Possible orthogonality of Bessel functions

Question

I was solving PDE using Laplace Transform and i was able to obtain solution in the next form

$$ \sum\limits_{n=1}^{\infty}A_nJ_1(B_nr)\exp(C_nt), $$ where $B_n$ such that

$$ CB_nJ_0(B_na) + \frac{B_n^2}{B_n^2+R}J_1(B_na) = 0, $$ where $C,R,a$ are non-negative constants.

This solution implies orthogonality in form of $\int_{0}^{a}w(r)J_1(B_nr)J_1(B_kr)dr$. I was trying to find $w(r)$ from Bessel equations, but i failed.

Maybe i'm wrong and such property doesn't exists? Or if so, can some one give me a hint? Any help would be greatly appreciated!

https://math.stackexchange.com/questions/204297/orthogonality-of-bessel-functions — A rural reader, Oct 14 '23 at 22:37
@Aruralreader This property would work if R was zero, but in my case R > 0. — Yakov Dunaev, Oct 15 '23 at 07:14

score 0 · Accepted Answer · answered Oct 20 '23 at 20:12

0

Functions $J_1(B_nr)$ are not orthogonal for any $R>0$.

answered Oct 20 '23 at 20:12

Yakov Dunaev

66

What if the $B_n$ are related to $a,C,R$ by the equation $$CB_nJ_0(B_na) + \frac{B_n^2}{B_n^2+R}J_1(B_na) = 0$$ where we want orthogonality on $[0,a]$ ? Presumably $w(r)$ involves $a,C,R$. – GEdgar Oct 20 '23 at 20:34
@GEdgar I thought so, but i think it's not possible due to my boundary condition. – Yakov Dunaev Oct 20 '23 at 21:47
My bad, i meant initial condition. I will not go into details, but my initial condition sugests that $$ \sum_{n=1}^{\infty} J_1(B_na) = 0. $$ If we try to calculate $\int_{0}^{a}w(r)J_1^2(B_na)dr$ we will get zero, which doesn't make sense for orthogonal functions. – Yakov Dunaev Oct 20 '23 at 22:07

Possible orthogonality of Bessel functions

1 Answers1