Numerical integration of linear hybrid multistep block method for third-order ordinary differential equations (IVPs)

In this article, insight into the usefulness of the linear hybrid multistep method for direct solution of initial value problems of third-order ordinary differential equations without reduction to the system of first-order ordinary differential equations is presented. The suggested method was derived using collocation and interpolation techniques, with power series as the basis function, to produce a system of algebraic equations. The unknown coefficients in the system of algebraic equations were gotten through the Gaussian elimination method. The values of the determined coefficient were substituted into the approximate polynomial and evaluated at different grid points which yields the expected continuous scheme. The technique produces a self-starting discrete system with better precision and a larger absolute stability interval. The fundamental features of the method were studied. The findings revealed that the approach is zero stable, consistent, convergent, and of order seven. The method's performance was tested by solving linear and nonlinear problems of general third-order ordinary differential equations. The results were found to compare favourably with some existing methods in the literature.