International Journal of


EISSN: 2313-3724, Print ISSN: 2313-626X

Frequency: 12

line decor
line decor

 Volume 11, Issue 1 (January 2024), Pages: 201-206


 Technical Note

Legendre operational differential matrix for solving ordinary differential equations


 Zainab I. Mousa 1, *, Mazin H. Suhhiem 2


 1Department of Mathematics, University of Kufa, Kufa, Iraq
 2Department of Mathematics, University of Sumer, Al-Rifai, Iraq

 Full text

  Full Text - PDF

 * Corresponding Author. 

  Corresponding author's ORCID profile:

 Digital Object Identifier (DOI)


In this paper, we used the Legendre operational differential matrix method based on the Tau method to find the approximate analytical solutions to the initial value problems and boundary value problems of ordinary differential equations. This method allows the solution of the ordinary differential equation to be computed in the form of an infinite series in which the components can be easily calculated. We introduced a comparison between the approximate solution that we computed and the exact solution of the selected problem, as we found the absolute error. According to the numerical results, the series of solutions we found are accurate and very close to the exact analytical solutions.

 © 2024 The Authors. Published by IASE.

 This is an open access article under the CC BY-NC-ND license (


 Legendre operational differential matrix, Tau method, Ordinary differential equations, Approximate analytical solutions, Absolute error comparison

 Article history

 Received 24 August 2023, Received in revised form 14 January 2024, Accepted 15 January 2024


No Acknowledgment.

 Compliance with ethical standards

 Conflict of interest: The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.


 Mousa ZI and Suhhiem MH (2024). Legendre operational differential matrix for solving ordinary differential equations. International Journal of Advanced and Applied Sciences, 11(1): 201-206

 Permanent Link to this page


 No Figure


 Table 1 Table 2 Table 3


 References (8)

  1. Edeo A (2019). Solution of second order linear and nonlinear two point boundary value problems using Legendre operational matrix of differentiation. American Scientific Research Journal for Engineering, Technology, and Sciences, 51(1): 225-234.   [Google Scholar]
  2. Geng F, Lin Y, and Cui M (2009). A piecewise variational iteration method for Riccati differential equations. Computers and Mathematics with Applications, 58(11-12): 2518-2522.   [Google Scholar]
  3. Hossain MJ, Alam MS, and Hossain MB (2017). A study on numerical solutions of second order initial value problems (IVP) for ordinary differential equations with fourth order and Butcher’s fifth order Runge-Kutta methods. American Journal of Computational and Applied Mathematics, 7(5): 129-137.   [Google Scholar]
  4. Islam MA (2015). Accurate solutions of initial value problems for ordinary differential equations with the fourth order Runge Kutta method. Journal of Mathematics Research, 7(3): 41-45.   [Google Scholar]
  5. Jung CY, Liu Z, Rafiq A, Ali F, and Kang SM (2014). Solution of second order linear and nonlinear ordinary differential equations using Legendre operational matrix of differentiation. International Journal of Pure and Applied Mathematics, 93(2): 285-295.   [Google Scholar]
  6. Masenge RP and Malaki SS (2020). Finite difference and shooting methods for two-point boundary value problems: A comparative analysis. Must Journal of Research and Development, 1(3): 160-170.   [Google Scholar]
  7. Moore TJ and Ertürk VS (2019). Comparison of the method of variation of parameters to semi-analytical methods for solving nonlinear boundary value problems in engineering. Nonlinear Engineering, 9: 1-13.   [Google Scholar]
  8. Sakka A and Sulayh A (2019). On Taylor differential transform method for the first Painlevé equation. Jordan Journal of Mathematics and Statistics, 12(3): 391-408.   [Google Scholar]