# Anyone have a "little" shooting-method function to share

Pauli Virtanen pauli.virtanen at iki.fi
Thu Nov 9 11:47:17 CST 2006

```ke, 2006-11-08 kello 16:08 -0800, David L Goldsmith kirjoitti:
> Hi!  I tried to send this earlier: it made it into my sent mail folder,
> but does not appear to have made it to the list.
>
> I need to numerically solve:
>     (1-t)x" + x' - x = f(t), x(0) = x0, x(1) = x1
> I've been trying to use (because it's the approach I inherited) an
> elementary finite-difference discretization, but unit tests have shown
> that that approach isn't working.
[clip]

You could try to use some of the pre-existing www.netlib.org codes for
solving BVPs. This only requires writing a wrapper.

However, I have written a wrapper for the COLNEW code, which is a
finite-difference method written by Ascher & Bader in '87, having an
adaptive mesh selection. You can find it here:

http://www.iki.fi/pav/bvp

As I understand, COLNEW does not have any special handling for singular
coefficients, but nevertheless it seems to be able to solve your
problem. The code goes as follows:

----------------------------------------------------
import scipy as N
import bvp

u0 = 1
u1 = 2

def f(t):
return N.sin(2*t)

def fsub(t, z):
u, du = z
return N.array([(f(t) + u - du)/(1-t)])

def gsub(z):
u, du = z
return N.array([u[0] - u0, u[1] - u1])

tol = [1e-5, 1e-5]
boundary_points = [0, 1]

solution = bvp.colnew.solve(boundary_points, [2], fsub, gsub,
is_linear=True, tolerances=tol,
vectorized=True, maximum_mesh_size=300)

import pylab

x = solution.mesh
pylab.plot(x, solution(x)[:,0])
pylab.savefig('solution.png')
----------------------------------------------------

BR,

Pauli Virtanen

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