Also, some illustrative examples along withan error analysis based on residual function are included to demonstrate thevalidity and applicability of the proposed method.
📌 3 Key Takeaways
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Objective & Hypothesis:In this study, by means of the matrix relations between the Laguerre polynomials, and their derivatives, a novel matrix method based on collocation pointsis modified and developed for solving a class of second-order nonlinear ordinarydifferential equations having quadratic and cubic terms, via mixed conditions.The method reduces the solution of the nonlinear equation to the solution of amatrix equation corresponding to system of nonlinear algebraic equations withthe unknown Laguerre coefficients.
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Key Findings & Data:In this study, by means of the matrix relations between the Laguerre polynomials, and their derivatives, a novel matrix method based on collocation pointsis modified and developed for solving a class of second-order nonlinear ordinarydifferential equations having quadratic and cubic terms, via mixed conditions.The method reduces the solution of the nonlinear equation to the solution of amatrix equation corresponding to system of nonlinear algebraic equations withthe unknown Laguerre coefficients.
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Conclusion & Impact:Also, some illustrative examples along withan error analysis based on residual function are included to demonstrate thevalidity and applicability of the proposed method.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Domain Parameters
Abstract
In this study, by means of the matrix relations between the Laguerre polynomials, and their derivatives, a novel matrix method based on collocation pointsis modified and developed for solving a class of second-order nonlinear ordinarydifferential equations having quadratic and cubic terms, via mixed conditions.The method reduces the solution of the nonlinear equation to the solution of amatrix equation corresponding to system of nonlinear algebraic equations withthe unknown Laguerre coefficients. Also, some illustrative examples along withan error analysis based on residual function are included to demonstrate thevalidity and applicability of the proposed method.
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Cite This Article
GÜRBÜZ, B., SEZER, M. (2022). 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), Cilt: 10 - Sayı: 2, 218–225. https://ajindex.com/articles/440585
Fried, I. (1979). Numerical solution of differential equations. Academic Press, New York
Kells, L.M. (1960). Elementary differential equations. ISBN 07-033530-3
Jordan, D.W. and Smith, P. (2007). Nonlinear ordinary differential equations: an introduction for Scientists and Engineers, Fourth Edition. Oxford University Press, New York
King, A.C., Billingham, J. and Otto, S.R. (2003). Differential equations: linear, nonlinear, ordinary, partial, Cambridge University Press, New York
Rawashdeh, M.S. and Maitama, S. (2015). Solving nonlinear ordinary differential equations using the NDM. Journal of Applied Analysis and Computation, 5(1), 77-88
Y¨uksel, G., G¨ulsu, 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
G¨urb¨uz, 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
G¨urb¨uz, B. and Sezer, M. (2014). Laguerre polynomial approach for solving Lane-Emdentype functional differential equations. Applied Mathematics and Computation, 242, 255-264
B¨ulb¨ul, B. and Sezer, M. (2013). Numerical solution of Duffing equation by using an improved Taylor matrix method. Journal of Applied Mathematics, 2013, 691614
Inc, M., Akgul, A. and Kılı¸cman, A. (2013). Numerical solutions of the second-order onedimensional telegraph equation based on reproducing kernel Hilbert space method. Abstract and Applied Analysis, 2013, Hindawi.