A new hybrid collocation method for solving nonlinear two-point boundary value problems

In this paper, numerical solution of boundary value problems (BVPs) of nonlinear ordinary differential equations (ODEs) by the collocation method is considered. Of course, to avoid solving systems of nonlinear algebraic equations resulting from the method, residual function of the boundary value problem is considered and an unconstrained optimisation model is suggested. Particle swami optimisation (PSO) algorithm is used for solving the unconstrained optimisation problem. In addition, convergence properties of the Chebyshev expansion are studied. The scheme is tested on some interesting examples and the obtained results demonstrate reliability and efficiency of the proposed hybrid method.