The Legendre Matrix-Collocation Approach for Some Nonlinear Differential Equations Arising in Physics and Mechanics

Öz

In this study, the Legendre operational matrix method based on collocation point is introduced to solve high order ordinary differential equations with some nonlinear terms arising in physics and mechanics. This technique transforms the nonlinear differential equation via mixed conditions into a matrix equation with unknown Legendre coefficients. This solution of this matrix equation yields the Legendre coefficients of the solution function. Thus, the approximate solution is obtained in terms of Legendre polynomials. Some test problems together with residual error estimation are given to show the usefulness and applicability of the method and the numerical results are compared.

Anahtar Kelimeler

• Legendre polynomials and series,nonlinear ordinary differential equation,matrix method,residual error

Kaynakça

1. Akyüz Daşcıoğlu A., Çerdik Yaslan H. 2011. The solution of high-order nonlinear ordinary differential equations by Chebyshev series. Appl. Math. and Comput. 217, 5658-5666.
2. Balcı M. A., Sezer M. 2016. Hybrid Euler-Taylor matrix method for solving of generalized linear fredholm integro-differential difference equations. Appl. Math. Comput. 273, 33-41.
3. El-Mikkawy M.E.A., Cheon G.S. 2005. Combinatorial and hypergeometric identities via the Legendre polynomials-a computational approach. Appl. Math. Comput. 166, 181-195.
4. Everitt W.N., Littlejohn R., Wellman L.L. 2002. Legendre polynomials, Legendre-Stirling numbers and the left-definite spectral analysis of the Legendre differential expressions. J. Comput. Appl. Math. 148, 213-238.
5. Gülsu M., Sezer M., Tanay B. 2009. A matrix method for solving high-order linear difference equations with mixed argument using hybrid Legendre and Taylor polynomials. Journal of the Franklin Institute 343, 647-659.
6. Gürbüz B., 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.
7. Gürbüz B., Sezer, M. 2017. A new computational method based on Laguerre polynomials for solving certain nonlinear partial integro differential equations. Acta Physica Polonica A 132, 561-563.
8. Gürbüz B., Sezer, M. 2017. Laguerre polynomial solutions of a class of delay partial functional differential equations, Acta Physica Polonica A 132, 558-560.Kreyszig E. 2013. Introductory functional analysis with applications, John-Wiley and Sons, New York.

9. Kürkçü Ö. K., Aslan E., Sezer M. 2017. A numerical method for solving some model problems arising in science and convergence analysis based on residual function. Appl. Num. Math. 121, 134-148.
10. Kürkçü Ö. K., Aslan E., Sezer M. 2016. A numerical approach with error estimation to solve general integro-differential difference equations using Dickson polynomials. Appl. Math. Comput. 276, 324-339.
11. Oğuz C., Sezer M. 2015. Chelyshkov collocation method for a class of mixed functional integro-differential equations. Appl. Math. Comput. 259, 943-954.
12. Sezer M., Gülsu M. 2010. Solving high-order linear differential equations by a Legendre matrix method based on hybrid Legendre and Taylor polynomials. Numer. Methods Partial Differential Eq. 26, 647-661.
13. Wazwaz A. M. 2005. Adomian decomposition method for a reliable treatment of the Bratu-type equations. Appl. Math. Comput. 166, 652-663.
14. Yüksel G., Gülsu M., 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.