Differential Equation with split boundary conditions

Question

So I have a problem where I'm trying to solve some differential equations with split boundary conditions and have ran into some problems.

Firstly I define some parameters:

λi = 2; r0 = {-100, 500}; r1 = {300, 300}; u = {0, -1}; v = {1/Sqrt[2], 1/Sqrt[2]};

Then I start from a Lagrangian:

L = Sqrt[x'[s]^2 + y'[s]^2] + λ (x'[s]*y''[s] -y'[s]*x''[s])^2/(x'[s]^2 + y'[s]^2)^3

and then generate two coupled differential equations:

eq1 = D[D[L, x'[s]], s] == D[D[L, x''[s]], {s, 2}] // FullSimplify
eq2 = D[D[L, y'[s]], s] == D[D[L, y''[s]], {s, 2}] // FullSimplify

which are very long and unpleasant to look at, but I'd like to be able to obtain a numerical solution with the following boundary conditions:

bc1 = x[0] == r0[[1]];
bc2 = x[1] == r0[[2]];
bc3 = y[0] == r1[[1]];
bc4 = y[1] == r1[[2]];
bc5 = x'[0] == u[[1]] *Sqrt[x'[0]^2 + y'[0]^2];
bc6 = x'[1] == v[[1]] *Sqrt[x'[1]^2 + y'[1]^2];
bc7 = y'[0] == u[[2]] * Sqrt[x'[0]^2 + y'[0]^2];
bc8 = y'[1] == v[[2]]*Sqrt[x'[1]^2 + y'[1]^2];

However when I try using NDSolve:

NDSolve[{eq1 , eq2, bc1, bc2, bc3, bc4, bc5, bc6, bc7, bc8} /. λ -> λi, {x, y}, {s, 0, 1}]

I get the errors:

NDSolve::ntdvdae: Cannot solve to find an explicit formula for the derivatives. NDSolve will try solving the system as differential-algebraic equations.

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

I tried to adapt an answer here and use parametricNDSolve to provide an initial value problem:

parSol2 = ParametricNDSolve[{eq1, eq2, bc1, bc3, bc5, bc7, x''[0] == xdd0, y''[0] == ydd0, 
    x'''[0] == xddd0, y'''[0] == yddd0}, {x, y}, {s, 0, 1}, {xdd0, ydd0, xddd0, yddd0}]

which works but then trying to solve for the root doesn't work, i.e.:

FindRoot[{bc2, bc4, bc6, bc8} /. {x -> x[xdd0, ydd0, xddd0, yddd0], 
    y -> y[xdd0, ydd0, xddd0, yddd0]} /. parSol // Evaluate, 
    {{xdd0, 0.1}, {ydd0, 0.1}, {xddd0, 0.1}, {yddd0, 0.1}}]

probably because I'm not providing a good initial guess (but I'm not sure how to go about improving this really with a 4 dimensional problem). Any suggestions?

Answer 1

NDSolve cannot solve the equations in the question, because they are ill-posed as written. As suggested by the first error message cited in the question, it cannot obtain expressions for the fourth derivatives of x and y. Specifically,

Solve[{eq1, eq2}, {x''''[s], y''''[s]}]
(* {} *)

So, there is no solution for {x''''[s], y''''[s]}. To see why this is so, extract the coefficients of these derivatives.

CoefficientArrays[Subtract @@@ {eq1, eq2}, {x''''[s], y''''[s]}] // Last;
Simplify[Normal[%]/(x'[s]^2 + y'[s]^2)/(2 λ)]
(* {{y'[s]^2, -x'[s] y'[s]}, {x'[s] y'[s], -x'[s]^2}} *)

Det[%]
(* 0 *)

This suggests that an algebraic identity may exist among the dependent variables or their derivatives. In any case, the difficulty lies with the mathematics of the problem, not with its attempted Mathematica solution.