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Re: NDSolve for catenary

  • To: mathgroup at smc.vnet.net
  • Subject: [mg111378] Re: NDSolve for catenary
  • From: "Kevin J. McCann" <Kevin.McCann at umbc.edu>
  • Date: Thu, 29 Jul 2010 06:44:57 -0400 (EDT)
  • References: <i2ok69$7sd$1@smc.vnet.net>

Your problem is likely in the attempt to solve a boundary value problem 
rather than an initial value problem. Here is a quick shooting method 
that gets the answer:

(* Define the shooter. Takes an initial slope of a and returns r[1] *)
Clear[shoot, a]
shoot[a_?NumberQ] := Module[{r, x, y},
   y = r /. NDSolve[{1 +
\!\(\*SuperscriptBox["r", "\[Prime]",
MultilineFunction->None]\)[x]^2 - r[x]
\!\(\*SuperscriptBox["r", "\[Prime]\[Prime]",
MultilineFunction->None]\)[x] == 0, r[0] == 1, r'[0] == a},
       r, {x, 0, 1}][[1]];
   y[1]
   ]

(* Find the correct initial slope *)

\[Alpha] = a /. FindRoot[shoot[a] == 1, {a, -1, 1}]

(* Now that we have the correct slope, solve the DE *)

R = r /. NDSolve[{1 +
\!\(\*SuperscriptBox["r", "\[Prime]",
MultilineFunction->None]\)[x]^2 - r[x]
\!\(\*SuperscriptBox["r", "\[Prime]\[Prime]",
MultilineFunction->None]\)[x] == 0, r[0] == 1, r'[0] == \[Alpha]},
     r, {x, 0, 1}][[1]]

(* Plot it *)

Plot[R[x], {x, 0, 1}, PlotRange -> {0.5, 1}]


Kevin

Sam Takoy wrote:
> Hi,
> 
> This:
> 
> NDSolve[{ 1 + r'[x]^2 - r[x] r''[x] == 0, r[0] == r[1] == 1}, r, {x,  0, 1}]
> 
> should solve the catenary problem, but it gives 1/0 errors. Can you 
> suggest a fix?
> 
> Many thanks!
> 
> Sam
> 


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