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Re: shooting method, boundary value problem
*To*: mathgroup at smc.vnet.net
*Subject*: [mg113775] Re: shooting method, boundary value problem
*From*: Alois Steindl <Alois.Steindl at tuwien.ac.at>
*Date*: Fri, 12 Nov 2010 05:26:43 -0500 (EST)
*References*: <ibgitq$e87$1@smc.vnet.net>
Hello,
as stated, your problem has really only the trivial solution. But if you
replace the second boundary condition by y(Pi/2)=0, you get a whole
family of solutions y=A Cos[t].
In order to pick one of these, you could replace the second boundary
condition by a proper initial condition, say y[0]=A. The second boundary
condition would still be satisfied.
If you are thinking about nonlinear problems, the period of the solution
could vary. To state the problem as a standard BVP, you have to rescale
your x variable, e.g. by setting
x = T*tau,
where T satisfies the trivial ODE T'=0. tau is the rescaled time.
So you could also specify your problem as
y''[tau]+T[tau]^2 y[tau] ==0,
T'[tau]==0,
y[0]==A,
y'[0]==0,
y[1]==0
with the initial guess T[0]=Pi/2.
With this method you could also solve more difficult problems, like
e.g. the pendulum equation:
y''[tau]+T[tau]^2 Sin[y[tau]] == 0.
Good luck
Alois
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