Centroidal Polygon: A New Enhancement of Euler to Improve Accuracy of First Order Non-Linear Ordinary Differential Equation

Abstract

The Euler method is one of the oldest methods to solve differential equation problems. The Euler method produces the simplest solution. However, although is not computationally expensive, the Eulermethod is lack of accuracy. To improve the Euler method, the researcher proposed a new scheme for better accuracy. The Euler method equation and the mean method were combined to enhance this method. As the improvement basis, the researcher used the Centroidal mean and the midpoint method or Polygon to improve the Euler method. The combination of the Euler and Centroidal mean is known as Centroidal Polygon (CP) scheme. The CP scheme was used to solve first-order non-linear Ordinary Differential Equations (ODE). The researcher used SCILAB 6.0 software to solve the equation and the CP scheme was tested in three different step sizes (0.1,0.01, and 0.001). Aside from that, the researcher had compared the CP scheme with previous schemes such as ZulZamri's Polygon (P) scheme, Nurhafizah's Harmoni-Polygon (HP) scheme, and Nooraida's Cube-Polygon (CuP) scheme to ensure that the CP scheme is more accurate than previous research. When the maximum error is calculated by subtracting the scheme and exact solution, the results show that the CP scheme delivers the highest accuracy results in the shortest amount of time. The new enhanced, modified Euler method is useful for other researchers to achieve good accuracy at low computational cost as an alternative to the more computationally expensive methods.