# How do I solve an ODE Two-Point Boundary Value Problem?

I have a feeling my question is a very basic one, but I am not at all well versed in computational sciences.

My equations are of the form:

$$y \in \mathbb{R}^3 \\ \dot{y}(t) = f(y(t)) \\ y_1(0) = a \\ y_2(T) = b \\ y_3(T) = c \\$$

Is there a known method to numerically solve such a set of equations?

• Is there a particular software package or language you're looking to solve this problem in? For someone not well versed in computational sciences, it might help with software recommendations. – Geoff Oxberry Dec 28 '12 at 7:05
• Any introductory text on numerical methods will have a chapter on this topic. – David Ketcheson Dec 28 '12 at 20:02
• What are $y_1$, $y_2$ and $y_3$ and how do they relate to $y(t)$? – John Alexiou Jan 4 '13 at 17:26

You should be able to solve this problem using a multiple shooting method; you need only find initial conditions $y_{2}(0)$ and $y_{3}(0)$ that yield a solution consistent with your stated "final conditions". These values are typically called "boundary values"; your problem is called a two-point boundary value problem. It is worth noting that multiple shooting methods are more numerically stable than single shooting methods.