nonlinear two point boundary problems

• To: mathgroup <mathgroup at yoda.physics.unc.edu>
• Subject: nonlinear two point boundary problems
• From: HAY at leicester.ac.uk
• Date: Wed, 25 NOV 92 15:45:07 GMT

```> I am in the process of trying to see if Mma can help me solve a nonlinear
> differential game problem and need to solve a nonlinear two point boundary
> value problem with a system of ODE's. Does anyone have any experience with
> trying to get Mma to solve two
> point boundary value problems. NDSolve doesn't seem to do the job. I would
> be very grateful for any advice. Many thanks

> Mark Salmon

The following is a re-write of the "shooting" method demonstrated at the
Rotterdam Mma Conference by Edward Lumsdaine and Jennifer Voitle for solving
Blasius' Equation.

f[n] f''[n] + 2 f'''[n]==0
f[0] == 0, f'[0] == 0, f'[10] == 1.

It takes about twice the time of their program because of using the system
function FindRoot rather than custom code.

Transform to a system of first order equations and construct the "gun";use
FindRoot to find the value of f''[0] (= w[0] below) needed to "hit" f'[10] = 1;
store the latest solution to the system on the way.

gun :=
(soln =   (* cache the latest trial solution *)
NDSolve[
{	2 w'[n] + f[n] w[n] == 0,
w[n] == v'[n],
v[n] == f'[n],
w[0] == #, v[0] == 0, f[0] == 0
},
{w,v,f},
{n,0,10}
]
)[[1,2,2]][10.]&

FindRoot[
gun[x] == 1, {x,1,0.5}
]

The current value of soln gives the solution required; it may be seen by, for
example,

Plot[
Evaluate[{f[n],v[n],w[n]}/.solution], {n,0,10},
PlotStyle -> Apply[RGBColor,IdentityMatrix[3],1]
]

Allan Hayes
hay at leicester.ac.uk

```

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