Partial Differential Equations

What Hacker News says

1. ezedvOct 2023
   Absolutely, if you're looking for advanced books on differential equations, I'd recommend "Partial Differential Equations" by Lawrence C. Evans for a comprehensive coverage of PDEs. For a more specialized focus on Ordinary Differential Equations, "Ordinary Differential Equations" by Vladimir I. Arnold is a fantastic choice. And for a deep dive into Stochastic Differential Equations, check out "Stochastic Differential Equations" by Bernt Øksendal.
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2. cpp_frogMar 2022
   > My goal here is to provide a roadmap for anyone interested in understanding mathematics at an advanced level. Anyone that follows and completes this curriculum will walk away with the knowledge equivalent to an undergraduate degree in mathematics.

   NO, NO, NO.

   There is no real way to go up to the real deal without having understood elementary Functional Analysis, which the article doesn't even mention. FA is roughly what Linear Algebra would look like if instead of finite dimensional vector spaces we considered infinite dimensional vector spaces. It opens the rigorous path to non-linear optimization, analysis of pdes, numerical analysis, control theory, an so on. What this article mentions is a way to work around things, but nowhere near an undergraduate degree in mathematics.

   I'm astonished that the PDE section has such books, they look like the calculus aspect of partial differential equations. A more appropriate book would be L. C. Evans' Partial Differential Equations. Same with ODEs, no mention of Barreira's or Coddington & Levinson's books.

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3. cpp_frogJul 2021
   It is easier to study from a theoretical point of view (easier that the heat or the wave equation: [1], Ch. 2), and it's easier to implement a solving method. When you are learning the finite element method, this is one of the first examples that people use to test if they got the right implementation.

   Now, I wonder if the author regards it important in his particular area (aerospace engineering), I'm new to the field so I don't see how. Right now I'm reading a book [0] on models to solve problems concerning aerospace applications and they mostly use a simplified form of the Navier-Stokes equations together with some elasticity assumptions.

   [0] Fluid Structure Interacion, Morand-Ohayon.

   [1] Partial Differential Equations, Evans.

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4. shooJul 2019
   in a past life i learned a little about PDE and the theory justifying why finite element methods work to approximate solutions.

   i never explicitly ran into anything about discrete differential geometry, but that doesn't mean it wasn't there all along, lurking beneath (or perhaps above?) my level of understanding.

   some reading: Evans -- Partial Differential Equations, Wendland -- Scattered Data Approximation, Wahba -- Spline models for observational data .

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5. johndcookDec 2017
   re book recommendation: Lawrence Evans's book Partial Differential Equations. Functional analysis etc.
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6. MrBraJan 2015
   "An overview of Inter-universal Teichmüller Theory and [...] how we can begin to understand this theory":

   http://www.quora.com/Joseph-Heavner/Posts/An-overview-of-Int...

   Also, from an anonymous post on 4chan @ http://boards.4chan.org/sci/thread/6931488/are-you-ready-for... :

   "How to understand IUTeich from the bottom up:

   - Algebra (Israel Gelfand)

   - The Method of Coordinates (Israel Gelfand)

   - How to Prove It: A Structured Approach (Daniel Velleman)

   - Kiselev's Geometry - Planimetry and Stereometry

   - Trigonometry (Israel Gelfand)

   - What Is Mathematics? An Elementary Approach to Ideas and Methods (Richard Courant)

   - A Course of Pure Mathematics (G.H. Hardy)

   - Linear Algebra (Kenneth Hoffman and Ray Kunze)

   - Elementary Differential Equations (William Boyce and Richard DiPrima)

   - Topology (James Munkres)

   - Calculus On Manifolds (Michael Spivak)

   - Principles of Mathematical Analysis (Walter Rudin)

   - Real and Complex Analysis (Walter Rudin)

   - Functional Analysis (Walter Rudin)

   - Partial Differential Equations (Lawrence Evans)

   - Analysis On Manifolds (James Munkres)

   - Abstract Algebra (David Dummit and Richard Foote)

   - Algebraic Topology (Allen Hatcher)

   - Introduction to Smooth Manifolds (John Lee)

   - Foundations of Differentiable Manifolds and Lie Groups - (Frank Warner)

   - Galois Theory (Harold Edwards)

   - Linear Representations of Finite Groups (Jean-Pierre Serre and Leonhard Scott)

   - A Classical Introduction to Modern Number Theory (Kenneth Ireland and Michael Rosen)

   - Introduction to Analytic Number Theory (Tom Apostol)

   - Modular Functions and Dirichlet Series in Number Theory (Tom Apostol)

   - Riemannian geometry (Peter Petersen)

   - The Theory of the Riemann Zeta-Function (Edward Charles Titchmarsh)

   - An Introduction to Teichmüller Spaces (Yoichi Imayoshi and Masahiko Taniguchi)

   - A Course in p-adic Analysis (Alain Robert)

   - Foundations of p-Adic Teichmuller Theory (Shinichi Mochizuki)

   - Mochizuki's papers at http://www.kurims.kyoto-u.ac.jp/~motizuki/papers-english.htm... "

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