Mastering PDEs with Similarity Solutions: Tips and Tricks from the Pros

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Discussion Overview

The discussion revolves around the challenges and methodologies involved in solving partial differential equations (PDEs) using similarity solutions. Participants express confusion about the process of identifying similarity variables and the rationale behind their selection, exploring both theoretical and practical aspects of the topic.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Homework-related

Main Points Raised

  • One participant expresses confusion regarding how to derive similarity variables, questioning whether it is based on scaling or intuition.
  • Another participant suggests that solving PDEs involves a degree of guesswork, which improves with experience.
  • A participant references a specific paper but struggles to understand the significance of certain mathematical relations presented in it, seeking clarification on how they lead to the formulation of similarity variables.
  • Concerns are raised about the perceived arbitrary nature of solutions learned in academic settings, with a participant expressing skepticism about their applicability in real-world scenarios, suggesting reliance on computer simulations instead.
  • One participant offers to assist if a specific problem is cited, mentioning a learned procedure for identifying similarity variables.
  • There is a follow-up where the same participant indicates they do not have a specific problem at the moment but may seek help later.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the methodology for deriving similarity variables, with multiple viewpoints on the nature of the process and its reliance on experience or intuition. The discussion remains unresolved regarding the significance of the mathematical relations mentioned.

Contextual Notes

Participants acknowledge the complexity of the topic and the potential limitations in understanding the significance of certain mathematical expressions. There is an indication of missing specific examples that could clarify the discussion.

member 428835
Hi PF!

I am confused about solving PDE's using a similarity solution. How do we come up with the similarity variable? Is it scaling or lucky guesses?

I've read several papers on it but haven't found the reasons on why they use the variables they do for the similarity to work. Any advice from you would be great!

Thanks!
 
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Solving PDEs is guesswork ... you get better at guessing as you gain experience.
http://web.iitd.ac.in/~vvksrini/Similar3.pdf
 
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This is one of the papers I read. I don't see the significance of either $$\frac{z}{s^{a/b}} = \frac{x}{t^{a/b}} \\ vs^{-c/b} = ut^{-c/b}$$ and how the above relations help us decide that ##\xi = x / t^{a/b}## and that ##u = t^{c/b} y ( \xi )##.

Could you help me out?
 
Simon Bridge said:
Solving PDEs is guesswork ... you get better at guessing as you gain experience.

I think that was always my main gripe with them, or at least when I learned about them. I totally understand that they are usually *the* way of describing most physical things. But the solutions one learned about always seemed cherry-picked to work.
In the end, I always had the feeling that if I ever were to encounter one in real life and needed the solution, a computer simulation was the only way.
 
joshmccraney said:
This is one of the papers I read. I don't see the significance of either $$\frac{z}{s^{a/b}} = \frac{x}{t^{a/b}} \\ vs^{-c/b} = ut^{-c/b}$$ and how the above relations help us decide that ##\xi = x / t^{a/b}## and that ##u = t^{c/b} y ( \xi )##.

Could you help me out?
If you can cite a specific problem, I think I can help you through this. There was a procedure we learned in school for identifying the similarity variable.

Chet
 
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Chestermiller said:
If you can cite a specific problem, I think I can help you through this. There was a procedure we learned in school for identifying the similarity variable.

Chet
So I'm not looking at one specifically now, but perhaps I can take you up on this offer later on (in a few weeks)?
 
joshmccraney said:
So I'm not looking at one specifically now, but perhaps I can take you up on this offer later on (in a few weeks)?
No problem.

Chet
 

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