A One-Step Multi-Derivative Hybrid Block Method with Modified-Picard Iteration for the Solution of Second Order IVPs

This study presents a one-step multi-derivative hybrid block method (OSMDHBM) of order ten, which incorporates third derivatives for the solution of linear and nonlinear second-order initial value problems (IVPs). The derivation incorporates a multi-step collocation and interpolation method, using an approximated power series as the basis function. The intra-step or off-step points are obtained from the derivative of a shifted Legendre polynomial (SLP) of degree four. The accuracy, consistency, and stability properties of the method are analyzed. The nonlinear IVPs are linearized using the modified Picard iteration method (MPIM). In order to demonstrate the superiority of the method, numerical experiments are presented. Comparisons are made between the numerical results obtained and results from other methods and similar schemes in the literature.