Mathematica Mathematica - Find max value NDSolve Plot

[carey1000]

Dear forum members:

After numerically solving a differential equation and plotting the results I would like to determine the single maximum value in the plotted range but do not know how.

The code below works for numerically solving the differential equation and plotting the results.

NDSolve[{x''[t] + x[t] - 0.167 x[t]^3 == 0.005 Cos[t + -0.0000977162*t^2/2], x[0] == 0, x'[0] == 0}, x, {t, 0, 1000}]

Plot[Evaluate[x[t] /. s], {t, 0, 1000},
Frame -> {True, True, False, False}, FrameLabel -> {"t", "x"}, FrameStyle -> Directive[FontSize -> 15], Axes -> False]

Thank you!
Carey

 

[Hepth]

I can't find an easy way to do it, with mapping or anything, but deconstructing the interpolating function can get it:

Max[s[[1]][[1]][[2]][[4]][[3]]]

This is basically the y-values of the array. Just found it by looking at what creates an interpolating function. I'm not sure if its always at this location, but I don't see why it wouldn't be.

 

[FunkyDwarf]

Tabulate the data and use Max@

 

[EnumaElish]

Maxdata = Evaluate[x[1]/.s];

Then, find a way to loop t = 2 thru 1000:
(Newmax = If[Evaluate[x[t]/.s]>Maxdata, Evaluate[x[t]/.s, Maxdata];
Maxdata = Newmax;)

 

[panzer7910]

Dear forum members:

..

Plot[Evaluate[x[t] /. s], {t, 0, 1000},
Frame -> {True, True, False, False}, FrameLabel -> {"t", "x"}, FrameStyle -> Directive[FontSize -> 15], Axes -> False]

Carey

To plot the results, why you need to divide by " .s "??

 

[kaiqi]

To plot the results, why you need to divide by " .s "??

In the help manual of Mathematica, it gives the example：

s = NDSolve[{y'[x] == y[x] Cos[x + y[x]], y[0] == 1}, y, {x, 0, 30}]
Plot[Evaluate[y[x] /. s], {x, 0, 30}, PlotRange -> All]

Also you could find the symbol /. means ReplaceAll (/.) in its help manual.

In[1]:= {x, x^2, y, z} /. x -> a

Out[1]= {a, a^2, y, z}

In[2]:= {x, x^2, y, z} /. x -> {a, b}

Out[2]= {{a, b}, {a^2, b^2}, y, z}

In[3]:= Sin[x] /. Sin -> Cos

Out[3]= Cos[x]

Combine these two, you could find that s represents the result of this numerical solution, and

Evaluate[y[x] /. s]

means using the numerical results s to represent the value of y[x] as y[x] itself does not have a definition in the ordinary form so the plot function cannot plot it unless you use something(here is the numerical results s) to represent it

 

[Bill Simpson]

Will this do?

In[1]:= f = x/.NDSolve[{x''[t]+x[t]-0.167x[t]^3==0.005 Cos[t+ -0.0000977162*t^2/2], x[0]==0, x'[0]==0}, x, {t,0,1000}][[1,1]];
Plot[f[x], {x, 0, 1000}]

Out[2]= ...PlotSnipped...

In[3]:= {*Looks like max is between 775 and 825, turn off warnings and hunt for it*)
Off[FindMaximum::eit];
Sort[Table[FindMaximum[{f[x], 0<x<1000}, {x,y,y-1,y+1}], {y,775,825}], OrderedQ[{First[#1], First[#2]}] &]

Out[4]= {
{-0.699441, {x -> 796.}}, {-0.695645, {x -> 803.}}, {-0.688966, {x -> 789.}},
{-0.681066, {x -> 810.}}, {-0.661475, {x -> 782.}}, {-0.66001, {x -> 817.}},
{-0.637412, {x -> 824.}}, {-0.614744, {x -> 775.}}, {0.094972, {x -> 820.}},
{0.108052, {x -> 778.}}, {0.121419, {x -> 813.}}, {0.144966, {x -> 785.}},
{0.145051, {x -> 806.}}, {0.16006, {x -> 799.}}, {0.161367, {x -> 792.}},
{0.412197, {x -> 795.}}, {0.414114, {x -> 788.}}, {0.423985, {x -> 802.}},
{0.432996, {x -> 781.}}, {0.444894, {x -> 809.}}, {0.469508, {x -> 816.}},
{0.470473, {x -> 774.}}, {0.492212, {x -> 823.}}, {1.02978, {x -> 779.}},
{1.05629, {x -> 821.}}, {1.05931, {x -> 786.}}, {1.0706, {x -> 814.}},
{1.07659, {x -> 793.}}, {1.08044, {x -> 807.}}, {1.08311, {x -> 800.}},
{1.16416, {x -> 780.}}, {1.17012, {x -> 787.}}, {1.17933, {x -> 794.}},
{1.19063, {x -> 801.}}, {1.20206, {x -> 808.}}, {1.21145, {x -> 815.}},
{1.21692, {x -> 822.}}, {1.22094, {x -> 779.646}}, {1.22094, {x -> 779.647}},
{1.23615, {x -> 786.619}}, {1.23615, {x -> 786.621}}, {1.24892, {x -> 793.61}},
{1.24892, {x -> 793.612}}, {1.25869, {x -> 800.615}}, {1.25869, {x -> 800.617}},
{1.26499, {x -> 807.63}}, {1.26499, {x -> 807.632}}, {1.26624, {x -> 821.672}},
{1.26624, {x -> 821.673}}, {1.26753, {x -> 814.65}}, {1.26753, {x -> 814.652}}}