Four step hybrid block method for the direct solution of fourth order ordinary differential equations

Abstract

‎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},,x_{n+2},,x_{n+3} $ while its forth derivative is collocated at all grid points‎, ‎i.e $ x_n‎, ‎,,x_{n+frac{1}{4}},,‎, ‎x_{n+1}‎ , ‎,,x_{n+2}‎, ‎,,x_{n+frac{5}{2}}‎, ‎,,x_{n+3}‎, ‎,,x_{n+frac{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.