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

[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!

 

[Simon Bridge]

Solving PDEs is guesswork ... you get better at guessing as you gain experience.
http://web.iitd.ac.in/~vvksrini/Similar3.pdf

 

[member 428835]

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?

 

[rumborak]

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.

 

[Chestermiller]

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

 

[member 428835]

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)?

 

[Chestermiller]

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