Trying to solve a transcendental equation involving Bessel functions

Question

I've never used Mathematica before and am trying to numerically solve equation (12) from this paper. Ideally I'd be able to find the smallest value of $x_{n\nu}$ for $\exp(-kr\pi)$ close to 1, and close to 0 for some range of $\nu$ (must be larger than 2) and then plot it.

I've tried a few methods I found after googling, but none seem to work.

I tried to use this by writing:

f= (2*BesselJ[a, x] + 
      x*(BesselJ[a + 1, x] + BesselJ[a - 1, x]))*(2*
       BesselY[a, x*exp[-Pi]] + 
      x*exp[-Pi]*(BesselY[a + 1, x*exp[-Pi]] + 
         BesselY[a - 1, x*exp[-Pi]])) - (2*BesselY[a, x] + 
      x*(BesselY[a + 1, x] + BesselY[a - 1, x]))*(2*
       BesselJ[a, x*exp[-Pi]] + 
      x*exp[-Pi]*(BesselY[a + 1, x*exp[-Pi]] + 
         BesselY[a - 1, x*exp[-Pi]]));
 sol[_a] = NSolve[f ==0 && x>0 && x < 10 && a > 2 && a < 12, x, Reals);

and was told that NSolve can't solve it.

I've also tried using

sol[a_] := x /. FindRoot[f, {{x, 0}, {a, 2}}]; 

which throws up an error:

SetDelayed::write : Tag Plus in *My definition of f here*[a_, x_] is Protected.

Answer 1

Well, first in Mathematica one should write Exp instead of exp. Besides that, do you know from some source that your equation has analytic solution? Then it would be helpful, if you write this along with the reason. If you have no such information, I would bet that it has not.

In this case you might want to solve it numerically. It may be done, for instance as follows.

First let us make sure that there are solutions in the interval you gave:

 f[x_, a_] := (2*BesselJ[a, x] +  x*(BesselJ[a + 1, x] + BesselJ[a - 1, x]))*(2*
   BesselY[a, x*Exp[-Pi]] + x*Exp[-Pi]*(BesselY[a + 1, x*Exp[-Pi]] + 
     BesselY[a - 1, x*Exp[-Pi]])) - (2*BesselY[a, x] + x*(BesselY[a + 1, x] + 
  BesselY[a - 1, x]))*(2* BesselJ[a, x*Exp[-Pi]] + x*Exp[-Pi]*(
  BesselY[a + 1, x*Exp[-Pi]] + BesselY[a - 1, x*Exp[-Pi]]));



Manipulate[Plot[f[x, a], {x, 0, 10}, PlotRange -> {-200, 1000}], 
           {a, 2, 12}]

You should see this after evaluating: Thus there are 2 solutions, the larger disappears at a > 3.74, while the smaller at a > 6.89

Let us now solve the equation for the smallest root:

lst1 = Table[{a, FindRoot[f[x, a] == 0, {x, 5.7}][[1, 2]]}, {a, 2, 6.7, 0.1}];

and with respect to the larger one:

lst2 = Table[{a, FindRoot[f[x, a] == 0, {x, 9}][[1, 2]]}, {a, 2, 3.6, 0.1}];

Now let us plot the solutions:

 ListPlot[{lst1, lst2}, AxesLabel -> {Style["a", 16, Italic], Style["x", 16, Italic]}]

This should appear on the screen: Have fun.