Construction of General Implicit-Block Method with Three-Points for Solving Seventh-Order Ordinary Differential Equations

Abstract

In order to solve general seventh-order ordinary differential equations (ODEs), this study will develop an implicit block method with three points of the form y(7)(ξ)=f(ξ,y(ξ),y′(ξ),y″(ξ),y‴(ξ),y(4)(ξ),y(5)(ξ),y(6)(ξ)) directly. The general implicit block method with Hermite interpolation in three points (GIBM3P) has been derived to solve general seventh-order initial value problems (IVPs) using the basic functions of Hermite interpolating polynomials. A block multi-step method is constructed to be suitable with the numerical approximation at three points. However, the construction of the new method has been presented while the numerical results of the implementations are used to prove the efficiency and the accuracy of the proposed method which compared with the RK and RKM numerical methods together to analytical method. We established the characteristics of the proposed method, including order and zero-stability. Applications of various IVP problems are also discussed, and the outcomes are very encouraging for the suggested approach. The proposed GIBM3P method yields more accurate numerical solutions to the test problems than the existing RK method, which are in good agreement with analytical and RKM method solutions.