Selecting the specific values from a table and saving in a table:

Question

I have the following code:

Clear["Global`*"]

K1 = 12;
K3 = 195/10;
eo = 885/100;
ea = 145/10;
L = 1;

ele[E2_] = -(1/(2 K3)) L (E2^2 ea eo Sin[2 g[z]] - (K1 - K3) Sin[
       2 g[z]] g'[z]^2 + (K1 + K3 + (K1 - K3) Cos[2 g[z]]) g''[z]) // 
  Simplify

pnd = ParametricNDSolveValue[{ele[E2] == 0, g[0] == 0, g[L] == 0}, 
  g, {z, 0, L}, {E2, gp0}, 
  Method -> 
   "BoundaryValues" -> {"Shooting", 
     "StartingInitialConditions" -> {g[0] == 0, g'[0] == gp0}}]

Plot[Evaluate@Table[pnd[E2, gp0][z], {gp0, 0, 15, 0.05}, {E2, 0, 2}],
 {z, 0, L},
 PlotRange -> All]

the output is:

I want to have only the plots with circles (I edit the output and put the circles just to indicate what I want) such as for values=0 or values>0 for {z=0,1} and store it in another table.

I tried to run a if statement on the

aa=Evaluate[Table[pnd[E2, gp0][z], {gp0, 0, 15, 0.05}, {E2, 0, 2}],
     {z, 0, L}]

such that for aa >=0 from {z=0,L} it can store in an another table.

I want the data series, in a table, for only those curves which lie entirely above the x-axis

Answer 1

We may calculate a small number of points for each function and then determine to which group it belongs.

First, store all your function in a table like:

funs=Flatten[Table[pnd[E2, gp0][z], {gp0, 0, 15, 0.05}, {E2, 0, 2}]];

To determine which functions result in zero, we must define an interval because machine numbers are not exact. I arbitrarily choose eps=10^-30. Now we may select the functions that are equal to zero and those above zeros:

eps = 10^-30.;
res = Reap[
   Do[mima = MinMax[Table[funs[[i]], {z, 0, 1, 0.2}]]; 
    Which[-eps < mima[[1]] < eps, Sow[i, x1],
     -eps < mima[[1]] && eps < mima[[2]], Sow[i, x2]
     ]
    , {i, Length[funs]}]
   ][[2]]

"res" contains now two lists, the first with indices of functions that are zero, the second from functions above zero. Here is the result:

Addendum

To get only the index of the first function in each group, we may proceed as:

We first choose a x-value where all the functions differ, e.g.: 0.2. Then we get all the function values there and delete duplicates. Then we get the position of the first occurrence of each value.

d02 = funs /. z -> 0.2;
vals = DeleteDuplicates[Select[d02, NonNegative], 
   Abs[#1 - #2] < 10^-10 &];
Position[d02, #, 1, 1] & /@ vals // Flatten

(* {1, 38, 75, 339} *)

Answer 2

Here's a way to investigate the solutions. In ParametricNDSolveValue, the parameters can be specified in the function description in the same order as the parameters supplied to ParametricNDSolveValue. So the solutions are pnd[E2,gp0][z]. You can start by just viewing the solutions one by one as E2 and gp0 are varied in a Manipulate construct and just pick out the ones you want manually. Next version you can check the values of the functions and pick them out programatically. Below is the first step:

gpTable = Table[gVal, {gVal, 0, 15, 1/4}];
Manipulate[
 Plot[pnd[E2, gpTable[[gIndex]]][z], {z, 0, L}, 
  PlotRange -> {{0, 1}, {-2, 2}}, PlotStyle -> Red],
 {{gIndex, 0}, 0, Length@gpTable, 1}, {{E2, 0}, 0, 2, 1}
 ]