Gfrktyl

I'm trying to plot the solution of a set of differential equations and see how the solution changes when the value of a certain variable de is changed. The equations have solutions to the values of de which I have gotten individually using NDSolve but I am unable to replicate the result for all required values of de in one single program. I code I have written is:

om = 1;

k = 1;

L = 0.001;

P = 1.3;

eqns = {

a'[t] == I*((de/om)*a[t] - (b[t] + Conjugate[b[t]])*a[t] 

-1/2) - (k/(2*om))*a[t], 

b'[t] == -I*((P*(Abs[a[t]])^2)/2 + b[t]) - (L/(2*om))*b[t], 

b[0] == 0, a[0] == 0};

s = ParametricNDSolve[eqns, {a[t], b[t]}, {t, 0, 100}, de, 

MaxSteps -> [Infinity]]

x[de] = b[de] + Conjugate[b[de]]

Manipulate[Plot[Evaluate[x[de][t] ], {t, 0, 99}, PlotRange 

-> All], {de, 0.1, 1}]

I have tried everything that I knew in my limited knowledge of Mathematica but I couldn't get a solution. I hope someone can help me with this.

Thank you very much for your help!

Clear["Global`*"]



om = 1;

k = 1;

L = 1/1000;

P = 13/10;



eqns = {a'[t] == 

    I*((de/om)*a[t] - (b[t] + Conjugate[b[t]])*a[t] - 1/2) - 

        (k/(2*om))*a[t], 

   b'[t] == -I*((P*(Abs[a[t]])^2)/2 + b[t]) - (L/(2*om))*b[t], 

   b[0] == 0, a[0] == 0};



s = ParametricNDSolve[eqns, {a, b}, {t, 0, 100}, de, MaxSteps -> ∞];



x[de_?NumericQ][t_?NumericQ] = (b[de][t] + Conjugate[b[de][t]]) /. s;



Manipulate[

 Plot[x[de][t], {t, 0, 99}, PlotRange -> {-4, 4}], {de, 0.1, 1, 

  Appearance -> "Labeled"}]

om = 1;

k = 1;

L = 0.001;

P = 1.3;

eqns = {a'[t] == 

    I*((de/om)*a[t] - (b[t] + Conjugate[b[t]])*a[t] - 

        1/2) - (k/(2*om))*a[t], 

   b'[t] == -I*((P*(Abs[a[t]])^2)/2 + b[t]) - (L/(2*om))*b[t], 

   b[0] == 0, a[0] == 0};

s = ParametricNDSolve[eqns, {a[t], b[t]}, {t, 0, 100}, de, 

   MaxSteps -> [Infinity]];

x = b[t]/. s;

Manipulate[

 Plot[x[de]+Conjugate[x[de]], {t, 0, 99}, PlotRange -> All], {de, 

  0.1, 1}]

When solving equations, Mathematica always returns solutions as substitution rules, then you have to use the /. operator to get a working function.

Also, there is no need to write [t] when calling x[de] in the Plot command, as x[*some value*] returns PInterpolatingFunction[{{0., 100.}}, <>][t], which already has the [t] argument.

Here is another variation that is somewhat more succinct than the other solutions.

m = 1;

k = 1;

L = 0.001;

P = 1.3;



pF = ParametricNDSolveValue[eqns, {a, b}, {t, 0, 100}, de];



Manipulate[

  With[{bF = pF[de][[2]]}, Plot[2 Re[bF[t]], {t, 0, 99}, PlotRange -> 4.1]],

  {de, .1, 1., .1, Appearance -> "Labeled"}]

Notes

• 
  

  Reduce[z + Conjugate[z] == 2 Re[z], z]

  
  
  

  
  

  True

  
  

  
  



• Specifying 4.1 for the plot range, fixes the y-axis scale and better demonstrates the growth of the curve as the parameter de varies.