Lie-Group Shooting/Boundary Shape Function Methods for Solving Nonlinear Boundary Value Problems

In the numerical integration of second-order nonlinear boundary value problem (BVP), right condition plays role as a target equation, which is solved either by half-interval method (HIM) or new derivative-free Newton (DFNM) to be presented in paper. With help shape function, we can transform BVP an initial (IVP) for variable. The terminal variable expressed function missing original variable, determined through few integrations IVP match equation. (NBSFM), solve equation obtain highly accurate value, and then compute precise solution. DFNM find more left values, whose performance superior than HIM. Apparently, converges faster Then, modify Lie-group shooting combine it BSFM solving with Robin conditions. Numerical examples are examined, assure that proposed methods together successfully BVPs high accuracy.