An Innovative Numerical Comparison of Euler and Taylor Series Methods in Solving Differential Equations with Application to the Logistic Growth Equation

The objective of this work is to numerically compare the two numerical techniques for solving ordinary differential equations, namely the Euler method and the Taylor series method. The special focus of this numerical comparison will be on solving both the linear and nonlinear ordinary differential equations that can be modelled using the logistic growth model. The methodology employed in this study to achieve these objectives includes a detailed mathematical formulation of the problem followed by the numerical implementation of each method using the MATLAB computing environment and the determination of differences between the two methods with respect to their accuracy and computational efficiency. Additionally, a new hybrid method (the hybrid Euler-Taylor method), which uses a standard Euler approximation and applies a second order Taylor correction is also presented. This hybrid method provides an average performance between these two methods in terms of both accuracy and computational efficiency. The results show that while the Euler method is easy to implement, it is not very accurate, whereas the Taylor series method is very accurate but requires more computational effort. The hybrid Euler-Taylor method occupies the “sweet spot” between accuracy and computational efficiency.