An Innovative Numerical Comparison of Euler and Taylor Series Methods in Solving Differential Equations with Application to the Logistic Growth Equation
DOI:
https://doi.org/10.59992/IJCI.2026.v5n7p2Keywords:
Mathematical Modeling, Approximate Solutions, Numerical Methods, Hybrid Euler–Taylor Method, Logistic Growth ModelAbstract
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.
References
• Ali, M. A. (2025). Development and Analysis of Advanced Numerical Algorithms for Solving Differential Equations Using Taylor, Euler, and Runge-Kutta Methods. Journal of Computational Analysis & Applications, 34(8).
• Ascher, U. M. (2008). Numerical methods for evolutionary differential equations. SIAM.
• Atkinson, K., Han, W., & Stewart, D. E. (2009). Numerical solution of ordinary differential equations. John Wiley & Sons.
• Butcher, J. C. (2021). B-series: algebraic analysis of numerical methods (Vol. 55). Springer.
• Golub, G. H., & Ortega, J. M. (2014). Scientific computing and differential equations: an introduction to numerical methods. Elsevier.
• Gupta, T., Utreja, K., & Sharma, A. (2024). Deciphering Complex Mathematics: Pythons Application of Numerical Methods. 2024 First International Conference on Technological Innovations and Advance Computing (TIACOMP),
• Henner, V., Nepomni︠a︡shchiĭ, A. A., Belozerova, T., & Khenner, M. (2023). Ordinary differential equations: analytical methods and applications. Springer.
• Jumaeva, S. (2024). Methods of Solving Differential Equations. Science and Education, 5(7), 14-19.
• Kiusalaas, J. (2010). Numerical methods in engineering with Python. Cambridge university press.
• Pelagalli, C., Faccio, S., & Casari, P. (2025). Mathematical Modeling and Simulation of Logistic Growth. Applied Sciences, 15(8), 4409.
• Racshitha, N., Manimaran, R., & Campus, V. (2025). Solving the Stochastic Delay Differential Equation by using Taylor’s and Modified Euler’s Methods. International Journal of Mathematics & Computer Science, 20(2).
• Salgado, G. H., & Romanelli, J. P. (2024). Numerical solutions of ordinary differential equations using Spline-Integral Operator. arXiv preprint arXiv:2409.20369.
• Sauer, T. (2025). Numerical analysis. BoD–Books on Demand.
• Slavova, A. (2024). New Trends in the Applications of Differential Equations in Sciences. Springer.
• Tumuluri, S. K. (2021). A first course in ordinary differential equations. Chapman and Hall/CRC.
