# Transforming boundary conditions into initial conditions…

Hello

Is there any methods available for transforming a 2nd order Boundary value problem such as

$F\left(x,y,\frac{\text{dy}}{\text{dx}},\frac{d^2y}{\text{dx}^2}\right)=0$

$y(a)=y_0$ and $y(b)=y_1$

into an initial value problem? I know this is possible for linear ODEs.

I also know of the shooting method (a numerical technique).

But I've often seen people make transformations or change of variables, which manage to convert the BVP into an IVP. I was wondering does anyone know or have any references to how one would go about finding such a transformation?

This author seems to have made some progress on the matter. http://www.jstor.org/stable/2027813

Is there any others people know of? Or transformations for that work for particular nonlinear problems people are aware of.

Regards

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This comes up in the citations and looks like it might be worth a peek: dx.doi.org/10.1016/0377-0427(94)90308-5 – Steve Huntsman Jan 29 '11 at 21:18
In theoretical chemistry, there is a trick known as the 'initial value representation' from W. H. Miller that is used in semiclassical quantum dynamics to solve the time-dependent Schrodinger equation (approximately). Perhaps some of the ideas in there might be of interest to a general analysis audience. – Jiahao Chen Jan 31 '11 at 1:28