How to find the roots of a transcendental equation that depends on Bessel functions?

Question

$I need to find the roots for m of this equation. Where l=10^{-4}mm and L=10^{-6}.$

I am very new in wolfram mathematica, so i am having trouble in trying to find the roots for this transcendental equation. I am looking for roots of m. Acctually the first non zero value for m. Here J is the Bessel function on first kind and N is the Bessel function of second kind. I have tried some codes but it did not work.

Please type your equation in Mathematica format. Do not post a picture. — yarchik, Jul 18 '20 at 18:58

score 3 · Answer 1 · answered Jul 18 '20 at 20:33

In Mathematica, a Bessel function of the second kind is BesselY

Clear["Global`*"]
l = 10^-4; L = 10^-6;
eqn = -BesselY[2, m*L]BesselJ[1, mL]*
     BesselJ[2, m*(l + L)]BesselY[1, m(l + L)] -
    BesselJ[2, m*L]BesselJ[1, mL]*
     BesselY[2, m*(l + L)]BesselJ[1, m(l + L)] +
    BesselY[2, m*L]BesselY[1, mL]*
     BesselJ[2, m*(l + L)]BesselJ[1, m(l + L)] +
    BesselJ[2, m*L]BesselJ[1, mL]*
     BesselY[2, m*(l + L)]BesselY[1, m(l + L)] == 0;
LogLinearPlot[Evaluate@eqn[[1]], {m, 0, 100000},
 PlotPoints -> 100, MaxRecursion -> 2]

LogLinearPlot[Evaluate@eqn[[1]], {m, 3*^4, 100000}]

The plot provides initial estimates to use with FindRoot

sol = FindRoot[eqn, {m, #}] & /@ {3.5*^4, 5*^4, 7*^4, 10^5}
{{m -> 37949.1}, {m -> 50847.7}, {m -> 69498.8}, {m -> 100805.}}

How to find the roots of a transcendental equation that depends on Bessel functions?

1 Answers1