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Topic Author
Posts: 4
Joined: September 9th, 2016, 3:47 am

### Similarity Transformation

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.

Cuchulainn
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### Re: Similarity Transformation

What kinds of PDEs? Are you aka 'similarity reduction'?
There's lot of stuff out there.
http://www.datasimfinancial.com
http://www.datasim.nl

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ppauper
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Joined: November 15th, 2001, 1:29 pm

### Re: Similarity Transformation

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

lovenatalya
Posts: 287
Joined: December 10th, 2013, 5:54 pm

### Re: Similarity Transformation

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.

frolloos
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Joined: September 27th, 2007, 5:29 pm
Location: Netherlands

### Re: Similarity Transformation

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-an ... 2-531680-4

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

ppauper
Posts: 70239
Joined: November 15th, 2001, 1:29 pm

### Re: Similarity Transformation

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

lovenatalya
Posts: 287
Joined: December 10th, 2013, 5:54 pm

### Re: Similarity Transformation

ppauper
Posts: 70239
Joined: November 15th, 2001, 1:29 pm

### Re: Similarity Transformation

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.

Cuchulainn
Posts: 61609
Joined: July 16th, 2004, 7:38 am
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### Re: Similarity Transformation

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?
http://www.datasimfinancial.com
http://www.datasim.nl

Every Time We Teach a Child Something, We Keep Him from Inventing It Himself
Jean Piaget

lovenatalya
Posts: 287
Joined: December 10th, 2013, 5:54 pm

### Re: Similarity Transformation

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.

Cuchulainn
Posts: 61609
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
Contact:

### Re: Similarity Transformation

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?
http://www.datasimfinancial.com
http://www.datasim.nl

Every Time We Teach a Child Something, We Keep Him from Inventing It Himself
Jean Piaget

ppauper
Posts: 70239
Joined: November 15th, 2001, 1:29 pm

### Re: Similarity Transformation

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

stanleylam
Posts: 1
Joined: March 24th, 2020, 6:30 am

### Re: Similarity Transformation

I would treat similarity transformation as a method of solving PDE. The point is understanding the structure implied by the PDE, finding the symmetry behind and it may help to reduce or transform the PDE into a simplified form. This would be more visible when we talk about PDE in physical system...

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