A new computational approach for the solutions of generalized pantograph-delay differential equations
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A new computational approach for the solutions of generalized pantograph-delay differential equations Lie-jun Xie1 · Cai-lian Zhou1 · Song Xu1
Received: 20 July 2016 / Revised: 29 December 2016 / Accepted: 23 January 2017 / Published online: 10 February 2017 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2017
Abstract In this study, a new computational approach is presented to solve the generalized pantograph-delay differential equations (PDDEs). The solutions obtained by our scheme represent by a linear combination of a special kind of basis functions, and can be deduced in a straightforward manner. Firstly, using the least squares approximation method and the Lagrange-multiplier method, the given PDDE is converted to a linear system of algebraic equations, and those unknown coefficients of the solution of the problem are determined by solving this linear system. Secondly, a PDDE related to the error function of the approximate solution is constructed based on the residual error function technique, and error estimation is presented for the suggested method. The convergence of the approximate solution is proved. Several numerical examples are given to demonstrate the accuracy and efficiency. Comparisons are made between the proposed method and other existing methods. Keywords Pantograph equation · Delay equation · Least squares approximation method · Lagrange-multiplier method · Residual error function technique Mathematics Subject Classification 34K28 · 65D15 · 41A10 · 65Q10
1 Introduction Delay differential equations have a wide application in science and engineering. The pantograph equation is one of the most important kinds of delay differential equations, and plays
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Lie-jun Xie [email protected]; [email protected] Cai-lian Zhou [email protected] Song Xu [email protected]
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Department of Mathematics, Faculty of Science, Ningbo University, Ningbo 315211, Zhejiang, People’s Republic of China
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A new computational approach for the solutions. . .
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an important role in explaining various problems in engineering and sciences such as biology, economy, control and electrodynamics. The name pantograph originated from the work of Ockendon and Tayler on the collection of current by the pantograph head of an electric locomotive (Ockendon and Tayler 1971). The PDDEs have been studied by many researchers analytically via using the different techniques. Buhmann and Iserles (1993) analyzed the stability of the numerical solution of discretized pantograph differential equation with trapezoidal rule discretizations. Li and Liu (2000) studied the structure of the exact solution sets of multi-pantograph delay differential equations and proved the existence and uniqueness of an exact solution. In Bellen et al. (1997), the behavior of the class of θ -methods was analyzed with relevance for the constant coefficient version of kind of pantograph equation. Furthermore, Guglielmi and Zennaro (2003) discussed the stability of one-leg θ -methods for the solution of the same pantograph eq
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