1
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I want to solve the following time dependent coupled differential equations numerically. Please guide me.

{
 {Derivative[1][Subscript[\[Rho], 1, 1]][t] == \[Gamma]/
    2 - \[Gamma] Subscript[\[Rho], 1, 1][t] - 
    I (1/2 \[CapitalOmega]Ra Subscript[\[Rho], 1, 3][t] - 
       1/2 \[CapitalOmega]Ra Subscript[\[Rho], 3, 1][t]) + 
    1/2 \[CapitalGamma] Subscript[\[Rho], 3, 3][t]},
 {Derivative[1][Subscript[\[Rho], 1, 2]][
    t] == -\[Gamma] Subscript[\[Rho], 1, 2][t] - 
    I (\[CapitalDelta]a Subscript[\[Rho], 1, 2][
         t] - \[CapitalDelta]b Subscript[\[Rho], 1, 2][t] + 
       1/2 \[CapitalOmega]Rb Subscript[\[Rho], 1, 3][t] - 
       1/2 \[CapitalOmega]Ra Subscript[\[Rho], 3, 2][t])},
 {Derivative[1][Subscript[\[Rho], 1, 3]][t] == 
   1/2 (-\[Gamma] Subscript[\[Rho], 1, 3][
         t] - (\[Gamma] + \[CapitalGamma]) Subscript[\[Rho], 1, 3][
         t]) - I (1/2 \[CapitalOmega]Ra Subscript[\[Rho], 1, 1][t] + 
       1/2 \[CapitalOmega]Rb Subscript[\[Rho], 1, 2][
         t] + \[CapitalDelta]a Subscript[\[Rho], 1, 3][t] - 
       1/2 \[CapitalOmega]Ra Subscript[\[Rho], 3, 3][t])},
 {Derivative[1][Subscript[\[Rho], 2, 1]][
    t] == -\[Gamma] Subscript[\[Rho], 2, 1][t] - 
    I (-\[CapitalDelta]a Subscript[\[Rho], 2, 1][
         t] + \[CapitalDelta]b Subscript[\[Rho], 2, 1][t] + 
       1/2 \[CapitalOmega]Ra Subscript[\[Rho], 2, 3][t] - 
       1/2 \[CapitalOmega]Rb Subscript[\[Rho], 3, 1][t])},
 {Derivative[1][Subscript[\[Rho], 2, 2]][t] == \[Gamma]/
    2 - \[Gamma] Subscript[\[Rho], 2, 2][t] - 
    I (1/2 \[CapitalOmega]Rb Subscript[\[Rho], 2, 3][t] - 
       1/2 \[CapitalOmega]Rb Subscript[\[Rho], 3, 2][t]) + 
    1/2 \[CapitalGamma] Subscript[\[Rho], 3, 3][t]},
 {Derivative[1][Subscript[\[Rho], 2, 3]][t] == 
   1/2 (-\[Gamma] Subscript[\[Rho], 2, 3][
         t] - (\[Gamma] + \[CapitalGamma]) Subscript[\[Rho], 2, 3][
         t]) - I (1/2 \[CapitalOmega]Ra Subscript[\[Rho], 2, 1][t] + 
       1/2 \[CapitalOmega]Rb Subscript[\[Rho], 2, 2][
         t] + \[CapitalDelta]b Subscript[\[Rho], 2, 3][t] - 
       1/2 \[CapitalOmega]Rb Subscript[\[Rho], 3, 3][t])},
 {Derivative[1][Subscript[\[Rho], 3, 1]][t] == 
   1/2 (-\[Gamma] Subscript[\[Rho], 3, 1][
         t] - (\[Gamma] + \[CapitalGamma]) Subscript[\[Rho], 3, 1][
         t]) - I (-(1/2) \[CapitalOmega]Ra Subscript[\[Rho], 1, 1][
         t] - 1/2 \[CapitalOmega]Rb Subscript[\[Rho], 2, 1][
         t] - \[CapitalDelta]a Subscript[\[Rho], 3, 1][t] + 
       1/2 \[CapitalOmega]Ra Subscript[\[Rho], 3, 3][t])},
 {Derivative[1][Subscript[\[Rho], 3, 2]][t] == 
   1/2 (-\[Gamma] Subscript[\[Rho], 3, 2][
         t] - (\[Gamma] + \[CapitalGamma]) Subscript[\[Rho], 3, 2][
         t]) - I (-(1/2) \[CapitalOmega]Ra Subscript[\[Rho], 1, 2][
         t] - 1/2 \[CapitalOmega]Rb Subscript[\[Rho], 2, 2][
         t] - \[CapitalDelta]b Subscript[\[Rho], 3, 2][t] + 
       1/2 \[CapitalOmega]Rb Subscript[\[Rho], 3, 3][t])},
 {Derivative[1][Subscript[\[Rho], 3, 3]][
    t] == -I (-(1/2) \[CapitalOmega]Ra Subscript[\[Rho], 1, 3][t] - 
       1/2 \[CapitalOmega]Rb Subscript[\[Rho], 2, 3][t] + 
       1/2 \[CapitalOmega]Ra Subscript[\[Rho], 3, 1][t] + 
       1/2 \[CapitalOmega]Rb Subscript[\[Rho], 3, 2][
         t]) - (\[Gamma] + \[CapitalGamma]) Subscript[\[Rho], 3, 3][t]}
}

while the boundary conditions are

{
 {Subscript[\[Rho], 1, 1][0] == 1},
 {Subscript[\[Rho], 1, 2][0] == 0},
 {Subscript[\[Rho], 1, 3][0] == 0},
 {Subscript[\[Rho], 2, 1][0] == 0},
 {Subscript[\[Rho], 2, 2][0] == 0},
 {Subscript[\[Rho], 2, 3][0] == 0},
 {Subscript[\[Rho], 3, 1][0] == 0},
 {Subscript[\[Rho], 3, 2][0] == 0},
 {Subscript[\[Rho], 3, 3][0] == 0}
}

using the numerical values of

{
    \[Gamma] = 0.1;
    
    \[CapitalGamma] = 1;
    
    \[CapitalOmega]Ra = 1;
    
    \[CapitalOmega]Rb = 8;
 
    \[CapitalDelta]a = 1;
    \[CapitalDelta]b = 0.5;
}

These equations are the elements of the Von Neuman equation for three level atomic system. I want to solve the density matrix elements time dependently to see the time dependent behavior of electromagnetically induced transparency.

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2
  • $\begingroup$ Your ode is complex and the boundary conditions are real valued. Are you looking for real solution? $\endgroup$ Aug 26, 2022 at 8:33
  • $\begingroup$ @Ulrich Neumann Thanks for your response. I am looking for complex solution. Later on i will use the real and imaginary parts to plot the susceptibility of the system. Regards $\endgroup$ Aug 26, 2022 at 9:15

1 Answer 1

4
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Edit: Added code to generate 3D plots of selected solution as function of the parameters below

First initialize the constants then flatten the array of equations and initial conditions:

\[Gamma] = 0.1;
\[CapitalGamma] = 1;
\[CapitalOmega]Ra = 1;
\[CapitalOmega]Rb = 8;
\[CapitalDelta]a = 1;
\[CapitalDelta]b = 0.5;

theEqns = {{Derivative[1][Subscript[\[Rho], 1, 1]][
      t] == \[Gamma] (-Subscript[\[Rho], 1, 1][t]) + 
      1/2 \[CapitalGamma] Subscript[\[Rho], 3, 3][t] - 
      I (1/2 \[CapitalOmega]Ra Subscript[\[Rho], 1, 3][t] - 
         1/2 \[CapitalOmega]Ra Subscript[\[Rho], 3, 1][t]) + \[Gamma]/
      2}, {Derivative[1][Subscript[\[Rho], 1, 2]][
      t] == -(\[Gamma] Subscript[\[Rho], 1, 2][t]) - 
      I (\[CapitalDelta]a Subscript[\[Rho], 1, 2][
           t] - \[CapitalDelta]b Subscript[\[Rho], 1, 2][t] - 
         1/2 \[CapitalOmega]Ra Subscript[\[Rho], 3, 2][t] + 
         1/2 \[CapitalOmega]Rb Subscript[\[Rho], 1, 3][
           t])}, {Derivative[1][Subscript[\[Rho], 1, 3]][t] == 
     1/2 (-((\[Gamma] + \[CapitalGamma]) Subscript[\[Rho], 1, 3][
             t]) - \[Gamma] Subscript[\[Rho], 1, 3][t]) - 
      I (\[CapitalDelta]a Subscript[\[Rho], 1, 3][t] + 
         1/2 \[CapitalOmega]Ra Subscript[\[Rho], 1, 1][t] - 
         1/2 \[CapitalOmega]Ra Subscript[\[Rho], 3, 3][t] + 
         1/2 \[CapitalOmega]Rb Subscript[\[Rho], 1, 2][
           t])}, {Derivative[1][Subscript[\[Rho], 2, 1]][
      t] == -(\[Gamma] Subscript[\[Rho], 2, 1][t]) - 
      I (-(\[CapitalDelta]a Subscript[\[Rho], 2, 1][
             t]) + \[CapitalDelta]b Subscript[\[Rho], 2, 1][t] + 
         1/2 \[CapitalOmega]Ra Subscript[\[Rho], 2, 3][t] - 
         1/2 \[CapitalOmega]Rb Subscript[\[Rho], 3, 1][
           t])}, {Derivative[1][Subscript[\[Rho], 2, 2]][
      t] == \[Gamma] (-Subscript[\[Rho], 2, 2][t]) + 
      1/2 \[CapitalGamma] Subscript[\[Rho], 3, 3][t] - 
      I (1/2 \[CapitalOmega]Rb Subscript[\[Rho], 2, 3][t] - 
         1/2 \[CapitalOmega]Rb Subscript[\[Rho], 3, 2][t]) + \[Gamma]/
      2}, {Derivative[1][Subscript[\[Rho], 2, 3]][t] == 
     1/2 (-((\[Gamma] + \[CapitalGamma]) Subscript[\[Rho], 2, 3][
             t]) - \[Gamma] Subscript[\[Rho], 2, 3][t]) - 
      I (\[CapitalDelta]b Subscript[\[Rho], 2, 3][t] + 
         1/2 \[CapitalOmega]Ra Subscript[\[Rho], 2, 1][t] + 
         1/2 \[CapitalOmega]Rb Subscript[\[Rho], 2, 2][t] - 
         1/
          2 \[CapitalOmega]Rb Subscript[\[Rho], 3, 3][
           t])}, {Derivative[1][Subscript[\[Rho], 3, 1]][t] == 
     1/2 (-((\[Gamma] + \[CapitalGamma]) Subscript[\[Rho], 3, 1][
             t]) - \[Gamma] Subscript[\[Rho], 3, 1][t]) - 
      I (-(\[CapitalDelta]a Subscript[\[Rho], 3, 1][t]) - 
         1/2 \[CapitalOmega]Ra Subscript[\[Rho], 1, 1][t] + 
         1/2 \[CapitalOmega]Ra Subscript[\[Rho], 3, 3][t] - 
         1/2 \[CapitalOmega]Rb Subscript[\[Rho], 2, 1][
           t])}, {Derivative[1][Subscript[\[Rho], 3, 2]][t] == 
     1/2 (-((\[Gamma] + \[CapitalGamma]) Subscript[\[Rho], 3, 2][
             t]) - \[Gamma] Subscript[\[Rho], 3, 2][t]) - 
      I (-(\[CapitalDelta]b Subscript[\[Rho], 3, 2][t]) - 
         1/2 \[CapitalOmega]Ra Subscript[\[Rho], 1, 2][t] - 
         1/2 \[CapitalOmega]Rb Subscript[\[Rho], 2, 2][t] + 
         1/2 \[CapitalOmega]Rb Subscript[\[Rho], 3, 3][
           t])}, {Derivative[1][Subscript[\[Rho], 3, 3]][
      t] == -((\[Gamma] + \[CapitalGamma]) Subscript[\[Rho], 3, 3][
          t]) - I (-(1/
           2) (\[CapitalOmega]Ra Subscript[\[Rho], 1, 3][t]) + 
         1/2 \[CapitalOmega]Ra Subscript[\[Rho], 3, 1][t] - 
         1/2 \[CapitalOmega]Rb Subscript[\[Rho], 2, 3][t] + 
         1/2 \[CapitalOmega]Rb Subscript[\[Rho], 3, 2][t])}} // Flatten

theInit = {{Subscript[\[Rho], 1, 1][0] == 
     1}, {Subscript[\[Rho], 1, 2][0] == 
     0}, {Subscript[\[Rho], 1, 3][0] == 
     0}, {Subscript[\[Rho], 2, 1][0] == 
     0}, {Subscript[\[Rho], 2, 2][0] == 
     0}, {Subscript[\[Rho], 2, 3][0] == 
     0}, {Subscript[\[Rho], 3, 1][0] == 
     0}, {Subscript[\[Rho], 3, 2][0] == 
     0}, {Subscript[\[Rho], 3, 3][0] == 0}} // Flatten

Then use NDSolveValue to solve the system. Variable solArray contains the nine numerical solutions. First element in solArray, $\texttt{solArray[[1]][t]}$ is the first function $\rho_{1,1}$ and so forth. I use {t,0,1} as an example plotting the real components of the solutions in one plot then a single plot : For example, if you want the value of say $\rho_{2,1}(1/2)$ then use solArray[4][1/2].

solArray = 
  NDSolveValue[
   Join[theEqns, theInit], {Subscript[\[Rho], 1, 1], Subscript[\[Rho],
     1, 2], Subscript[\[Rho], 1, 3], Subscript[\[Rho], 2, 1], 
    Subscript[\[Rho], 2, 2], Subscript[\[Rho], 2, 3], 
    Subscript[\[Rho], 3, 1], Subscript[\[Rho], 3, 2], 
    Subscript[\[Rho], 3, 3]}, {t, 0, 1}];
Length@solArray
Plot[Re[#[t]] & /@ solArray, {t, 0, 1}, PlotRange -> All]
Plot[Re[solArray[[1]][t]], {t, 0, 1}]

enter image description here

Also, if the array of functions is a bit awkward to deal with, you can individually specify each function as the following and then use them separately (have to use different names on the left though):

{p11, p12, p13, p21, p22, p23, p31, p32, p33} = 
  NDSolveValue[
   Join[theEqns, theInit], {Subscript[\[Rho], 1, 1], Subscript[\[Rho],
     1, 2], Subscript[\[Rho], 1, 3], Subscript[\[Rho], 2, 1], 
    Subscript[\[Rho], 2, 2], Subscript[\[Rho], 2, 3], 
    Subscript[\[Rho], 3, 1], Subscript[\[Rho], 3, 2], 
    Subscript[\[Rho], 3, 3]}, {t, 0, 1}];
Length@solArray
Plot[Re@p11[t], {t, 0, 1}]

Edit: Added code to write DE as a function of parameters

First write the system as a function of parameters. Now the 9 equations are functions of the parameters of $\texttt{theEqnsF}$:

ClearAll["Global`*"]
theEqns2[\[Gamma]_, \[CapitalGamma]_, \[CapitalOmega]Ra_, \
\[CapitalOmega]Rb_, \[CapitalDelta]a_, \[CapitalDelta]b_] := \
{{Derivative[1][Subscript[\[Rho], 1, 1]][
      t] == \[Gamma] (-Subscript[\[Rho], 1, 1][t]) + 
      1/2 \[CapitalGamma] Subscript[\[Rho], 3, 3][t] - 
      I (1/2 \[CapitalOmega]Ra Subscript[\[Rho], 1, 3][t] - 
         1/2 \[CapitalOmega]Ra Subscript[\[Rho], 3, 1][t]) + \[Gamma]/
       2}, {Derivative[1][Subscript[\[Rho], 1, 2]][
      t] == -(\[Gamma] Subscript[\[Rho], 1, 2][t]) - 
      I (\[CapitalDelta]a Subscript[\[Rho], 1, 2][
           t] - \[CapitalDelta]b Subscript[\[Rho], 1, 2][t] - 
         1/2 \[CapitalOmega]Ra Subscript[\[Rho], 3, 2][t] + 
         1/2 \[CapitalOmega]Rb Subscript[\[Rho], 1, 3][
           t])}, {Derivative[1][Subscript[\[Rho], 1, 3]][t] == 
     1/2 (-((\[Gamma] + \[CapitalGamma]) Subscript[\[Rho], 1, 3][
             t]) - \[Gamma] Subscript[\[Rho], 1, 3][t]) - 
      I (\[CapitalDelta]a Subscript[\[Rho], 1, 3][t] + 
         1/2 \[CapitalOmega]Ra Subscript[\[Rho], 1, 1][t] - 
         1/2 \[CapitalOmega]Ra Subscript[\[Rho], 3, 3][t] + 
         1/2 \[CapitalOmega]Rb Subscript[\[Rho], 1, 2][
           t])}, {Derivative[1][Subscript[\[Rho], 2, 1]][
      t] == -(\[Gamma] Subscript[\[Rho], 2, 1][t]) - 
      I (-(\[CapitalDelta]a Subscript[\[Rho], 2, 1][
             t]) + \[CapitalDelta]b Subscript[\[Rho], 2, 1][t] + 
         1/2 \[CapitalOmega]Ra Subscript[\[Rho], 2, 3][t] - 
         1/2 \[CapitalOmega]Rb Subscript[\[Rho], 3, 1][
           t])}, {Derivative[1][Subscript[\[Rho], 2, 2]][
      t] == \[Gamma] (-Subscript[\[Rho], 2, 2][t]) + 
      1/2 \[CapitalGamma] Subscript[\[Rho], 3, 3][t] - 
      I (1/2 \[CapitalOmega]Rb Subscript[\[Rho], 2, 3][t] - 
         1/2 \[CapitalOmega]Rb Subscript[\[Rho], 3, 2][t]) + \[Gamma]/
       2}, {Derivative[1][Subscript[\[Rho], 2, 3]][t] == 
     1/2 (-((\[Gamma] + \[CapitalGamma]) Subscript[\[Rho], 2, 3][
             t]) - \[Gamma] Subscript[\[Rho], 2, 3][t]) - 
      I (\[CapitalDelta]b Subscript[\[Rho], 2, 3][t] + 
         1/2 \[CapitalOmega]Ra Subscript[\[Rho], 2, 1][t] + 
         1/2 \[CapitalOmega]Rb Subscript[\[Rho], 2, 2][t] - 
         1/2 \[CapitalOmega]Rb Subscript[\[Rho], 3, 3][
           t])}, {Derivative[1][Subscript[\[Rho], 3, 1]][t] == 
     1/2 (-((\[Gamma] + \[CapitalGamma]) Subscript[\[Rho], 3, 1][
             t]) - \[Gamma] Subscript[\[Rho], 3, 1][t]) - 
      I (-(\[CapitalDelta]a Subscript[\[Rho], 3, 1][t]) - 
         1/2 \[CapitalOmega]Ra Subscript[\[Rho], 1, 1][t] + 
         1/2 \[CapitalOmega]Ra Subscript[\[Rho], 3, 3][t] - 
         1/2 \[CapitalOmega]Rb Subscript[\[Rho], 2, 1][
           t])}, {Derivative[1][Subscript[\[Rho], 3, 2]][t] == 
     1/2 (-((\[Gamma] + \[CapitalGamma]) Subscript[\[Rho], 3, 2][
             t]) - \[Gamma] Subscript[\[Rho], 3, 2][t]) - 
      I (-(\[CapitalDelta]b Subscript[\[Rho], 3, 2][t]) - 
         1/2 \[CapitalOmega]Ra Subscript[\[Rho], 1, 2][t] - 
         1/2 \[CapitalOmega]Rb Subscript[\[Rho], 2, 2][t] + 
         1/2 \[CapitalOmega]Rb Subscript[\[Rho], 3, 3][
           t])}, {Derivative[1][Subscript[\[Rho], 3, 3]][
      t] == -((\[Gamma] + \[CapitalGamma]) Subscript[\[Rho], 3, 3][
          t]) - I (-(1/2) (\[CapitalOmega]Ra Subscript[\[Rho], 1, 3][
             t]) + 1/2 \[CapitalOmega]Ra Subscript[\[Rho], 3, 1][t] - 
         1/2 \[CapitalOmega]Rb Subscript[\[Rho], 2, 3][t] + 
         1/2 \[CapitalOmega]Rb Subscript[\[Rho], 3, 2][t])}} // Flatten

theInit = {{Subscript[\[Rho], 1, 1][0] == 
     1}, {Subscript[\[Rho], 1, 2][0] == 
     0}, {Subscript[\[Rho], 1, 3][0] == 
     0}, {Subscript[\[Rho], 2, 1][0] == 
     0}, {Subscript[\[Rho], 2, 2][0] == 
     0}, {Subscript[\[Rho], 2, 3][0] == 
     0}, {Subscript[\[Rho], 3, 1][0] == 
     0}, {Subscript[\[Rho], 3, 2][0] == 
     0}, {Subscript[\[Rho], 3, 3][0] == 0}} // Flatten

Can now supply the system function $\texttt{theEqnsF}$ to NDSolveValue where now must supply values for the parameters:

   (* theEqns[\[Gamma],\[CapitalGamma],\[CapitalOmega]Ra,\[CapitalOmega]\
Rb,\[CapitalDelta]a,\[CapitalDelta]b] *)
{p11, p12, p13, p21, p22, p23, p31, p32, p33} = 
  NDSolveValue[
   Join[theEqnsF[0.1, 1, 1, 8, 1, 0.5], 
    theInit], {Subscript[\[Rho], 1, 1], Subscript[\[Rho], 1, 2], 
    Subscript[\[Rho], 1, 3], Subscript[\[Rho], 2, 1], 
    Subscript[\[Rho], 2, 2], Subscript[\[Rho], 2, 3], 
    Subscript[\[Rho], 3, 1], Subscript[\[Rho], 3, 2], 
    Subscript[\[Rho], 3, 3]}, {t, 0, 1}];
Plot[Re@p33[t], {t, 0, 1}]

enter image description here

Now generate a series of plots for $\texttt{p33}$ when $\Delta b$ varies from 0.1 to 5:

plotTable = Table[
   (* theEqns[\[Gamma],\[CapitalGamma],\[CapitalOmega]Ra,\
\[CapitalOmega]Rb,\[CapitalDelta]a,\[CapitalDelta]b] *)
   {p11, p12, p13, p21, p22, p23, p31, p32, p33} = 
    NDSolveValue[
     Join[theEqns[0.1, 1, 1, 8, 1, currentB], 
      theInit], {Subscript[\[Rho], 1, 1], Subscript[\[Rho], 1, 2], 
      Subscript[\[Rho], 1, 3], Subscript[\[Rho], 2, 1], 
      Subscript[\[Rho], 2, 2], Subscript[\[Rho], 2, 3], 
      Subscript[\[Rho], 3, 1], Subscript[\[Rho], 3, 2], 
      Subscript[\[Rho], 3, 3]}, {t, 0, 1}];
   ParametricPlot3D[{t, currentB, Re@p33[t]}, {t, 0, 1}],
   {currentB, 0.1, 5, 0.1}
   ];
selectedFunction = "p33";
Show[plotTable, BoxRatios -> {1, 1, 1}, PlotRange -> All, 
 AxesLabel -> {Style["t", 14], , Style["\[CapitalDelta]b", 14], 
   Style[selectedFunction, 14]}]

enter image description here

Finally generate a table of values for $\texttt{p33}$ in the form needed for the function ListPlot3D and generate a smooth function of $\texttt{p33}$ over the indicated range:

plotTable = Table[
   (* theEqns[\[Gamma],\[CapitalGamma],\[CapitalOmega]Ra,\
\[CapitalOmega]Rb,\[CapitalDelta]a,\[CapitalDelta]b] *)
   {p11, p12, p13, p21, p22, p23, p31, p32, p33} = 
    NDSolveValue[
     Join[theEqns[0.1, 1, 1, 8, 1, currentB], 
      theInit], {Subscript[\[Rho], 1, 1], Subscript[\[Rho], 1, 2], 
      Subscript[\[Rho], 1, 3], Subscript[\[Rho], 2, 1], 
      Subscript[\[Rho], 2, 2], Subscript[\[Rho], 2, 3], 
      Subscript[\[Rho], 3, 1], Subscript[\[Rho], 3, 2], 
      Subscript[\[Rho], 3, 3]}, {t, 0, 1}];
           if = p33;
   Table[{t, currentB, if[t]} // N, {t, 0, 1, 1/50}],
   {currentB, 0.1, 5, 0.1}
   ];

ListPlot3D[Flatten[plotTable, 1], BoxRatios -> {1, 1, 1}, 
 AxesLabel -> {Style["t", 14], , Style["\[CapitalDelta]b", 14], 
   Style[selectedFunction, 14]}]

enter image description here

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3
  • $\begingroup$ Thank you for your reply and help. can we use different value of [CapitalDelta]b to plot the solution against time and [CapitalDelta]b $\endgroup$ Aug 26, 2022 at 17:46
  • $\begingroup$ @Assad Hafiz: added some code to generate smooth funtion of a selected solution as a function of the parameters. You will need to review the code carefully and adjust it to your needs. $\endgroup$
    – josh
    Aug 26, 2022 at 21:21
  • $\begingroup$ Again thank you for your reply and help. Can we use finite difference method or runge kutta method to solve these equations. $\endgroup$ Aug 27, 2022 at 5:09

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