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enter image description here

I'm trying to solve the above differential equation using the shooting method. When i set Method -> {"Shooting", "StartingInitialConditions" -> {z[0] == 0, z'[0] == 1} i get a solution, but it is equal to zero. I need a solution that is different of zero. When i increase the value of z'[0], say to z'[0]=1000 Mathematica stay running for a long time and i get no solution. How can i solve this?

Here is the code:

n = 3.;
xf = 10000.;
s = NDSolve[
   {
    -z''[t] == (1 + (n - 2) t)^(1/(2 - n) (2 (n - 1) + 0.001)) z[
        t]^2 (Sin[z[t]+2])
    , z[0] == 0, z'[xf] == 0}, z, {t, 0, xf},
   Method -> {"Shooting", 
     "StartingInitialConditions" -> {z[0] == 0, z'[0] == 1}
     }
   ] // Chop
Plot[Evaluate[z[t] /. s], {t, 0., 10}, PlotRange -> All]
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1 Answer 1

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This problem appears to satisfy the boundary condition z'[t] == 0 at large t only for z[t] == 0, as can be seen from.

s = ParametricNDSolve[{-z''[t] == (1 + (n - 2) t)^(1/(2 - n) (2 (n - 1) + 10^-3))
    z[t]^2 (Sin[z[t] + 2]), z[0] == 0, z'[0] == sl}, {z, z'}, {t, 0, xf}, {sl}]
Plot[Table[Evaluate[z'[n][t] /. s],{n, -1, 1, 1/4}], {t, 0, 100}, PlotRange -> All]

enter image description here

or

Plot[z'[sl][100] /. s, {sl, -10, 10}, PlotRange -> All, AxesLabel -> {"z'[0]", "z'[100}"}]

enter image description here

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4
  • $\begingroup$ how did you got the first graphic? $\endgroup$
    – Stratus
    Commented Apr 20, 2017 at 3:02
  • $\begingroup$ ParametricNDSolve provides solutions to differential equations for various parameters, in this case z'[0]. The first graphic displays solutions for z'[0] with values given by Range[-1, 1, 1/4], $\endgroup$
    – bbgodfrey
    Commented Apr 20, 2017 at 3:51
  • $\begingroup$ n = 3.; p = 2.; xf = 1000000.; s = ParametricNDSolveValue[{-z''[ t] == (1 + (n - 2) t)^(1/(2 - n) (2 (n - 1) + 10^-5)) z[ t]^2 Sin[z[t] + 2] , z[0] == 0, z'[xf] == bc2}, z'[xf], {t, 0, xf}, bc2] do you know why after solving this equation the following plot take so long? Plot[s[bc2], {bc2, 0, 2.1}] $\endgroup$
    – Stratus
    Commented Apr 23, 2017 at 18:22
  • $\begingroup$ ParametricNDSolve does not actually solve the ODE until the parameter bc2 is specified. Plot probably requires several tens of solutions, and each is slow, because xf is very large. $\endgroup$
    – bbgodfrey
    Commented Apr 23, 2017 at 22:34

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