On the Numerical Solution of Generalized Pantograph Equation

Absrtact: In this study, a numerical algorithm for solving a generalization of a functional differential equation known as the pantograph equation is presented. Firstly, the proposed algorithm produces an approximate polynomial solution as a power series for the problem. Then, we transform the obtained power series into Pade series form to obtain an approximate polynomial of an arbitrary order for solving pantograph equation. The structure and advantages of using of the proposed method are presented. To show the validity and applicability of the numerical method some linear and nonlinear experiments are examined. The results reveal the high accuracy and efficiency of the proposed method. 2010 mathematics subject classification: 34K50 • 34Kxx • 41A21

[1]  Nicola Guglielmi,et al.  Asymptotic stability properties of T-methods for the pantographs equation , 1997 .

[2]  R. D. Driver,et al.  Ordinary and Delay Differential Equations , 1977 .

[3]  D. Li,et al.  Runge-Kutta methods for the multi-pantograph delay equation , 2005, Appl. Math. Comput..

[4]  Arieh Iserles,et al.  The Pantograph Equation in the Complex Plane , 1997 .

[5]  Emiko Ishiwata,et al.  On the attainable order of collocation methods for pantograph integro-differential equations , 2003 .

[6]  Mustafa Bayram,et al.  Numerical solutions of chemical differential-algebraic equations , 2003, Appl. Math. Comput..

[7]  M. Z. Liu,et al.  Properties of analytic solution and numerical solution of multi-pantograph equation , 2004, Appl. Math. Comput..

[8]  Lawrence Stark.  Stability and Oscillations , 1968 .

[9]  V. Kolmanovskii,et al.  Applied Theory of Functional Differential Equations , 1992 .

[10]  K. Gopalsamy.  Stability and Oscillations in Delay Differential Equations of Population Dynamics , 1992 .

[11]  John Ockendon,et al.  The dynamics of a current collection system for an electric locomotive , 1971, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[12]  Mehdi Dehghan,et al.  Variational iteration method for solving a generalized pantograph equation , 2009, Comput. Math. Appl..

[13]  V. Kolmanovskii,et al.  Stability of Functional Differential Equations , 1986 .

[14]  Elçin Yusufoglu,et al.  An efficient algorithm for solving generalized pantograph equations with linear functional argument , 2010, Appl. Math. Comput..

[15]  Y. Kuang.  Delay Differential Equations: With Applications in Population Dynamics , 2012 .

[16]  S. Karimi Vanani,et al.  Multiquadric approximation scheme on the numerical solution of delay differential systems of neutral type , 2009, Math. Comput. Model..

[17]  David J. Evans,et al.  The Adomian decomposition method for solving delay differential equation , 2005, Int. J. Comput. Math..

[18]  B. Asner.  Ordinary and Delay Differential Equations (R. D. Driver) , 1978 .

[19]  Mustafa Bayram,et al.  Numerical method to solve chemical differential‐algebraic equations , 2002 .

[20]  Mehmet Sezer,et al.  A Taylor method for numerical solution of generalized pantograph equations with linear functional argument , 2007 .