RE: Transcendental equation

[Allen.Windhorn]

Guido,

> -----Original Message-----
> From: address@hidden
> [mailto:address@hidden On
> Behalf Of guido.bonalumi
> 
> Hi everyone!
> I am working on an assignment involving heat transfer and I need to
> find the firsts N roots of a transcendental equation. In particular
> this is the
> equation:
> 
> x*(J1(x))/(J0(x))=Bi
> 
> where Bi is the Biot number (Bi=h*L/k, we can assume that is equal
> to 0.02) and J0 and J1 are Bessel functions of the first kind
> (Jn(n, x)).
> I know how to solve the Bessel equation (besselj(n, x)) but I do
> not have any clue on how I can find the roots I need.
> Can anyone help me?

Well, the J1 and J0 functions have zeros alternating out to infinity,
so the ratio has zeros where J1 is zero and singularities between
the zeros, where J0 is zero, so if the Bi value is constant or at
least doesn't get too high, the roots of your equation will be
pretty close to the zeros of J1(x).  These are spaced at intervals
of about pi and since the zeros of J1 are about the same as the
zeros of (sin(x)-cos(x)) which is to say 5pi/4, 9pi/4, 13pi/4...,
you can just use those as starting points and use fzero (which does
a secant method search or something similar) to find the root near
that point.  You would need to pick a couple points either side of
the guess that have opposite signs.  Or there are probably other
rootfinders available that are simpler for well-behaved functions
(anybody?).

Regards,
Allen