A collocation method using Hermite polynomials for approximate solution of pantograph equations. (English) Zbl 1221.65187

Summary: A numerical method based on polynomial approximation, using Hermite polynomial basis, to obtain the approximate solution of generalized pantograph equations with variable coefficients is presented. The technique we have used is an improved collocation method. Some numerical examples, which consist of initial conditions, are given to illustrate the reality and efficiency of the method. In addition, some numerical examples are presented to show the properties of the given method; the present method has been compared with other methods and the results are discussed.


65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
65L03 Numerical methods for functional-differential equations
34K28 Numerical approximation of solutions of functional-differential equations (MSC2010)
Full Text: DOI


[1] Ockendon, J.R.; Tayler, A.B., The dynamics of a current collection system for an electric locomotive, Proc. roy. soc. London A, 322, 447-468, (1971)
[2] Derfel, G.A.; Vogl, F., On the asymptotics of solutions of a class of linear functional – differential equations, eur., J. appl. math., 7, 511-518, (1996) · Zbl 0859.34049
[3] G.R. Morris, A. Feldstein, E.W. Bowen, The Phragmen-Lindel’ of principle and a class of functional – differential equations, in: Proceedings of the NRL-MRC Conference on Ordinary Differential Equations, 1972, pp. 513-540.
[4] Derfel, G.; Iserles, A., The pantograph equation in the complex plane, J. math. anal. appl., 213, 117-132, (1997) · Zbl 0891.34072
[5] Derfel, G., On compactly supported solutions of a class of functional – differential equations, Modern problems of functions theory and functional analysis, (1980), Karaganda University Press, (in Russian)
[6] Derfel, G.; Dyn, N.; Levin, D., Generalized refinement equation and subdivision process, J. approx. theory, 80, 272-297, (1995) · Zbl 0823.45001
[7] Fox, L.; Mayers, D.F.; Ockendon, J.A.; Tayler, A.B., On a functional differential equation, J. inst. math. appl., 8, 271-307, (1971) · Zbl 0251.34045
[8] Ajello, W.G.; Freedman, H.I.; Wu, J., A model of stage structured population growth with density depended time delay, SIAM J. appl. math., 52, 855-869, (1992) · Zbl 0760.92018
[9] Buhmann, M.D.; Iserles, A., Stability of the discretized pantograph differential equation, Math. comput., 60, 575-589, (1993) · Zbl 0774.34057
[10] Gülsu, M.; Sezer, M., The approximate solution of high-order linear difference equation with variable coefficients in terms of Taylor polynomials, Appl. math. comput., 168, 1, 76-88, (2005) · Zbl 1082.65592
[11] Gülsu, M.; Sezer, M., A Taylor polynomial approach for solving differential-difference equations, J. comput. appl. math., 186, 2, 349-364, (2006) · Zbl 1078.65551
[12] Kanwal, R.P.; Liu, K.C., A Taylor expansion approach for solving integral equations, Int. J. math. educ. sci. technol., 20, 3, 411-414, (1989) · Zbl 0683.45001
[13] Nas, Ş.; Yalçınbaş, S.; Sezer, M., A Taylor polynomial approach for solving high-order linear Fredholm integro-differential equations, Int. J. math. educ. sci. technol., 31, 2, 213-225, (2000) · Zbl 1018.65152
[14] Sezer, M., A method fort he approximate solution of the second order linear differential equations in terms of Taylor polynomials, Int. J. math. educ. sci. technol., 27, 6, 821-834, (1996) · Zbl 0887.65084
[15] Sezer, M.; Gülsu, M., A new polynomial approach for solving difference and Fredholm integro-difference equation with mixed argument, Appl. math. comput., 171, 1, 332-344, (2005) · Zbl 1084.65133
[16] Yalçınbaş, S.; Sezer, M., The approximate solution of high-order linear volterra – fredholm integro-differential equations in terms of Taylor polynomials, Appl. math. comput., 112, 291-308, (2000) · Zbl 1023.65147
[17] Sezer, M.; Yalçınbaş, S.; Şahin, N., Approximate solution of multi-pantograph equation with variable coefficients, J. comput. appl. math., 214, 406-416, (2008) · Zbl 1135.65345
[18] Liu, M.Z.; Li, D., Properties of analytic solution and numerical solution and multi-pantograph equation, Appl. math. comput., 155, 853-871, (2004) · Zbl 1059.65060
[19] Evans, D.J.; Raslan, K.R., The adomain decomposition method for solving delay differential equation, Int. J. comput. math., 82, 1, 49-54, (2005) · Zbl 1069.65074
[20] Sezer, M.; Akyüz-Daşcıoğlu, A., A Taylor method for numerical solution of generalized pantograph equations with linear functional argument, 200, 217-225, (2007) · Zbl 1112.34063
[21] Rao, G.P.; Palanisamy, K.R., Walsh stretch matrices and functional differential equation, IEEE trans. autom. control, 27, 272-276, (1982) · Zbl 0488.34060
[22] Hwang, C.; Shih, Y.P., Laguerre series solution of a functional differential equation, Int. J. syst. sci., 13, 7, 783-788, (1982) · Zbl 0483.93056
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.