Separation of variables in MATLAB

[SeM]

Hello, I haven't found any program that can be used to perform separation of variables on difficult PDEs. Is there such a method somewhere?

 

[NFuller]

Programs can numerically solve PDEs. Separation of variables is a trick that you can use to solve some types of PDEs on paper.

 

[SeM]

Hi, I wish it was so easy. If you see on literature database, there are several emerging methods for simplifying very complex PDEs using matrix reps, linear algebra and other approaches . I was wondering if anyone has the knowledge on how to to do some, or any other methods using MATLAB Or computational math software.

Thanks for the answer anyway!

 

[SeM]

Programs can numerically solve PDEs. Separation of variables is a trick that you can use to solve some types of PDEs on paper.

This is the one I wanted to simplify further, but I suspect it is not possible:

$$ r^2 \frac{1}{R}\frac{\partial^2 R}{\partial r^2} - (r+2i r^2) \frac{1}{R}\frac{\partial R}{\partial r} - \frac{r^2}{RY} = \frac{1}{Y} \frac{\partial^2 Y}{\partial \theta^2}+2ir \frac{1}{Y}\frac{\partial Y}{\partial \theta}$$

I have already done separation of variables, and can't get rid of the 1/RY term.

 

[NFuller]

I have already done separation of variables, and can't get rid of the 1/RY term.

Not all PDEs are separable, if you can't do it then a program can't do it either. Where does this PDE come from? There may already be special methods for solving it.

Hi, I wish it was so easy. If you see on literature database, there are several emerging methods for simplifying very complex PDEs using matrix reps, linear algebra and other approaches

What database? There are various methods for dealing with certain classes of PDEs, boundary layer rescaling comes to my mind, but no method works for everything.

 

[SeM]

It comes from a system I have solved and writing about. But I am not sure it makes physical meaning, and in order to find that out I need to solve it, and find out what that 1/RY factor does and can be trated.

I divided this into three components:

The first tw dR/dr parts as one ODE equal zero.

The strange term r^2/RY = 0

and the right hand side of the equation equal to zero. I have solved the former and the latter, but the central term, r^2/RY = 0 is not an ODE, so I am not sure its valid to extract that as an own term and treat the rest as two ODEs..

Is this OK as a method?

 

[SeM]

Not all PDEs are separable, if you can't do it then a program can't do it either. Where does this PDE come from? There may already be special methods for solving it.

What database? There are various methods for dealing with certain classes of PDEs, boundary layer rescaling comes to my mind, but no method works for everything.

scholar.google.com

 

[NFuller]

The strange term r^2/RY = 0

How do you know this is zero? This implies that either $r$ is zero or the denominator goes to infinity.

Is this OK as a method?

It would help to see the original PDE before you attempt separation of variables. The fact that you are having trouble likely means that separation of variables will not work here i.e. the solution cannot be written as the product of two functions with only $r$ and $\theta$ dependence.

 

[SeM]

Hi Nfueller, I don't. I was trying to find a way to work around it, as r can be zero.
The whole PDE is:$$ r^2 \frac{1}{R}\frac{\partial^2 R}{\partial r^2} - (r+2i r^2) \frac{1}{R}\frac{\partial R}{\partial r} - \frac{r^2}{RY} = \frac{1}{Y} \frac{\partial^2 Y}{\partial \theta^2}+2ir \frac{1}{Y}\frac{\partial Y}{\partial \theta}$$