Fermat Collocation Method for Solvıng a Class of the Second Order Nonlinear Differential Equations

Salih Yalçınbaş^1, and Dilek Taştekin¹

¹Department of Mathematics Celal Bayar University, Muradiye, Manisa, Turkey

Cite this paper:
Salih Yalçınbaş and Dilek Taştekin. Fermat Collocation Method for Solvıng a Class of the Second Order Nonlinear Differential Equations. Applied Mathematics and Physics. 2014; 2(2):33-39. doi: 10.12691/amp-2-2-2

Abstract

In this paper, a matrix method based on collocation points on any interval [a,b] is proposed for the approximate solution of some second order nonlinear ordinary differential equations with the mixed conditions in terms of Fermat 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 Fermat coefficients. Also, the method can be used for solving Riccati equation. The numerical results show the effectiveness of the method for this type of equation. Comparing the methodology with some known techniques shows that the present approach is relatively easy and high accurate.

Keywords:
Nonlinear ordinary differential equations Riccati equation Fermat polynomials collocation points

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References:

[1]	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).
[2]	H. Bulut, D.J. Evens, on the solution of Riccati equation by the Decomposition method, Intern. J. Computer Math., 79(1) (2002) 103-109.
[3]	G. Adomian, Solving frontier problems of physics the decomposition method, Kluwer Academic Publisher Boston, (1994).
[4]	V. Hernandez, J.J. Ibonez, J. Peinodo, E. Arias, A GMRES-based BDF method for solving Differential Riccati Equations, Appl. Math. Comput. amc. 2007. 06. 021.
[5]	M. A. El-Tawil, A.A. Bahnasawi, A. Abdel-Naby, Solving riccati differential equation using adomian’s decomposition method, Appl. Math. Comput. 157 (2004) 503-514.
[6]	D. Taştekin, S. Yalçınbaş, M. Sezer, Taylor collocation method for solving a class of the first order nonlinear differential equations, Mathematical and Computational Applications, vol.18, no.3, pp. 383-391, 2013.
[7]	A. F. Horadam, University of New England, Armidale, Australia, 1983.
[8]	L. Carliztz, Restricted Compositions, The Fibonacci Quarterly 14, 3, 254-264, 1976.
[9]	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), 821-834, 1996.
[10]	S. Yalçınbaş, Taylor polynomial solutions of nonlinear Volterra-Fredholm integral equations, Applied Mathematics and Computation 127, 195-206, 2002.
[11]	S. Yalçınbaş, K. Erdem, Approximate solutions of nonlinear Volterra integral equation systems, International Journal of Modern Physics B 24(32), 6235-6258, 2010.
[12]	L.M. Kells, Elemetary Differential Equations, McGraw-Hill Book Company, Newyork, (1965).
[13]	S.L. Ross, Differential Equations, John Wiley and Sons, Inc. Newyork, (1974).
[14]	R.A. Adams, Calculus, Addison-wesley Publisher, 562 (1990).