Modifying 'Bessel second-order ordinary differential equation' in matlab library.

Hello. I'm trying to use "besseli" to solve "Bessel second-order ordinary differential equation"

In the page of matlabe function explanation, https://kr.mathworks.com/help/matlab/ref/besseli.html?lang=en#d122e79187

besseli can solve the bessel differential equation like the form below.

"This differential equation, where ν is a real constant, is called the modified Bessel's equation:

"

Assuming that i want to get zero order, so the code for this is

besseli(0,z);

I can understand for this.

What I really want to apply is the equation below actually.

Note that k=√jwμσ. r is radius J is current density. this equation is for solving current density in cylinder type wire.

I'm trying to get J(0) and J(1) which is zero order and first order of bessel function.

so the bessel equation form introduced in matlab function explanation looks simmilar but different.

In this case, how can I adjust the form of matlab besseli's bessesl equation?

Answer 1

[David Goodmanson]

Hi K^2,

Picking kr as the argument, take a look at J(kr) and do the differentiations on

d^2/dr^2 J(kr) + (1/r) d/dr(J(kr).

Take out a factor of k for each derivative, e.g.

d/dr(J(kr)) = k*J'(kr)

where the prime always denotes differentiation by the entire argument.

So for kr = z, then formally

J'(kr) =  d/dz J(z)

Proceeding like this (and not forgetting that 1/r is going to become k/z) you can match the top equation for nu = 0, so the solution really is I0(kr). All the In(kr) work similarly, including I1(kr).

*******

You can also do a more dimensional analysis style derivation. Every term in the desired equation

d^2/dr^2 J + (1/r) d/dr J - k^2 J = 0

has dimension length^-2. Divide through by k^2 and combine k with r

d^2/d(kr)^2 J + (1/kr) d/d(kr) J - J = 0

This equation is dimensionless and matches the correct differential equation for I0, on the condition that you use the dimensionless quantity kr as the argument for I0.

[Kyoungtak Kim]

Thank you for your kind answer.

I still can't understand the equation, but with your help, I understood how the function in matlab works.

Thank you again.