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I would greatly appreciate your help in implementation of Mathematica for solving a 2nd order differential equation expressing some important engineering problem.

Apparently, this equation has no analytical solution. Therefore I attempt to solve it numerically with maximal possible approach to the analytical effect, while using the ParametricNDSolve command.

The equation has 3 variables: n, Ω and ψ and differentiation is by z. Actually, z = 1. n can be above zero and below it, however the values rarely exceed ±3. Ω and ψ are positive numbers and they can reach big values. As the 1st approach I have defined n = 3. The expression is:

Solution = ParametricNDSolve[{Ω f''[z] - f'[z] - ψ f[z]^n == 0,
                              f[0] - Ω f'[0] == 1,f'[1] == 0}, f, {z, 1, 1}, {Ω, ψ}];

Unfortunately Mathematica is unable restrict itself to z = 1 and produces the singularity messages at z < 1.

My question is thus following: is it possible to restrict the solution to z=1 without attempting to solve at z<1?

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  • $\begingroup$ Is it f''[z][z] or f''[z] or z f''[z] ?? $\endgroup$ – thils Nov 9 '15 at 10:44
  • $\begingroup$ it is f''[z], sorry for this typo $\endgroup$ – Sitra Ahra Nov 9 '15 at 11:05
  • $\begingroup$ Because one boundary condition is at z = 0 and the other at z = 1, ParametricNDSolve must integrate from 0 to 1. $\endgroup$ – bbgodfrey Nov 10 '15 at 3:26
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Because boundary conditions are specified at z = 0 and at z = 1, ParametricNDSolve must integrate between those two endpoints. However, this does not in general prevent obtaining a solution.

s = ParametricNDSolveValue[{Ω f''[z] - f'[z] - ψ f[z]^3 == 0, 
    f[0] - Ω f'[0] == 1, f'[1] == 0}, f, {z, 0, 1}, {Ω, ψ}];

which then can be evaluated at s[Ω, ψ][1]. For instance,

Plot[Evaluate[Table[s[Ω, ψ][1], {ψ, 5}]], {Ω, 1, 5}, AxesLabel -> {Ω, f}]

enter image description here

It is the case that stiffness is encountered at small Ω for ψ == 5.

Addendum

The parameter ψ can be eliminated by rescaling f by ψ^(-1/(n - 1)), which reduces parameter space from two to one dimensions. With the rescaled f designated g,

t = ParametricNDSolveValue[{Ω g''[z] - g'[z] - g[z]^3 == 0, g[0] - Ω g'[0] == 1, 
    g'[1] == 0}, g, {z, 0, 1}, {Ω}]
Plot[t[Ω][1], {Ω, 1, 20}, PlotRange -> {{0, 20}, All}, AxesLabel -> 
    {"Ω", "f \!\(\*SuperscriptBox[\(ψ\), FractionBox[\(1\), \(n - 1\)]]\)"

enter image description here

| improve this answer | |
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  • $\begingroup$ Thank you a lot! I have the following question: perhaps the singularity could be avoided or reduced by defining other solution method? $\endgroup$ – Sitra Ahra Nov 10 '15 at 8:03
  • $\begingroup$ @SitraAhra Although there probably are several ways to avoid this issue, I recommend the rescaling shown in the addendum above. $\endgroup$ – bbgodfrey Nov 11 '15 at 2:53
  • $\begingroup$ Thank you a lot! Indeed the rescaling solves the problem, for all positive values of Ω and n. Thank you a lot again! $\endgroup$ – Sitra Ahra Nov 13 '15 at 8:52

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