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Re: NDSolve with a constraint : how ?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg72597] Re: NDSolve with a constraint : how ?
*From*: dh <dh at metrohm.ch>
*Date*: Wed, 10 Jan 2007 04:15:22 -0500 (EST)
*References*: <en82l6$6pp$1@smc.vnet.net>
Hi Cham,
here is a 2 dim. example for your problem. Imagine that you throw a
stone with a angle p and given velocity upward. What is the angle to
come as near as possible to a given point (the stone has not enough
energy to reach the point)?
Method: we define a function that calculates the distance of closest
approach and use this function in FindMinimum:
v=5; (*velocity*)
g=10; (*acceleration*)
reference={2,1}; (*point to approach*)
f:={fx[#],fy[#]}&; (*general 2 dim function*)
dist[p_Real]:=(
eq=Sequence@@Thread[#]&/@{
f''[t]=={0,-g}, (*these are Newtons equations *)
f[0]=={0,0},(*initial conditions*)
f'[0]==v{Sin[p],Cos[p]}
};
res=f[t] /. NDSolve[eq,f[t],{t,0,1}][[1]]; (*the trajectory*)
t0=FindRoot[
D[Plus@@((res-reference)^2),
t]==0,{t,.5}];(*parameter t for closets approach*)
di=Plus@@((res/.t0)- reference)^2; (*calculate the distance of the
closest approach*)
Print["p=",p,", dist= ",di]; ParametricPlot[res,{t,0,1}]; (*make
a picture*)
di
);
FindMinimum[dist[p],{p,0.5}]
Daniel
Cham wrote:
> I need to find a proper way to solve an equation with the NDSolve operation. I'm looking for a solution { x[ t ], y[ t ], z[ t ] } which should obey some constraint (the initial conditions { x[0], y[0], z[0] } are not well known and I only have some very approximate values). How should I do this ?
>
> More specifically, I'm using a simple code like this :
>
> NDSolve[
> {
> x'[t] == Fx[ x[t], y[t], z[t] ],
> y'[t] == Fy[ x[t], y[t], z[t] ],
> z'[t] == Fz[ x[t], y[t], z[t] ],
>
> x[0] == x0,
> y[0] == y0,
> z[0] == z0
> },
>
> {x, y, z}, {t, 0, 100}
> ]
>
> Mathematica then finds easily a solution. But the solution I'm looking for must obey a constraint, and the inital conditions {x0, y0, z0} aren't well known. I need to find the initial values {x0, y0, z0} for which the horizontal distance is the closest to some constant, for an unknown "t" :
>
> rho = Sqrt[x[t]^2 + y[t]^2] = cste, for some unknown "t".
>
> How can I program Mathematica so it could find the right set of numbers {x0, y0, z0} ?
>
> For the moment, all I can do is find a solution from some approximate values {x0, y0, z0}, then check by trial and errors if there's a "t" which gives rho = ctse (approximately). If not, I have to use the NDSolve again, and again (changing a bit the inital values after each trial), until it works. This can be very long, especially since I don't know what value of "t" will satisfy the constraint.
>
> Any better idea ?
>
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