Explicit Solution of First-Order Differential Equation Using Aitken’s and Newton’s Interpolation Methods

Author: Salisu Ibrahim 1
1Mathematics Education Departments, Faculty of Education, Tishk International University-Erbil, Kurdistan Region, Iraq

Abstract: The struggle to find the analytic solution of several differential equations leads to several issues like difficulty in finding solutions, singularities, convergent issues, and stability. Because of these problems, most of the researchers come up with explicit approaches such as the Runge Kutta method, Euler’s method, and Taylor’s polynomial method for finding numerical solutions to the ordinary differential equation. In this work, we combine both the Aitken methods and Newton’s interpolation method (NIM) to solve first-order differential equations. The numerical results obtained provide minimal error. The result is supported by solving an example.

Keywords: Ordinary Differential Equation, Aitken’s Method, Newton Interpolation Polynomial, Numerical Approximation

Download the PDF Document

Doi: 10.23918/eajse.v9i1p46

Published: January 11, 2023


Al Din, I. N. (2020). Solving Bernoulli Differential Equations by using Newton’s Interpolation and Aitken’s Methods. ‏

Al Din, I. N. (2020). Using Newton’s Interpolation and Aitken’s Method for Solving First Order Differential Equation. World Applied Sciences Journal, 38(3), 191-194. ‏

Atkinson, K., Han, W., & Stewart, D. E. (2009). Numerical solution of ordinary differential equations. John Wiley & Sons, 10. ‏

Chasnov, J. R. (2014). Differential Equations with YouTube Examples-eBooks and textbooks from bookboon. com.‏

Faith, C. (2018). Solution of first order differential equation using numerical newton’s interpolation and lagrange method. Int. J. Dev. Res, 8, 18973-18976.

Kreyszig, E. et al (2011). Advanced Engineering Mathematics 10th Edition. John Wiley & Sons. ‏
Ibrahim, S. Numerical Approximation Method for Solving Differential Equations. Eurasian Journal of Science & Engineering, 6(2), 157-168, 2020.

Ibrahim, S., & Isah, A. (2021). Solving System of Fractional Order Differential Equations Using Legendre Operational Matrix of Derivatives. Eurasian Journal of Science & Engineering, 7(1), 25-37.

Ibrahim, S., & Isah, A. (2022) Solving Solution for Second-Order Differential Equation Using Least Square Method. Eurasian Journal of Science & Engineering, 8(1), 119-125.

Ibrahim, S., & Koksal, M. E. (2021a). Commutativity of Sixth-Order Time-Varying Linear Systems. Circuits Syst Signal Process. https://doi.org/10.1007/s00034-021-01709-6

Ibrahim, S., & Koksal, M. E. (2021b). Realization of a Fourth-Order Linear Time-Varying Differential System with Nonzero Initial Conditions by Cascaded Two Second-Order Commutative Pairs. Circuits Syst Signal Process. https://doi.org/10.1007/s00034-020-01617-1.

Isah, A., & Ibrahim, S. (2021). Shifted Genocchi Polynomial Operational Matrix for Solving Fractional Order System. Eurasian Journal of Science & Engineering, 7(1) 25-37.

Mbagwu, J. P., & Ide, N. A. D. (2021). Comparison of Newton’s Interpolation and Aitken’s Methods with Some Numerical Methods for Solving System of First and Second Order Differential Equation. World Scientific News, 164, 108-121. ‏

Rababah, A., & Ibrahim, S. (2016a). Weighted G^1-Multi-Degree Reduction of Bézier Curves. International Journal of Advanced Computer Science and Applications, 7(2), 540-545. https://thesai.org/Publications/ViewPaper?Volume=7&Issue=2&Code=ijacsa&SerialNo=70

Rababah, A., & Ibrahim, S. (2016b). Weighted Degree Reduction of Bézier Curves with G^2-continuity. International Journal of Advanced and Applied Science, 3(3), 13-18.

Rababah, A., & Ibrahim, S. (2018). Geometric Degree Reduction of Bézier curves, Springer Proceeding in Mathematics and Statistics, Book Chapter 8. 2018. https://springer.com/us/book/9789811320941

Salisu, I. (2022a,). Commutativity of high-order linear time-varying systems. Advances in Differential Ð Equations and Control Processes, 27(1) 73-83. Ð https://dx.doi.org/10.17654/0974324322013

Salisu, I. (2022b). Discrete least square method for solving differential equations, Advances and Applications in Discrete Mathematics 30, 87-102. Ð https://dx.doi.org/10.17654/0974165822021

Salisu, I. (2022c). Commutativity Associated with Euler Second-Order Differential Equation. Advances in Differential Equations and Control Processes, 28(2022), 29-36. Ð https://dx.doi.org/10.17654/0974324322022

Salisu, I., & Abedallah R. (2022). Decomposition of Fourth-Order Euler-Type Linear Time-Varying Differential System into Cascaded Two Second-Order Euler Commutative Pairs, Complexity, 2022, https://doi.org/10.1155/2022/3690019.

Zill, D. G. (2012). A first course in differential equations with modeling applications: Cengage Learning.

Zou, L., Song, L., Wang, X., Weise, T., Chen, Y., & Zhang, C. (2020). A new approach to Newton-type polynomial interpolation with parameters. Mathematical Problems in Engineering, 2020.