Modified operational matrix method for second-order nonlinear ordinary differential equations with quadratic and cubic terms

Burcu Gürbüz; Mehmet Sezer

doi:10.11121/ijocta.01.2020.00827

Modified operational matrix method for second-order nonlinear ordinary differential equations with quadratic and cubic terms

Authors

Burcu Gürbüz Üsküdar University http://orcid.org/0000-0002-4253-5877
Mehmet Sezer Manisa Celal Bayar University http://orcid.org/0000-0002-7744-2574

DOI:

https://doi.org/10.11121/ijocta.01.2020.00827

Keywords:

Nonlinear ordinary differential equations, Laguerre polynomials and series, collocation points, residual error estimation

Abstract

In this study, by means of the matrix relations between the Laguerre polynomials, and their derivatives, a novel matrix method based on collocation points is modified and developed for solving a class of second-order nonlinear ordinary differential equations having quadratic and cubic terms, via mixed conditions. The method reduces the solution of the nonlinear equation to the solution of a matrix equation corresponding to system of nonlinear algebraic equations with the unknown Laguerre coefficients. Also, some illustrative examples along with an error analysis based on residual function are included to demonstrate the validity and applicability of the proposed method.

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Author Biographies

Burcu Gürbüz, Üsküdar University

Burcu Gurbuz received her Ph.D. from the Department of Mathematics of Celal Bayar University in 2017 under the supervision of Prof. Dr. Mehmet SEZER in Applied Mathematics. Recently, she has been awarded as "2019 Young Visiting Research Fellow" from the Embassy of France and currently, she is a post-doctoral researcher at Institution of Mathematics in Johannes Gutenberg-University Mainz. She has worked as investigator in several national and international scientific research projects. Her main research interests ordinary and partial differential equations, integral and integro differential-difference equations, numerical methods, mathematical biology models and scientific computation. Ms. Gurbuz is a member of Turkish Math Society (TMD), Society for Mathematical Biology and European Society for Mathematical and Theoretical Biology (ESMTB).

Mehmet Sezer, Manisa Celal Bayar University

Mehmet Sezer received his Ph.D. under the supervision of Prof. Dr. Sueda MORALI in the Department of Applied Mathematics at Ege University in 1982 and studied in the field of "Parabolic partial differential equations". In 1982, he was Assistant Professor at Faculty of Education (Balikesir) of Uludag} University, Associate Professor in 1989, in 1995 Professor at Faculty of Education of Dokuz Eylul University. In 2004, Mathematics Professor at Faculty of Science at Mu\u{g}la University and since 2012, he has been Applied Mathematics Professor at Faculty of Science at Celal Bayar University. His main research interests are ordinary and partial differential equations, integral and integro differential-difference equations, delay differential equations and their numerical solutions. Prof. Sezer has been reviewer for numerous influential journals, has published research articles related to differential equations, linear algebra, analytic geometry and calculus; and has been authored over 100 papers.

References

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Yuksel, G., Gulsu, M. and Sezer, M. (2011). Chebyshev polynomial solutions of a class of second-order nonlinear ordinary differential equations. Journal of Advanced Research in Scientific Computing, 3(4), 11-24.

Gurbuz, B. and Sezer, M. (2016). Laguerre polynomial solutions of a class of initial and boundary value problems arising in science and engineering fields. Acta Physica Polonica A, 130(1), 194-197.

Gurbuz, B. and Sezer, M. (2014). Laguerre polynomial approach for solving Lane-Emden type functional differential equations. Applied Mathematics and Computation, 242, 255-264.

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Published

2020-07-01

CITATION

DOI: 10.11121/ijocta.01.2020.00827

Published: 2020-07-01

How to Cite

Gürbüz, B., & Sezer, M. (2020). Modified operational matrix method for second-order nonlinear ordinary differential equations with quadratic and cubic terms. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 10(2), 218–225. https://doi.org/10.11121/ijocta.01.2020.00827

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Issue

Vol. 10 No. 2 (2020): IJOCTA

Section

Research Articles

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