Modified operational matrix method for second-order nonlinear ordinary differential equations with quadratic and cubic terms
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.
[1] Fried, I. (1979). Numerical solution of di扩er- ential equations. Academic Press, New York.
[2] Kells, L.M. (1960). Elementary di扩erential equations. ISBN 07-033530-3.
[3] Jordan, D.W. and Smith, P. (2007). Nonlin- ear ordinary di扩erential equations: an intro- duction for Scientists and Engineers, Fourth Edition. Oxford University Press, New York.
[4] King, A.C., Billingham, J. and Otto, S.R.(2003). Di扩erential equations: linear, nonlin- ear, ordinary, partial, Cambridge University Press, New York.
[5] Rawashdeh, M.S. and Maitama, S. (2015). Solving nonlinear ordinary diferential equa- tions using the NDM. Journal of Applied Analysis and Computation, 5(1), 77-88.
[6] Y¨uksel, G., G¨ulsu, M. and Sezer, M. (2011). Chebyshev polynomial solutions of a class of second-order nonlinear ordinary diferential equations. Journal of Advanced Research in Scientific Computing, 3(4), 11-24.
[7] 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.
[8] G¨urb¨uz, B. and Sezer, M. (2014). Laguerre polynomial approach for solving Lane-Emden type functional diferential equations. Applied Mathematics and Computation, 242, 255-264.
[9] B¨ulb¨ul, B. and Sezer, M. (2013). Numerical solution of Du伍ng equation by using an im- proved Taylor matrix method. Journal of Ap- plied Mathematics, 2013, 691614.
[10] Inc, M., Akgul, A. and K1l1>cman, A. (2013). Numerical solutions of the second-order one- dimensional telegraph equation based on re- producing kernel Hilbert space method. Ab- stract and Applied Analysis, 2013, Hindawi.