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# Kindly look for help with solving an eigenvalue equation, thanks :)

Posted 9 years ago
 Hi, I am trying to solve the following eigenvalue equation for \Beta as the propagation constant in my problem: f[\[Beta]_] := BesselJ[0, h[\[Beta]]*a]/h[\[Beta]]/a/ BesselJ[1, h[\[Beta]]*a] + (n1^2 + n2^2)* KD1[q[\[Beta]]*a]/q[\[Beta]]/a/BesselK[1, q[\[Beta]]*a] - 1/h[\[Beta]]^2/a^2 + Sqrt[((n1^2 - n2^2)* KD1[q[\[Beta]]*a]/2/n1^2/q[\[Beta]]/a/ BesselK[1, q[\[Beta]]*a])^2 + \[Beta]^2*(1/q[\[Beta]]^2/a^2 + 1/h[\[Beta]]^2/a^2)^2/n1^2/k^2] be letting f[[Beta]_] ==0. The parameters in the above equation are defined as follows: a = 0.2*10^-6; n1 = 1.4469; n2 = 1.0; \[Lambda] = 1.3*10^-6; k = 2 \[Pi]/\[Lambda]; h[\[Beta]_] := Sqrt[n1^2*k^2 - \[Beta]^2]; q[\[Beta]_] := Sqrt[\[Beta]^2 - n2^2*k^2]; JD1[x_] := BesselJ[0, x] - BesselJ[1, x]/x; KD1[x_] := -(BesselK[0, x] + BesselK[2, x])/2; I tried NSolve and FindRoot but they do not work. Does anyone know what would be the best way to solve this type of problem? For your convenience, I also attached the .nb file with this discussion so you can directly download and have a try if you want :) Thanks a lot! Di Attachments:
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Posted 9 years ago
 Sorry I just saw it. Thank you very much for the above help. I got the problem solved by finding an error in the equation itself so no problem now. Thank you all for the time and help! :)
Posted 9 years ago
 Thanks, but seems that it also has problem with plotting...the vertical axis is not plotting to the right scale actually :(
Posted 9 years ago
 Hint: Plot[Re[f[\[Beta]]], {\[Beta], -10, 10}] Seems to plot without problems showing that there is no f[beta]==0 but rather a minimum that is far from zero.The function is so near constant over -10 {-1, 3} to the plot if you prefer to see it that way.One other possible problem is that you are using floating point numbers with only a single digit of precision in your notebook.If you change all those constants to exact rational numbers then you can see this In[8]:= Table[{N[\[Beta]], N[f[\[Beta]], 20]}, {\[Beta], -5, 5, 1}] Out[8]= { {-5., 2.6546952405717024423 + 2.7166456587854339284 I}, {-4., 2.6546952405704768852 + 2.7166456587851677216 I}, {-3., 2.6546952405695236741 + 2.7166456587849606719 I}, {-2., 2.6546952405688428091 + 2.7166456587848127792 I}, {-1., 2.6546952405684342900 + 2.7166456587847240436 I}, { 0., 2.6546952405682981170 + 2.7166456587846944650 I}, { 1., 2.6546952405684342900 + 2.7166456587847240436 I}, { 2., 2.6546952405688428091 + 2.7166456587848127792 I}, { 3., 2.6546952405695236741 + 2.7166456587849606719 I}, { 4., 2.6546952405704768852 + 2.7166456587851677216 I}, { 5., 2.6546952405717024423 + 2.7166456587854339284 I}} and hopefully all those digits are correct.