Graphical Interpretation of the Slopes Used in the Derivation of Classical Fourth Order Runge-Kutta (RK4) Formula

International Journal of Research and Scientific Innovation (IJRSI) | Volume IX, Issue IV, April 2022 | ISSN 2321–2705

Graphical Interpretation of the Slopes Used in the Derivation of Classical Fourth Order Runge-Kutta (RK4) Formula

Md. Azmol Huda, Mohammad Wahiduzzaman, Munnujahan Ara
Mathematics Discipline, Khulna University, Khulna, Bangladesh

Abstract: Many practical issues in science and engineering are formulated by ordinary differential equations (ODE) that require their own numerical solution. There are a variety of numerical approaches, e.g. the Euler method, the modified Euler method, the Heun’s method, the Adam-Bashforth method, and so on, that exist in the context of numerical analysis. Amongst them, the classical fourth order Runge-Kutta (RK4) technique is the most reliable and most used. The objective of this paper is twofold. The first goal is to derive the value of different parameters in the formulation of the fourth order Runge-Kutta method, and the second goal is to give details of the geometrical interpretation of this method, principally explaining the role of the increment parameters in the formula. The whole discussion will facilitate perception of the key mechanism of the Runge-Kutta method.

Keywords: Runge-Kutta method, Euler method, Heun’s method, Increment parameter.

I.INTRODUCTION:

Around 1900, two German mathematicians, Curl Runge and Wilhelm Kutta, devised the Runge-Kutta techniques in numerical analysis [1]. In 1895, C. Runge presented a work that was a more complex development and was an extension of the Euler method’s approximation. To determine the numerical solution of differential equations, various order Runge-Kutta techniques have been widely utilized [2-3]. To solve second-order fuzzy differential equations, a novel version of the enhanced Runge-Kutta Nystrom technique is used [4]. For the numerical solution of n-th order fuzzy differential equations based on the Seikkala derivative with initial value issue [5-8], the Runge-Kutta technique of order five is utilized. Also, the fourth and fifth-order Runge-Kutta techniques [9-11] are used to the specific Lorenz equation. In [12-14], implicit and multistep Runge-Kutta techniques are investigated. Euler and Coriolis [15-16] explore the fundamental concepts of differential equation theory and their numerical solution. In the articles of Runge [17], Heun [18], and Nystrom [19], the early works of the Runge-Kutta technique are examined. Adams and Bashforth, Dahlquist [21-22], and Moulton have published the foundations of multistep Runge-Kutta techniques. Recently, some work on the Runge-Kutta technique has been published, for example, Mechee and Yasen, Geeta and Varun [27-28] used extended RK integrators to solve ordinary differential equations. Vijeyata and Pankaj describe computational approaches for solving

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