A Modified Fourth Derivative Block Method and its direct applications to third-order initial value problems

Abstract

A theoretical order eight Modified Fourth Derivative four-step block method (MFDFBM) has been derived, analysed and numerically applied to solve multiple problems originating from Fluid Dynamics, engineering and other sciences. The MFDFBM was derived by applying collocation and interpolation techniques to a power series approximation. Further introducing fourth derivative terms at each of the collocating points yields a block method with an improved order of accuracy. It was observed that the order of the block method increases with the number of fourth derivative terms introduced into the integration interval. Numerical experiments are presented to test MFDFBM on numerical examples, including non-linear homogeneous thin film flow (NHTFF) problems and two non-linear initial value problems(IVPs). The experiments confirm the good impact of adding the fourth derivative terms, which help improve the order of accuracy of the derived MFDFBM, thereby minimising error and agreeing with analytical solution up to at least seven decimal places.