What do I need to know before studying PDE?

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    Pde Prerequisites
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Discussion Overview

The discussion revolves around the prerequisites and foundational knowledge necessary before studying Partial Differential Equations (PDEs). Participants explore various mathematical concepts and skills that may be beneficial or essential for understanding PDEs, including multivariable calculus, ordinary differential equations (ODEs), vector analysis, and functional analysis.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant mentions having learned multivariable calculus and ODEs but seeks additional recommendations for preparation before studying PDEs.
  • Another participant suggests that vector analysis is relevant, particularly in relation to problems involving integrals and equations like the Navier-Stokes equations.
  • A participant expresses difficulty understanding concepts in a PDE textbook's appendix, such as measure and Lipschitz continuity, and seeks resources to learn these topics.
  • Some participants argue that the material in the appendix is not a prerequisite for understanding the main content of PDEs.
  • It is proposed that basic functional analysis is necessary for understanding certain methods in PDEs, particularly in relation to Fourier coefficients and Sturm-Liouville problems.
  • One participant emphasizes the importance of knowing the specific interest in PDEs, whether it be theoretical properties, numerical issues, or modeling applications.
  • A suggestion is made to engage with practical PDEs, such as the Poisson equation, through numerical solutions to enhance understanding.
  • Another participant critiques the focus of many math courses on theoretical properties of differential equations, advocating for a more application-oriented approach to demonstrate their utility.
  • Concerns are raised about the balance between theoretical and practical aspects in teaching differential equations, with some participants advocating for a stronger emphasis on theoretical understanding.

Areas of Agreement / Disagreement

Participants express a range of views on the necessary background for studying PDEs, with no clear consensus on specific prerequisites. Some emphasize the importance of practical applications, while others highlight the need for a solid theoretical foundation.

Contextual Notes

Participants mention various mathematical concepts and areas of study that may be relevant but do not reach a definitive agreement on which are essential or in what order they should be learned. The discussion reflects differing educational experiences and perspectives on the balance between theory and application in the study of PDEs.

pyfgcr
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Hi, I'm really interesting of PDE, but I don't really know what I have to learn before start with PDE.
I have learn multivariable calculus and ODE, but are there something need to learn before PDE?
Thanks in advance.
 
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How about vector analysis? PDEs appear in a lot vector analysis problems path, surface volume inegrals, Javier stokes eqn...
 
I have bought a PDE book by Evan and I don't understand even the notion and inequalities in the appendix such as measure, support, Lipschitz continuous, convolution, norm,... and many other things.
I wonder where I can learn all this stuff.
 
what about the beginning chapters? I wouldn't judge a book by reading what's in the appendix. That stuff is there in case you need it and isn't a prerequisite to understanding PDEs.
 
For basic PDE You need to know basic Functional Analysis (i.e for stuff like Fourier coefficients in problems of Sturm-Liouville in separation of variables). Other than that also ODE and Multivariable and Vector calculus are essential.

The more advanced you get in the book the more you need to know more results from Analysis. It's up to you if you want to understand the methods to understand the theory behind them. It's a long road, that's for sure...
 
You should ask yourself what is your interest on PDEs? Theoretical properties, numerical issues, modeling of e.g. biological phenomena, … ?

What was your interest when studying ODEs?
 
Pick a practical PDE, like one of those already suggested ( say, Poisson Equation in 2D)

and try to solve it numerically.

Yes, numerical attempt will be like designing an experiment and you will learn a lot along the way.

Once you solve it numerically, in order to make sure you did correctly, try to get an analytical 1D solution with no y-variation and see if you solved it correctly.Thing is, I had been educated for more than 10 years in different schools, in different contexts, but when I had to solve a PDE correctly,

that's when I learned all that is necessary. My 2 cents. Good luck.
 
That’s it!

People, especially pure math orientated have to understand that Des (ODE, PDE, DDE,…) are more than just objects to study from a mathematical point of view. Unfortunately, until now most math courses about Des are about theoretical properties. No doubt, this is important, but these guys (most of them are really not able to this) have to show students what one could do with these Des.

I am talking to you as a math PhD.

To start with Des either from a numerical point of view or a modeling point of view, it is much more important in order to understand the power of this tool than just concentrating on mathematical properties.

To understand why the solution of an ODE exists and is unique is important for the examination but it has nothing to do with the real value of an ODE.

As an example, I know money exists and it is important to have, but what you could do with it is a complete different field. The same with Des….
 
Unstable said:
That’s it!

People, especially pure math orientated have to understand that Des (ODE, PDE, DDE,…) are more than just objects to study from a mathematical point of view. Unfortunately, until now most math courses about Des are about theoretical properties. No doubt, this is important, but these guys (most of them are really not able to this) have to show students what one could do with these Des.

I am talking to you as a math PhD.

To start with Des either from a numerical point of view or a modeling point of view, it is much more important in order to understand the power of this tool than just concentrating on mathematical properties.

To understand why the solution of an ODE exists and is unique is important for the examination but it has nothing to do with the real value of an ODE.

As an example, I know money exists and it is important to have, but what you could do with it is a complete different field. The same with Des….

Eh, in my experience most courses/texts devote so much time to numerical methods and applications that student's walk away without understanding how to solve any equation that doesn't precisely match one that they're seen before. If anything, intro courses should be far more theoretically oriented.
 
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  • #10
Wow… that is different to my course… I learned only theoretical stuff!

Nevertheless, I want to say that I am an expert in ODE modeling with biological background and we have to go back to the basics.

The traditional ideas of DES are what?
 
  • #11
EDIT:

DEs are the perfect example where theoretical math fails (for students and most users, don't get me wrong theoretical stuff
is important but should not be only thing)

The question is not why my car is red and has 4 wheels (and not 7 maybe), the question is why I could drive from A to B with this (maybe perfect) car under several circumstances!
 
Last edited:

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