Question

I am trying to plot roots of a function that is composed of multiple bessel functions being added and multiplied in Matlab. The equation is Jm(omega)*Ik(omega)+Im(omega)*Jk(omega) where Jm is the bessel function of the first kind of order m (besselj). Im is the modified bessel function of the first kind of order m (besseli). For each mode m=o,1,2,…and n=1,2,3… The frequency omega(mn) is the corresponding root of the listed equation. m=0,1,2 n-1,2,3,4. I need to solve the equation for the 12 roots. I am new to Matlab and this is a little out of my league. So far I have this code but I wasn’t sure if I needed the variable omega in the script or not. I have also looked at other people’s questions on the suject but didn’t see any quite like this. The plots I have seen look nothing like mine which tells me I am probably wrong. Thanks for any help.

m=(0:2); k=(1:3); n=(1:4);
Jm=besselj(m,n');
Ik=besseli(k,n');
Jk=besselj(k,n');
Im=besseli(m,n');
g=Jm.*Ik+Im.*Jk
plot(g)

Answer 1

Plotting

besselj and besseli take what you call omega as their second parameter, so to plot your function you should try something like

m=0; k=1; omega=0:0.02:10;
Jm=besselj(m,omega);
Ik=besseli(k,omega);
Jk=besselj(k,omega);
Im=besseli(m,omega);
g=Jm.*Ik+Im.*Jk;
plot(omega,g);
hold all;
plot(omega,0,'k');
axis([min(omega) max(omega) -100 100]);

This shows you that for m=1, k=1 the first zeros are around 3.2, 6.3 and 9.4:

Finding the roots numerically

You could implement Halley’s method for your function g, similar to how the roots of besselj are determined in the MatlabCentral file linked by Cheery.