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How to solve the Warning: Unable to solve symbolically. Returning a numeric solution using vpasolve. of Bessel function

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I had used Matlab to solve the Bessel equation like this, but it came out that warning Warning: Unable to solve symbolically. Returning a numeric solution using vpasolve. But it should have the result. So may I ask it's there anyway to get the result?
Here's my code:
clc; clear all; close all;
syms a
n2 = 1;
n1 = 1.445;
y = n2/n1;
d = y.^2+1;
a1 = besselj(0,a);
c1= a1==0;
solve(c1,a);

Accepted Answer

Walter Roberson
Walter Roberson on 13 Mar 2020
The BesselJ function is an oscillating function that has an infinite number of zeros. There is no known closed form for the solutions for the equation, so solve() is not able to return a symbolic solution.
However I would also point out that you have used beselj(0,a) . That corresponds to but the equation in the image is which would correspond to besselj(1,a) rather than besselj(0,a)
What can you do? Well, you cannot find a useful symbolic solution for . Numeric solutions exist, and you can approximate how far apart they are, but you cannot give a formula for them except in terms of infinite summations or infinite integrals.
vpasolve() can find numeric roots without complaining about it. Because there are an infinite number of roots, you should pass a starting point into vpasolve(), such as
vpasolve(besselj(1,a), 3)
vpasolve(besselj(1,a), 6)

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