2
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H[t_] = {{0, 1}, {1, t}};
tmax=5;
fun[X_?ArrayQ, t_] := -I (H[t].X - X.H[t]);

sol = NDSolve[{σ'[t] == fun[σ[t], t], σ[0] == 
Table[If[i == 1 \[And] j == 1, 1, 0], {i, 1, 2}, {j, 1, 
  2}]}, σ[t], {t, 0, tmax}]

Flatten[Evaluate[σ[t] /. sol /. {t -> tmax/100}], 1] // MatrixForm

 Plot[Table[Flatten[Evaluate[σ[t] /. sol], 1][[i, i]], {i, 1, 2}], {t, 0,tmax}, PlotRange -> All]

Table plot

How to change the color of two lines in plot... and why 

Flatten[Evaluate[σ[t] /. sol], 1][[1, 1]]

gives 

t

while 

 Flatten[Evaluate[σ[t] /. sol], 1][[2, 2]]

gives

 {InterpolatingFuction[{{0.,5.}},<>}[t]}[[2,2]]
| improve this question | |
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  • $\begingroup$ Why would you write H[t_] = {{0, 1}, {1, 1}}; ? what is t doing here? $\endgroup$ – Nasser Jan 4 '14 at 0:18
  • $\begingroup$ I can change H[t_] and consider that it is time dependent. You can take it like H[t_] = {{0, 1}, {1, t}}; $\endgroup$ – santosh Jan 4 '14 at 0:21
2
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Using the solutions you have derived:

y[x_] := Diagonal[\[Sigma][t] /. sol /. t -> x];
u[x_] := Chop[y[x][[1]]]
v[x_] := Chop[y[x][[2]]]

Then plotting:

Plot[{u[t], v[t]}, {t, 0, 5}, 
 PlotStyle -> {{Thick, Blue}, {Thick, Red}}]

gives:

enter image description here

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  • $\begingroup$ Thanks for your answer. For my First question...if number of evaluated functions are very large means {i,1,n}, where n considers as 20 then its very inconvenient to write in your way... $\endgroup$ – santosh Jan 5 '14 at 20:14
  • $\begingroup$ Also consider H[t] is 20 x 20 matrix... $\endgroup$ – santosh Jan 5 '14 at 23:17
  • $\begingroup$ @santosh Sorry for too narrow focus. I agree for larger systems this would be very invonvenient. In that case I would tabulete the results and use ListPlot. You could still use Diagonal to extract diagonal elements of matrix but that is up to you. $\endgroup$ – ubpdqn Jan 6 '14 at 2:33
  • $\begingroup$ Is it possible to only evaluate Diagonal element inside the NDsolve to make it faster? something like...sol = NDSolve[{σ'[t] == fun[σ[t], t], σ[0] == Table[If[i == 1 [And] j == 1, 1, 0], {i, 1, 2}, {j, 1, 2}]}, Diagonal[σ[t]], {t, 0, tmax},DependentVariables-> σ[t]] $\endgroup$ – santosh Jan 6 '14 at 23:29
3
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You need to evaluate the data series separately. E.g.

Plot[{
  Flatten[σ[t] /. sol, 1][[1, 1]],
  Flatten[σ[t] /. sol, 1][[2, 2]]},
 {t, 0, tmax}, PlotRange -> All]

enter image description here

See here for explanation. (Your data is in an unevaluated form within Plot.)

Addendum

For the second part of your question, you have this :-

Flatten[Evaluate[σ[t] /. sol], 1][[1, 1]]

t

Presumably you want the curve data: Evaluate is unnecessary, but you need values for t, i.e.

ListPlot[Table[
  Flatten[σ[t] /. sol, 1][[1, 1]],
  {t, 0, tmax, 0.1}], PlotRange -> All]

enter image description here

Further to OP's comment

While using Plot would be ideal for automatic x-axis (t) sample intervals - i.e. more dense where y changes more rapidly - a suitable solution using the original function is to use a sample interval of 0.01 in Table and use ListPlot or ListLinePlot:-

ListLinePlot[Table[
  Flatten[σ[t] /. sol, 1][[i, i]],
  {i, 1, 2}, {t, 0, tmax, 0.01}],
 PlotRange -> All, PlotStyle -> {Red, Green}]

enter image description here

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  • $\begingroup$ Thanks for your answer. For my First question...if number of evaluated functions are very large means {i,1,n}, where n considers as 20 then its very inconvenient to write in your way... $\endgroup$ – santosh Jan 5 '14 at 20:14
  • $\begingroup$ Also consider H[t] is 20 x 20 matrix... $\endgroup$ – santosh Jan 5 '14 at 23:18
  • $\begingroup$ That should work ok with Table and ListLinePlot. $\endgroup$ – Chris Degnen Jan 5 '14 at 23:46

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