# Nonlinear shooting (BVPs)

1. Feb 28, 2009

Given the boundary value problem (primes denote differentiation w.r.t x):
$$\begin{array}{l} y'' = f(x,y,y') \\ y(a) = \alpha \\ y(b) = \beta \\ \end{array}$$

the nonlinear shooting method may be implemented to solve the problem. A bisection algorithm may be used or, with a little more effort, Newton's method may be implemented (in which case one solves a fourth order, and not a second order, IVP - http://www.math.utah.edu/~pa/6620/shoot.pdf" [Broken]).

But what happens if the form of the boundary conditions changes to:
$$\begin{array}{l} y'(a) = \alpha \\ y(b) = \beta \\ \end{array}$$

Is one still justified in reducing the BVP to a 2nd order IVP, but this time with initial SLOPE fixed at $$y'(a) = \alpha$$ and then trying different values of $$y(a)$$ in order to achieve the condition $$y(b) = \beta$$? (As opposed to varying initial slope to achieve second condition).

Furthermore, if this will indeed work, then a bisection method should be easy to implement, but what about adapting Newton's method for this case? Can anyone point me to an appropriate reference that discusses this matter?

Thank you! :-)

Last edited by a moderator: May 4, 2017
2. Feb 28, 2009