. |
^  45  | | **B**  12.1–12.4  | @hsl(240,100%,93%): Theory of distributions|
^ :::  |   11 Nov. | ::: | @hsl(240,100%,98%): Thursday: Distributions, Dirac's Delta|
^ :::  |   12 Nov. | ::: | @hsl(240,100%,98%): Friday: Distributions, Fundamental Solutions |
^  46  | | **B** 12.5  | @hsl(240,100%,93%): Distributions, Green's Functions |
^ :::  |   18 nov  | ::: | @hsl(240,100%,98%): Thursday: Green's function, fundamental solutions |
^ :::  |   19 nov  | ::: | @hsl(240,100%,98%): 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  | |    | @hsl(240,100%,93%): Last Week with lectures. Repetiton, examples |
^ :::  |   25 Nov  | ::: | @hsl(240,100%,98%): Thursday: Repetition|
^ :::  |   26 Nov  | ::: | @hsl(240,100%,98%): Friday: **Last Lecture** Examples, Repetition|
^ :::  | |  :::    | @hsl(100,100%,98%):  From week 46: Section 2.1 in [[https://link.springer.com/book/10.1007/978-1-4615-9966-1|Fritz John]] (SpringerLink; du må være innenfor NTNUs nett fysisk eller på VPN. ) |