Legendre collocation method for solving a class of the second order nonlinear differential equations with the mixed non-linear conditions

In this paper, a matrix method based on Legendre collocation points on interval [-1, 1] is proposed for the approximate solution of some second order nonlinear ordinary differential equations with the mixed nonlinear conditions in terms of Legendre polynomials. The method, by means of collocation points, transforms the differential equation to a matrix equation which corresponds to a system of nonlinear algebraic equations with unknown Legendre coefficients. The numerical results show the effectiveness of the method for this type of equation. When this method is compared with the other usual techniques, results would be easier and have higher accuracy.

Keywords:

Nonlinear ordinary differential equations, Legendre polynomials,

PDF

___

G.F.Corliss, Guarented error bounds for ordinary differential equations, in theory and numeric of ordinary and partial equations (M.Ainsworth, J.Levesley, W.A Light, M.Marletta, Eds), Oxford Universty press, Oxford, pp. 342 (1995).
H. Bulut, D.J. Evens, On the solution of Riccati equation by the Decomposition method, Intern. J. Computer Math., 79(1) (2002) 103-109.
G. Adomian, Solving frontier problems of physics the decomposition method, Kluwer Academic Publisher Boston, (1994).
L.M. Kells, Elemetary Differential Equations, McGraw-Hill Book Company, Newyork, (1965).
S.L. Ross, Differential Equations, John Wiley and Sons, Inc. Newyork, (1974).
A. Gurler,S. Yalcinbas, Legendre collocation method for solving nonlinear differential equations, Mathematical & Computational Applications, 18 (3) (2013) 521-530.
S. Yalcinbas, Taylor polynomial solutions of nonlinear Volterra-Fredholm integral equations, Applied Mathematics and Computation 127, (2002) 195-206.
S. Yalcinbas, K. Erdem, Approximate solutions of nonlinear Volterra integral equation systems, International Journal of Modern Physics B 24(32), (2010) 6235-6258.
M. Sezer, A method for the approximate solution of the second order linear differential equations in terms of Taylor polynomials, International Journal of Mathematical Education in Science and Technology 27(6), (1996) 821-834.
S. Yalcinbas, Taylor polynomial solutions of nonlinear Volterra-Fredholm integral equations, Applied Mathematics and Computation, 127, (2002) 195-206.
D. Tas¸tekin,S. Yalcinbas, M. Sezer, Taylor collocation method for solving a class of the first order nonlinear differential equations, Mathematical & Computational Applications, 18 (3) (2013) 383-391.
S. Yalcinbas, D. Tas¸tekin, Fermat collocation method for solving a class of the second order nonlinear differential equations, Applied Mathematics and Physics, 2 (2) (2014) 33-39.
D. Tas¸tekin,S. Yalcinbas, Fermat collocation method for solving a class of the first order nonlinear differential equations, Journal of Mathematical Sciences and Applications, 2 (1) (2014) 4-9.
H. Gurler, S. Yalcinbas, A new algorithm for the numerical solution of the first order nonlinear differential equations with the mixed non-linear conditions by using Bernstein polynomials, New Trends in Mathematical Sciences, 3 (4) (2015) 114-124.
S. Yalcinbas, H. Gurler, Bernstein collocation method for solving the first order nonlinear differential equations with the mixed non-linear conditions, Mathematical & Computational Applications, 20 (3) (2015) 160-173.
S. Yalcinbas, K. Erdem Bicer, D. Tas¸tekin, Fermat collocation method for the solutions of nonlinear system of second order boundary value problems, New Trends in Mathematical Sciences, 4 (1) (2016) 87-96.