Prep for PDEs: Math Staple Requirements

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    Pdes Preparation
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Discussion Overview

The discussion centers around the mathematical preparation required for studying partial differential equations (PDEs), including the necessary background knowledge in various mathematical disciplines and the relevance of numerical methods.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant expresses interest in PDEs and seeks advice on the necessary mathematical background, mentioning their current study of analysis and linear algebra.
  • Another participant emphasizes the importance of numerical methods for solving real-world PDEs, suggesting that programming skills, particularly in C++, are essential.
  • A participant asserts that a foundational understanding of ordinary differential equations (ODEs) is necessary before tackling PDEs.
  • One participant mentions the common analytical technique of separation of variables for solving PDEs and lists various numerical methods relevant to PDEs.
  • Another participant argues that the prerequisites for an undergraduate PDE course are primarily calculus and ODEs, challenging the notion that advanced mathematics is required.
  • A participant provides a link to a Wikipedia page on numerical methods for PDEs, indicating its usefulness for further exploration.

Areas of Agreement / Disagreement

Participants express differing views on the level of mathematical preparation required for studying PDEs. Some suggest advanced topics are necessary, while others argue that a basic understanding of calculus and ODEs suffices for undergraduate study.

Contextual Notes

There are varying opinions on the necessity of advanced mathematics such as abstract algebra, topology, and numerical analysis, with no consensus on the exact requirements for studying PDEs.

l'Hôpital
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DEs in general are something that I find very interesting. Though my knowledge of DEs are very rudimentary to say the least, I find them fascinating. In particular, I want to learn about PDEs and obtain a deeper understanding for ODEs.

My question is, then, what kind of math preparation would I need before attempting to tackle PDEs? Currently, I'm trying to at least obtain a basic understanding Analysis (reading some Spivak as well as other things to get my proof skills up). I've also got a basic understanding of Linear Algebra. How far in advanced mathematics must I be to truly be ready for PDEs? Must I have completed most of the Math 'staples' (Abstract algebra, Topology, analysis) before tackling PDEs?
 
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The problem with most real world PDE's is that they have no closed form analytical solution. Therefore, they have to be solved numerically and that involves computer programming. So if I were you I would take some time to study C++.
 
ODE is a must.
 
I haven't studied PDEs,
but for analytical solutions, separations of variables is usually used.

As for numeric methods Wikipedia provides quite an extensive list of methods:

http://en.wikipedia.org/wiki/List_o...pics#Numerical_partial_differential_equations


For fundamentals I'd consider, programing (c++ and/or FORTRAN), linear algebra, vector calculus, tensor calculus, and any numeric analysis .

For real advanced mathematics, a course on fractals could be interesting as a possible method to deal with cases, where there will be non negligible dynamics on all scales. For instance in Navier Stokes equations, you can have turbulence on multiple scales, people try and account for this by adding a fudge factor to the viscountcy.

Then consider application courses that are computationally oriented toward solving some class of of PDEs (e.g. computational fluid mechanics) or that involve extensive USE of a lot of vector and tensor calculus.
 
lol what are you guys talking about. this guy's essentially asking what he needs to know to take an undergrad course in pdes and you guys are telling him he needs to be an applied mathematician.

to study pdes at an undergrad level you need calculus and odes. i just finished the first semester of a 2 semester pdes class and the only we trick we repeatedly used is integration by parts.

we used walter strauss's book which is ok but i don't have any other suggestions.
 

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