Bessell equation/functions of order 0

[AStaunton]

just trying to solve PDEs by sep of variables;

am looking at laplace heat equation cylindrical problem and the equation that results is bessell function order 0...

in my notes, I have that the general solution to the equation is: R=AJ_0(r)+BY_0(r)

where J_0 and Y_0 apparently denote two independent functions to the equation and I think that both of these functions are 0 order...

My question is, how do J_0 and Y_0 differ from each other...looking on wikipedia page (my go to guide), it only shows a graph of one 0 order bessel, that is a max it 0 and has infinitely many roots..I think..

so for example if the function I just described corresponded to J_0 what would Y_0 look like...would it be pretty much the same but out of phase? I reasoned that this might be the case because this is how cos and sine are related...

Also advice on the key properties of bessell 0 order functions would be appreciated especially those properties that are relevant to finding eigenvalues in PDE sep variable problems...ie, At what values are the roots found and so on...

 

[LCKurtz]

just trying to solve PDEs by sep of variables;

am looking at laplace heat equation cylindrical problem and the equation that results is bessell function order 0...

in my notes, I have that the general solution to the equation is: R=AJ_0(r)+BY_0(r)

where J_0 and Y_0 apparently denote two independent functions to the equation and I think that both of these functions are 0 order...

My question is, how do J_0 and Y_0 differ from each other...looking on wikipedia page (my go to guide), it only shows a graph of one 0 order bessel, that is a max it 0 and has infinitely many roots..I think..

so for example if the function I just described corresponded to J_0 what would Y_0 look like...would it be pretty much the same but out of phase? I reasoned that this might be the case because this is how cos and sine are related...

Also advice on the key properties of bessell 0 order functions would be appreciated especially those properties that are relevant to finding eigenvalues in PDE sep variable problems...ie, At what values are the roots found and so on...

Perhaps you didn't look far enough down the Wikipedia page. If you weren't looking at:

http://en.wikipedia.org/wiki/Bessel_function

look there now. Scroll down for a picture of Y₀. They don't look pretty much the same near x = 0. As far as finding the roots, you would have to find them numerically. In a separation of variables problem perhaps you can get by symbolically, like x₁ is the first positive root of Y₀ or some such. Hard to say more without more particulars.

 

[AStaunton]

thanks..

would I be correct in saying that Y_0 blows up at x=0?

Also, I think J_0 is an even function, is Y_0 odd?

 

[AStaunton]

on an unrelated topic...you give a very good tip about putting the Greek alphabet and other ubiquitous symbols in your signature...can you please tell me how to edit my signature?

 

[LCKurtz]

thanks..

would I be correct in saying that Y_0 blows up at x=0?

Yes.

Also, I think J_0 is an even function, is Y_0 odd?

Yes, J₀ is even as you can see form its series. If my memory serves, if you are considering real functions only then Y₀ is defined for x > 0 only.

on an unrelated topic...you give a very good tip about putting the Greek alphabet and other ubiquitous symbols in your signature...can you please tell me how to edit my signature?

Go to my PF at the top left and you will see a place to edit your .sig file.