Cattmul-Rom spline approach and the order of convergence of Green’s functional method for functional differential equations

The purpose of this work is to investigate the convergence properties of Green’s function method applied to boundary value problems for functional differential equations. Recently, involving Picard and Mann iterations, a Green’s function technique was developed (in Int. J. Computer Math. 95, no. 10 (2018) 1937-1949) for third order functional differential equations, but without specifying the order of convergence of the proposed method. In order to improve this aspect, here we establish the maximal order of convergence of Green’s function method applied to two-point boundary value problems associated to second and third order functional differential equations. In this context, by using suitable quadrature rule and appropriate spline interpolation procedure, the Picard iterations are approximated by a sequence of cubic splines on uniform mesh. Some numerical experiments are presented in order to test the theoretical results and to illustrate the accuracy of the method.