Riccati Transformations in the Back-And-Forth Shooting Method for Solving Two-Point Boundary-Value Problems

Abstract

Certain alternatives in the so-called Riccati transformation used in The back-and-forth shooting method are introduced. The alternatives are obtained in the following two main respects. First, the transformations in question are, in the nonlinear case, approximate in a local sense so that their parameters, certain coefficients, may be estimated in alternative nominal trajectories or by alternative procedures. Second, only a linear transformation was employed in the basic version of the method; however, the frame of the method also admits higher degree expansions. Based on these alternatives four different versions of the method are introduced and a convergence theorem for them is given. Also the functioning and convergence properties of them are experimented in a test example from magnetohydrodynamics.