How Does Changing Boundary Conditions Affect Nonlinear Shooting Methods in BVPs?

[radiogaga35]

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

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" ).

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

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! :-)

 

[radiogaga35]

Ok, I applied the aforementioned approach (with appropriately-adapted Newton-method implementation) to a trial problem and it worked perfectly.

In principle it seems like a sensible enough approach, but I'm not very clued up on BVP methods, so I'm not sure if there is any theoretical reason to avoid this approach? I.e. varying the initial "displacement" instead of slope. Certainly I've only ever seen the initial slope being varied in simple, single shooting methods