How to find all roots of an equation?

Questions
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1

In[162]:= b = 1;

In[163]:= Df = 0.05;

In[164]:= r0 = 0.02;

In[165]:= rx = 2;

In[166]:= m = 0.05;

In[167]:= jrmax = 5.6*10^(-7);

In[168]:= k = 2.5;

In[169]:= v0 =  1;

In[170]:= S = 1.4*10^(-8);

In[171]:= cs0 = S/m

Out[171]= 2.8*10^-7

In[172]:= t = 0.008

Out[172]= 0.008

In[173]:= R0 = r0/Sqrt[Df*b/m]

Out[173]= 0.02

In[174]:= Rx = rx/Sqrt[Df*b/m]

Out[174]= 2.

In[175]:= Cs0 = cs0/cs0

Out[175]= 1.

In[176]:= T = t/(r0^2/Df)

Out[176]= 1.

In[177]:= Rr =  r0*k/(b*Df)

Out[177]= 1.

In[178]:= Rv =  r0*v0/(b*Df)

Out[178]= 0.4

In[179]:= Jrmax = jrmax/(r0*S)

Out[179]= 2000.

In[180]:= h1 =  Rv/R0

Out[180]= 20.

In[181]:= h2 = Rv/Rx

Out[181]= 0.2

In[182]:= (*characteristic root*)

In[188]:= eq1 = ((h1 + (1 - Rv/2)/R0)*
      BesselY[Rv/2, \[Lambda] R0] + \[Lambda]*
      BesselY[Rv/2 - 1, \[Lambda] R0])*((h2 + (1 - Rv/2)/Rx)*
      BesselJ[Rv/2, \[Lambda]*Rx] +
     \[Lambda]*
      BesselJ[Rv/2 - 1, \[Lambda]*Rx]) - ((h1 + (1 - Rv/2)/R0)* 
      BesselJ[Rv/2, \[Lambda]*R0] +
     \[Lambda]*BesselJ[Rv/2 - 1, \[Lambda] R0])*((h2 + (1 - Rv/2)/Rx)*
      BesselY[Rv/2, \[Lambda]*Rx] +
     \[Lambda]*BesselY[Rv/2 - 1, \[Lambda] Rx])

Out[188]= (\[Lambda] BesselJ[-0.8, 2. \[Lambda]] + 
    0.6 BesselJ[0.2, 2. \[Lambda]]) (\[Lambda] BesselY[-0.8, 
      0.02 \[Lambda]] + 
    60. BesselY[0.2, 0.02 \[Lambda]]) - (\[Lambda] BesselJ[-0.8, 
      0.02 \[Lambda]] + 
    60. BesselJ[0.2, 0.02 \[Lambda]]) (\[Lambda] BesselY[-0.8, 
      2. \[Lambda]] + 0.6 BesselY[0.2, 2. \[Lambda]])

Find all roots of eq1.

2

Hi Jaques

You could do this by iteration using the fact that three successive roots have closely the same distance.

Below the code to find the first 20 positive roots:

r1 = \\[Lambda] /. FindRoot[eq1 == 0, {\\[Lambda], 1}];

r2 = \\[Lambda] /. FindRoot[eq1 == 0, {\\[Lambda], 2}];


roots = Join[{r1, r2}, 
   Table[r = 2 r2 - r1; r1 = r2; 
    r2 = \\[Lambda] /. FindRoot[eq1 == 0, {\\[Lambda], r}], {20}]];

Plot[eq1, {\\[Lambda], 0, 35},  Epilog -> {Red, Point[{#, 0}] & /@ roots}]

Best Regards
Robert

3

(1) Copying and pasting line by line is tedious. Supplying only the input, sans In[…]=, is preferred.

(2) It can be solved if the variable is restricted to a finite range.

NSolve[{eq1 == 0, -10 <= \[Lambda] <= 10}, \[Lambda]]

(* Out[26]= {{\[Lambda] -> -8.76828}, {\[Lambda] -> -7.16348}, \
{\[Lambda] -> -3.96149}, {\[Lambda] -> -2.38046}, {\[Lambda] -> \
-0.901317}, {\[Lambda] -> 0.901317}, {\[Lambda] -> 
   2.38046}, {\[Lambda] -> 3.96149}, {\[Lambda] -> 
   7.16348}, {\[Lambda] -> 8.76828}} *)
4

Hi Daniel,

NSolve[eq1 == 0,  \\[Lambda]]

Did not work, now I am aware that it is possible to specify additional conditions.
Thank you.

5

In[214]:= eq3 = NSolve[{eq1 == 0, 0 < [Lambda] < 100}, [Lambda]]

Out[214]= {{\[Lambda] -> 0.901317}, {\[Lambda] -> 
   2.38046}, {\[Lambda] -> 3.96149}, {\[Lambda] -> 
   7.16348}, {\[Lambda] -> 8.76828}, {\[Lambda] -> 
   10.3734}, {\[Lambda] -> 11.9785}, {\[Lambda] -> 
   13.5832}, {\[Lambda] -> 15.1875}, {\[Lambda] -> 
   16.7915}, {\[Lambda] -> 18.3949}, {\[Lambda] -> 
   19.9979}, {\[Lambda] -> 21.6003}, {\[Lambda] -> 
   23.2024}, {\[Lambda] -> 24.8039}, {\[Lambda] -> 
   26.405}, {\[Lambda] -> 28.0056}, {\[Lambda] -> 
   29.6057}, {\[Lambda] -> 31.2055}, {\[Lambda] -> 
   32.8047}, {\[Lambda] -> 34.4036}, {\[Lambda] -> 
   36.002}, {\[Lambda] -> 37.6001}, {\[Lambda] -> 
   39.1977}, {\[Lambda] -> 40.795}, {\[Lambda] -> 
   42.392}, {\[Lambda] -> 43.9885}, {\[Lambda] -> 
   45.5848}, {\[Lambda] -> 47.1807}, {\[Lambda] -> 
   48.7762}, {\[Lambda] -> 50.3715}, {\[Lambda] -> 
   51.9665}, {\[Lambda] -> 53.5612}, {\[Lambda] -> 
   55.1557}, {\[Lambda] -> 56.7499}, {\[Lambda] -> 
   58.3438}, {\[Lambda] -> 59.9375}, {\[Lambda] -> 
   61.531}, {\[Lambda] -> 63.1242}, {\[Lambda] -> 
   64.7172}, {\[Lambda] -> 66.3101}, {\[Lambda] -> 
   67.9027}, {\[Lambda] -> 69.4951}, {\[Lambda] -> 
   71.0874}, {\[Lambda] -> 72.6795}, {\[Lambda] -> 
   74.2714}, {\[Lambda] -> 75.8632}, {\[Lambda] -> 
   77.4548}, {\[Lambda] -> 79.0463}, {\[Lambda] -> 
   80.6376}, {\[Lambda] -> 82.2288}, {\[Lambda] -> 
   83.8198}, {\[Lambda] -> 85.4108}, {\[Lambda] -> 
   87.0016}, {\[Lambda] -> 88.5923}, {\[Lambda] -> 
   90.1828}, {\[Lambda] -> 91.7733}, {\[Lambda] -> 
   93.3637}, {\[Lambda] -> 94.954}, {\[Lambda] -> 
   96.5441}, {\[Lambda] -> 98.1342}, {\[Lambda] -> 99.7242}}