A shooting reproducing kernel Hilbert space method for multiple solutions of nonlinear boundary value problems

In this work an iterative method is proposed to predict and demonstrate the existence and multiplicity of solutions for nonlinear boundary value problems. In addition, the proposed method is capable of calculating analytical approximations for all branches of solutions. This method is a combination of reproducing kernel Hilbert space method and a shooting-like technique which takes advantage of two powerful methods for solving nonlinear boundary value problems. The formulation and implementation of this iterative method is discussed for nonlinear second order with two and three-point boundary value problems. Also, the convergence of the proposed method is proved. To demonstrate the computational efficiency, the mentioned method is implemented for some nonlinear exactly solvable differential equations including strongly nonlinear Bratu equation and nonlinear reaction–diffusion equation. It is also applied successfully to two nonlinear three-point boundary value problems with unknown exact solutions. In the last example a new branch of solutions is found which shows the power of the method to search for multiple solutions and indicates that the method may be successful in cases where purely analytic methods are not.