FUW Trends in Science & Technology Journal, www.ftstjournal.com e-ISSN: 24085162; p-ISSN: 20485170; December, 2021: Vol. 6 No. 3 pp. 868 – 876 868 NUMERICAL APPLICATION OF ORDINARY DIFFERENTIAL EQUATIONS USING POWER SERIES FOR SOLVING HIGHER ORDER INITIAL VALUE PROBLEMS T. Y. Kyagya 1 , D. Raymond 1 and J. Sabo 2 1 Department of Mathematics and Statistic, Federal University, Wukari-Nigeria 2 Department of Mathematics, Adamawa University Mubi-Nigeria Received: August 16, 2021 Accepted: October 07, 2021 Abstract: In this research, we have proposed the numerical application of second derivative ordinary differential equations using power series for the direct solution of higher order initial value problems. The method was derived using power series, via interpolation and collocation procedure. The analysis of the method was studied, and it was found to be consistent, zero-stable and convergent. The derived method was able to solve highly stiff problems without converting to the equivalents system of first order ODEs. The generated results showed that the derived methods are notable better than those methods in literature. We further sketched the solution graph of our method and it is evident that the new method convergence toward the exact solution. Keywords: Numerical application, ODEs, higher order IVPs, power series, collocation Introduction Mathematical modeling of real-life problems usually result into functional equation, for example, Ordinary differential equation and Partial differential equation, Integro and Integral differential equation, Stochastic differential equation and others. Not all ordinary differential equations such as those used to model real life problems can be solved analytically, Omar (2004). Most of the problems in science, mathematical physics and engineering are formulated by differential equations. The solution of differential equations is a significant part to develop the various modeling in science and engineering. There are many analytical methods for finding the solution of ordinary differential equations. But a few numbers of differential equations have analytic solutions where a large numbers of differential equations have no analytic solutions. In recent years, mathematical modeling of processes in biology, physics and medicine, particular in dynamic problems, cooling of a body and simple harmonic motion has led to significant scientific advances, both in mathematics and biosciences (Brauer & Castillo, 2012; Elazzouzi et al., 2019). The applications of mathematics in biology and physics are completely opening new pathways of interactions, and this is certainly true, particular in areas like dynamic problems and cooling of a body. This research considered the solution of high order initial value problems (IVPs) of ODEsof the form 1 0 ' , , ' , , ' ' a y y a y y y t f y (1.1) Equation (1.1) occurs in deferent fields of applied mathematics, among which are elasticity, fluid mechanics, and quantum mechanics as well as in engineering and physics. The existence and uniqueness of the solution for these equation have been discussed in Wend (1969). In general, finding the exact solutions of these equation is not easy. Over the years, deferent numerical methods have been developed in order to approximate the solution of equation (1.1). Among these methods are block method, linear multistep method, hybrid method and rung-kutta method, etc. (Lambert, 1973; Gear, 1966, 1971, 1978; Suleiman, 1979, 1989). Recently, some efforts have been made to develop hybrid block method for solving (1.1) directly; among others are Kuboye & Omar (2015), Omar & Abdelrahim (2016), Abdelrahim & Omar (2016), Alkasassbeh & Omar (2017), Skwame et al. (2019a, 2019b). However, these methods are focused on specific points (specifically, second order IVPs). Mathematical Formulation of the Method Power series polynomial of the form j q p j j t a t y 1 0 (2.1) is considered as a basis function to approximate the solution of the initial value problems of general second order ordinary differential equation of the form 1 0 ' , , ' , , ' ' y a y y a y y y t f y (2.2) method is derived by the introduction of off-mesh points through one-step scheme following the method of Gragg and Stetter (1964), Gear (1964), Butcher (1965), and recently Omar & Adeyeye (2016), Omole & Ogunware (2018), Kamo et al. (2018), Skwame et al. (2019b). Using (2.1) with 2 p and 7 q , the polynomial is as follows; j j j t a t y 8 0 (2.3) Differentiating (2.3) twice, to yield 2 8 0 1 ' ' j j j t a j j t y (2.4) Substituting (2.3) into (2.1) to yield ' , , 1 2 8 0 y y t f t a j j j j j (2.5) Now, interpolating (2.3) at 9 5 9 4 and and collocating (2.5) at 1 9 8 , 9 7 , 3 2 , 9 5 , 9 4 , 0 and lead to a system of equation written in a matrix form below; U TA (2.6) Supported by