# TMA4305 Partial Differential Equations 2019

## Messages

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### Lectures; past, present, and future

Week | Chapter/section | Remarks and notes |
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34 | Ch 1–3 Notes | Read Ch 1 on your own. Also have a look at Ch 2 on your own – but we will return to it when needed (especially the higher dimensional calculus). For needed background on well-posedness theory for ODEs, consult for example my notes on dynamical systems, Ch 1, pp. 2–7. (No need to study the proofs.) |

Covered introductory and preliminary material, including some notation. The concept of well-posed problem [Jaques Hadamard 1902]: Solutions exist, are unique, and depend continuously on data.Basic theory of transport equations and conservations laws in one space dimension, balance law in multiple dimensions. |
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Notes: Collisions of characteristics (A5 for the screen, A4 for printing) |
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35 | Ch 3 end Ch 4: 4.1–4.4 | Start on the wave equation (Ch 4 in Borthwick): Characteristics and d'Alembert's solution (§4.2), and Duhamel's principle (§4.4) (if we get that far) |

Wednesday: Finished Ch 3 with a look at a traffic model. Also proved that the characteristic method applied to the IVP \(u_t+a(t,x,u)u_x=f(t,x,u)\) with \(u(0,x)=g(x)\) actually solves the problem. I used differentials to handle the algebra; much easier than transforming partial derivatives between coordinate systems. |
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Thursday: Wave equation, D'Alembert's solution, Duhamel's principle. | ||

36 | Ch 4: 4.6, 4.7 Notes | The wave equation in higher dimensions, energy method for uniqueness |

Wednesday: Solution formulas for the wave equation in 3 and 2 space dimensions. Huygens' principle in 3D, not in 2D. (A similar technique works for \(2k+1\) dimensions, with subsequent descent to \(2k\) dimensions.) |
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Thursday: Proof that the solution formula in three dimensions actually solves the problem. Energy methods for uniqueness and domains of dependence in the wave equation. |
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Notes: On the energy method for wave equations (A5 for the screen, A4 for printingThese notes may be thought of as an alternative to, and expansion of, §4.7) |
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37 | Notes Ch 6 | A bit more from the above notes on the energy method. Start chapter 6 (heat equation) |

Wednesday: Proved Theorem 2 from the end of the note (above) on the energy method for wave equations. The current version in that note is too complicated. I will update it to match the way I lectured it.Started on the heat equation, and derived the formula for the standard heat kernel in \(n\) space dimensions. |
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Thursday: The solution of the initial value problem for the heat equation, and Duhamel's principle for the non-homogeneous heat equation. As part of the proof I did a partial integration, moving the Laplacian from one term to another. This is really an application of Green's second identity ( B Theorem 2.11): The boundary integrals vanish when we integrate over a large enough region containing the support of \(f\). |
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38 | Notes Ch 9: 9.5 and more | Notes on the parabolic maximum principle (see below) Perhaps start working on Ch 9 and my associated notes. |

Wednesday: The weak maximum principle for the heat equation, on bounded and unbounded domains. Corollaries are continuous dependence of data, uniqueness. | ||

Notes: Weak maximum principle for the heat equation (A5 for the screen, A4 for printing)These notes are an alternative to §9.5 Updated 2019-09-18 with an even better version of the maximum principle on \(\mathbb{R}^n\) than the one in the lecture!Updated 2019-09-19 (removed an erroneous example) |
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Thursday: Started notes on the Laplace equation (below). Covered the mean value property, and proved its equivalence to harmonicity. Regularity (harmonic functions are \(C^\infty\), and the strong and weak maximum principle. | ||

New notes: Harmonicfunctionology (A5 for the screen, A4 for printing)These notes are related to §§9.1–9.4 |
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39 | Notes Ch 9: 9.1–9.4 | Notes on the Laplace equation (continued) |