A simple, efficient and accurate new Lie--group shooting method for solving nonlinear boundary value problems

The present paper provides a new method for numerical solution of nonlinear boundary value problems. This method is a combination of group preserving scheme (GPS) and a shooting--like technique which takes advantage of two powerful methods for solving nonlinear boundary value problems. This method is very effective to search unknown initial conditions. To demonstrate the computational efficiency, the mentioned method is implemented for some nonlinear exactly solvable differential equations including strongly nonlinear Bratu equation, nonlinear reaction--diffusion equation and one singular nonlinear boundary value problem. It is also applied successfully on two nonlinear three--point boundary value problems and a third--order nonlinear boundary value problem which the exact solutions of this problems are unknown. The examples show the power of method to search for unique solution or multiple solutions of nonlinear boundary value problems with high computational speed and high accuracy. In the test problem 5 a new branch of solutions is found which shows the power of the method to search for multiple solutions and indicates that the method is successful in cases where purely analytic methods are not.