Message Boards

WOLFRAM COMMUNITY

3695 Views

3 Replies

1 Total Likes

View groups...

Follow this post

Share this post:

GROUPS:

Mathematics Physics

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

Di Chang

Di Chang, University of Calgary

Posted 10 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 = 0.210^-6; n1 = 1.4469; n2 = 1.0; \[Lambda] = 1.310^-6; k = 2 \[Pi]/\[Lambda]; h[\[Beta]_] := Sqrt[n1^2k^2 - \[Beta]^2]; q[\[Beta]_] := Sqrt[\[Beta]^2 - n2^2k^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:*

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

POSTED BY: Di Chang

3 Replies

Sort By:

Bill Simpson

Posted 10 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<beta<10 that 5 digits is inadequate to show the scale with enough precision. What do you want the plot to be? You can add PlotRange -> {-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.

Hint:

Plot[Re[f[\[Beta]]], {\[Beta], -10, 10}]

enter image description here

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<beta<10 that 5 digits is inadequate to show the scale with enough precision.

What do you want the plot to be? You can add PlotRange -> {-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.

POSTED BY: Bill Simpson

Di Chang

Di Chang, University of Calgary

Posted 10 years ago

Thanks, but seems that it also has problem with plotting...the vertical axis is not plotting to the right scale actually :(

POSTED BY: Di Chang

Di Chang

Di Chang, University of Calgary

Posted 10 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 BY: Di Chang

Reply to this discussion

Reply Preview

Attachments

Remove Add a file to this post

Follow this discussion

or Discard

Group Abstract

Feedback