American Journal of Engineering Research (AJER) 2019 American Journal of Engineering Research (AJER) e-ISSN: 2320-0847 p-ISSN : 2320-0936 Volume-8, Issue-8, pp-40-53 www.ajer.org Research Paper Open Access www.ajer.org Page 40 Analysis and Comparative Study of Numerical Solutions of Initial Value Problems (IVP) in Ordinary Differential Equations (ODE) With Euler and Runge Kutta Methods Anthony Anya Okeke 1,* , Pius Tumba 1 , Onyinyechi Favour. Anorue 2 , Ahmed Abubakar Dauda 1 1 Department of Mathematics, Federal University Gashua, P.M.B. 1005 Gashua, Yobe State, Nigeria 2 Department of Mathematics, Michael Okpara University of Agriculture, Umudike, Abia State, Nigeria Corresponding Author: Anthony Anya Okeke ABSTRACT : In this paper, we present Euler’s method and fourth-order Runge Kutta Method (RK4) in solving initial value problems (IVP) in Ordinary Differential Equations (ODE). These two proposed methods are quite efficient and practically well suited for solving these problems. For us to obtain and verify the accuracy of the numerical outcomes, we compared the approximate solutions with the exact solution. We found out that there is good agreement between the exact and approximate solutions. We also compared the performance and the computational effort of the two methods. In addition, to achieve more accuracy in the solutions, the step size needs to be very small. Lastly, the error terms have been analyzed for these two methods for different steps sizes and compared also by appropriate examples to demonstrate the reliability and efficiency. KEYWORDS Ordinary Differential Equations (ODE), Initial value Problems (IVP), Euler Method, Runge Kutta Method, Error Analysis. --------------------------------------------------------------------------------------------------------------------------------------- Date of Submission: 20-07-2019 Date of Acceptance: 06-08-2019 --------------------------------------------------------------------------------------------------------------------------------------- I. INTRODUCTION It is a common truth that Differential Equations are among the most important Mathematical tools used in producing models in the engineering, mathematics, physics, aeronautics, elasticity, astronomy, dynamics, biology, chemistry, medicine, environmental sciences, social sciences, banking and many other areas [1]. Many researchers have studied the nature of Differential Equations and many complicated systems that can be described quite precisely with mathematical expressions. A differential equation that has only one independent variable is called an Ordinary Differential Equation (ODE), and all derivatives in it are taken with respect to that variable. Most often, the variable is time, t; although, some authors use x as the independent variable. The differential equation where the unknown function depends on two or more variables is referred to as Partial Differential Equations (PDE). Although there are many analytic methods for finding the solution of differential equations, there exist quite a number of differential equations that cannot be solved analytically [2]. This means that the solution cannot be expressed as the sum of a finite number of elementary functions (polynomials, exponentials, trigonometric, and hyperbolic functions). For simple differential equations, it is possible to find closed form solutions [3]. But many differential equations arising in applications are so complicated that it is sometimes impractical to have solution formulas; even when a solution formula is available, it may involve integrals that can be calculated only by using a numerical quadrature formula. In either case, numerical methods provide a powerful alternative tool for solving the differential equations under the prescribed initial condition or conditions [3]. There are many types of practical numerical methods for solving initial value problems for ordinary differential equations. Historically, the ancestor of all numerical methods in use today was developed by Leonhard Euler between 1768 and 1770 [4], improved Euler’s me thod and Runge Kutta methods described by Carl Runge and Martin Kutta in 1895 and 1905 respectively [5]. We have excellent and exhaustive books on this can be consulted, such as [1-3, 6-15]. In this work, we shall consider two standard numerical methods Euler and Runge Kutta for solving initial value problems of ordinary differential equations.