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cprasad
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Similarity Transformation

November 26th, 2017, 2:50 pm

I am looking for a book (or any resource such as an article, website, etc.) for learning Similarity Transformation for transforming
differential equations. I would appreciate it if you let me know of any reference.
 
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Cuchulainn
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Re: Similarity Transformation

November 26th, 2017, 4:18 pm

What kinds of PDEs? Are you aka 'similarity reduction'?
There's lot of stuff out there. 
 
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ppauper
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Re: Similarity Transformation

November 27th, 2017, 9:05 pm

as a starting point, I believe the similarity transformation for the heat equation is in at least some of Paul's books.
 
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lovenatalya
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Re: Similarity Transformation

April 27th, 2018, 5:37 pm

The similarity transform is but one form of symmetry that can be used to solve PDE's. When we talk about symmetry, we usually refer to group transform. PDE is differentiable, so most of the symmetry should also be differentiable. So that gives you a differentiable group transform which is a Lie group. So if you really want to study symmetry for PDE, you should search for the application of Lie group/algebra to PDE. Peter Olver's book is a great one to learn. A more technical example is this paper that may be too involved for you but may be worth a look.
 
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frolloos
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Re: Similarity Transformation

April 28th, 2018, 10:04 am

In addition to Peter Olver's book, I've been told every serious student of Lie group applications to DEs should have Ovsiannikov's book:

https://www.elsevier.com/books/group-analysis-of-differential-equations/ovsiannikov/978-0-12-531680-4

But perhaps the OP can start with Bluman & Cole's paper:

http://www.jstor.org/stable/24893142?se ... b_contents
 
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ppauper
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Re: Similarity Transformation

April 28th, 2018, 10:33 am

frolloos wrote:
But perhaps the OP can start with Bluman & Cole's paper:

http://www.jstor.org/stable/24893142?se ... b_contents


indeed, if he's working with black-scholes, the heat equation is the way to go
 
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lovenatalya
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Re: Similarity Transformation

April 28th, 2018, 8:46 pm

 
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ppauper
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Re: Similarity Transformation

April 29th, 2018, 5:49 am

lovenatalya wrote:

that's maybe a little advanced for someone to for learn Similarity Transformation for transforming differential equations, but it's of historical interest in that george bluman essentially derived some of the formulas for barrier options a couple of decades before the math finance folk rediscovered them. And I doubt if they credited him for it.
 
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Cuchulainn
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Re: Similarity Transformation

April 29th, 2018, 7:48 pm

I have Olver's book, I did Lie Groups and Lie Representations in undergrad and I don't understand what these PDEs deliver.

The link to classical mathematical physics (where all the interesting PDEs come from) is tenuous. Am I missing something?
 
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lovenatalya
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Re: Similarity Transformation

April 30th, 2018, 7:20 pm

Cuchulainn wrote:
I have Olver's book, I did Lie Groups and Lie Representations in undergrad and I don't understand what these PDEs deliver.

The link to classical mathematical physics (where all the interesting PDEs come from) is tenuous. Am I missing something?

I do not quite understand your statement. Are you saying Lie group/algebra and its representations have tenuous bearings on mathematical physics? Noether's theorem that gives all the conservation laws is but one example of its bearing. Many eigenvalue problems, for energy levels, in quantum mechanics either are too tedious or cannot be solved without Lie group or discrete group representations. I must have misunderstood your statement.
 
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Cuchulainn
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Re: Similarity Transformation

May 1st, 2018, 10:21 am

lovenatalya wrote:
Cuchulainn wrote:
I have Olver's book, I did Lie Groups and Lie Representations in undergrad and I don't understand what these PDEs deliver.

The link to classical mathematical physics (where all the interesting PDEs come from) is tenuous. Am I missing something?

I do not quite understand your statement. Are you saying Lie group/algebra and its representations have tenuous bearings on mathematical physics? Noether's theorem that gives all the conservation laws is but one example of its bearing. Many eigenvalue problems, for energy levels, in quantum mechanics either are too tedious or cannot be solved without Lie group or discrete group representations. I must have misunderstood your statement.

I'm saying I don't understand how this stuff is useful in let's say Navier Stokes or Black Scholes PDE. Caveat: I haven't tried and wouldn't know where to start.
It is constructive in the Bishop sense?
 
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ppauper
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Re: Similarity Transformation

May 1st, 2018, 11:52 am

I've come across papers in the past where people claim to have used it to find new classes of solutions to various PDEs.
It seemed to me at the time that (at least for those particular solutions) there were easy ways to find them: if you understand  the problem, you have a gut feeling of what the solutions you want would look like and you can go fishing for them

not particularly relevant to the very specific request of the original poster
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