Abstract
In this paper, the classical fourth-order Runge-Kutta methodis presented for solving the first-order ordinary differential equation. First, the given solution domain is discretizedby using a uniform discretization grid point. Next by applyingthe forward difference method, we discretized the given ordinary differential equation. And formulating a difference equation. Then using this difference equation, the given first-order ordinary differential equation is solved by using the classicalfourth-order Runge-Kutta method at each specified grid point. To validate the applicability of the proposed method, two model examples are considered and solved at each specific grid point on its solution domain. The stability and convergent analysis of the present method is worked by supportedthe theoretical and mathematical statementsand the accuracy of the solution is obtained. The accuracy of the present methodhas been shown in the sense ofmaximumabsolute error and the local behavior of the solution is captured exactly. Numerical and exact solutions have been presented in tables and graphs and the corresponding maximumabsolute errorisalso presented in tables and graphs. The present method approximates the exact solution very well and it is quite efficient and practically well suitedfor solving first-order ordinary differential equations. The numerical result presented in tables and graphsindicates that the approximate solution is in good agreement with the exact solution. Hence the proposed method is accruable to solve ordinary differential equations.
Highlights
Numerical analysis is a subject that involves computational methods for studying and solving mathematical problems
Numerical methods for the solution of ordinary differential equations may be put in two categories-numerical integration methods and Runge-Kutta methods
We propose a numerical method for solving the first-order ordinary differential equation and the convergence has been shown in the sense of absolute errors that the local behavior of the solution is captured exactly
Summary
Numerical analysis is a subject that involves computational methods for studying and solving mathematical problems. When predictor-corrector methods are used, RungeKutta methods still find application in starting the computation and in changing the interval of integration [11] One of this method Runge-Kutta 4th order method is a numerical technique used to solve ordinary differential equations [9]. Runge-Kutta methods are considered in the context of using them for starting and for changing the interval, matters such as stability and minimization of roundoff errors are not significant [11] Methods such as the Taylor series method, Euler method, RungeKutta second & third order methods, and finite difference method, have been used to approximate first-order ordinary differential equation.
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