# Boundary value problem: complicated functionals

I am trying to solve a non-linear boundary value problem where the differential equation is a convoluted function of the variable I am trying to solve for. The equation below presents the final form of the differential equation (without simplifications) that needs to be solved.

$$\frac{\partial\Phi}{\partial l} -\frac{\mathrm d}{\mathrm d s}\frac{\partial\Phi}{\partial l'} +\frac{\mathrm d^2}{\mathrm d s^2}\frac{\partial\Phi}{\partial l''} =0$$

Here's how I'm trying to define the differential equation I would like to solve:

f[{l_, s_}] := (Rv = 50;
hst = 0.75;
w = Rv/hst;
kb = 5;
psist = 14.3;
V0 = 17/(4*Pi*50*50);
S0 = 3*10^6/(4*Pi*5000*5000);
nu0 = V0*Rv*Rv;
sigma = S0/V0;
h = w*(Sqrt[s*s + (l - 1/w - 1)^2] - 1);
psi = psist*(h^(-12) - h^(-6));
K = Exp[-psi];
fact = nu0*(1 + sigma) + 1/K;
chi = 0.5*(fact - Sqrt[fact^2 - 4*nu0*nu0*sigma]);
curv =
D[l, s]/(s*(Sqrt[1 + (D[l, s])^2])) +
D[l, s]/Sqrt[(1 + (D[D[l, s], s])^2)^3];
fun = (0.5*kb*curv^2 - chi*psi)*s*Sqrt[1 + (D[l, s])^2]);
soln = First[
NDSolve[{D[f[{l[s], s}], l[s]] - Dt[D[f[{l[s], s}], l'[s]], s] +
Dt[Dt[D[f[{l[s], s}], l''[s]], s], s] == 0, l[0] == 0,
l'[0] == 0, l'[10] == 0, l''[10] == 0}, l, {s, 0, 10},
Method -> {"Shooting"}]]

• fun in the pure function f(l,s) gives the expression for the functional phi in the above equation.
• Derivatives with respect to s, l(s), l'(s), l''(s) are taken to get the final differential equation (the one shown above).
• I would like to obtain the solution in terms of an interpolating function object l(s).

The difficulty is that the BVP solver is running into singularities when I try to evaluate the solution. The error message displayed is:

Power::infy: Infinite expression 1/0. encountered. >>
Infinity::indet: Indeterminate expression 0. ComplexInfinity encountered. >>
Power::infy: Infinite expression 1/0. encountered. >>
Infinity::indet: Indeterminate expression 0. ComplexInfinity encountered. >>
Power::infy: Infinite expression 1/0. encountered. >>
General::stop:Further output of Power::infy will be suppressed during this calculation. >>
Infinity::indet: Indeterminate expression 0. ComplexInfinity encountered. >>
General::stop: Further output of Infinity::indet will be suppressed during this calculation. >>
NDSolve::ndnum: Encountered non-numerical value for a derivative at s == 0.. >>


And when I try to make the solver avoid these singularities by adding

SolverDelayed->True


in the NDSolve routine after specifying the method of solution, it says,

NDSolve::bvdae: Differential-algebraic equations must be given as initial value problems. >>


I am not sure whether I am proceeding in the right direction or even if this is the right way to solve this problem but so far I haven't found anything similar in the Documentation Center or the forum here. I will be grateful to anyone who can tell me where I am going wrong.

• Shooting method usually need a "StartingInitialConditions", but I failed to, and I'm afraid it won't be easy to, find a proper one for your equation. See here for an example. – xzczd Dec 8 '14 at 8:36
• @xzczd I have initial conditions from solving an approximate version of the equations above. But they too aren't working. I believe Mathematica is struggling with the equations as they are, because when I tried to simplify the expression for the differential equation to be solved using FullSimplify`, it couldn't spit out the result after a day (I quit the kernel then, so don't know if it could have done the job given more time). Thanks for trying anyway. – Sayani Majumdar Dec 13 '14 at 13:35
• @SayaniMajumdar As late as this may be, out of curiosity, this looks like the Euler-Lagrange equation. May I ask what you are trying to minimize via this Euler-Lagrange equation? What physics are you trying to capture? A Lennard-Jones 6-12 potential? – dearN Aug 17 '15 at 11:57