Simple shooting-projection method for numerical solution of two-point Boundary Value Problems

This paper presents a novel shooting algorithm for solving two-point boundary value problems (BVPs) for differential equations of one independent variable. This algorithm includes the following steps: First, a shooting step is performed, in which a guess for the initial condition is made and a forward numerical integration of the differential equation is performed so that an Initial Value Problem (IVP) solution, called a shooting trajectory, is obtained. The shooting trajectory starts from one of the boundary constraints but typically does not end at the other boundary constraint. Next, a projection step is performed, in which the shooting trajectory is transformed into a new, projection trajectory that satisfies both constraints and has the same second derivative as the shooting trajectory. The projection trajectory is an approximate BVP solution. Finally, a correction step is performed, in which the projection trajectory is used to obtain a new guess for the initial condition, and the procedure is repeated until convergence.