Four Step Hybrid Block Method for the Direct Solution of Fourth Order Ordinary Differential Equations

This paper proposes a direct four-step implicit hybrid block method for directly solving general fourth-order initial value problems of ordinary differential equations. In deriving this method, the approximate solution in the form of power series is interpolated at four points, i.e x(n), x(n+1/4), x(n+2), x(n+3) while its forth derivative is collocated at all grid points, i.e x(n), x(n+1/4), x(n+1), x(n+2), x(n+5/2), x(n+3), x(n+7/2), and x(n+4) to produce the main continuous schemes. In order to verify the applicability of the new method, the properties of the new method such as local truncation error, zero stability, order and convergence are also established. The performance of the newly developed method is then compared with the existing methods in terms of error by solving the same test problems. The numerical results reveal that the proposed method produces better accuracy than several existing methods when solving the same initial value problems (IVPs) of second order ODEs.