The long time behavior of a spectral collocation method for delay differential equations of pantograph type

In this paper, we propose an efficient numerical method for delay differential equations with vanishing proportional delay qt (0 < q < 1). The algorithm is a mixture of the Legendre-Gauss collocation method and domain decomposition. It has global convergence and spectral accuracy provided that the data in the given pantograph delay differential equation are sufficiently smooth. Numerical results demonstrate the spectral accuracy of this approach and coincide well with theoretical analysis.

[1]  Ben-yu Guo,et al.  Legendre–Gauss collocation methods for ordinary differential equations , 2009, Adv. Comput. Math..

[2]  J. McLeod,et al.  The Functional-Differential Equation y'(x) = ay(lambda x) + by(x). , 1971 .

[3]  J. B. McLeod,et al.  The functional-differential equation $y'\left( x \right) = ay\left( {\lambda x} \right) + by\left( x \right)$ , 1971 .

[4]  Arieh Iserles,et al.  On the generalized pantograph functional-differential equation , 1993, European Journal of Applied Mathematics.

[5]  Tang,et al.  ON SPECTRAL METHODS FOR VOLTERRA INTEGRAL EQUATIONS AND THE CONVERGENCE ANALYSIS , 2008 .

[6]  Hehu Xie,et al.  Discontinuous Galerkin Methods for Delay Differential Equations of Pantograph Type , 2010, SIAM J. Numer. Anal..

[7]  H. Brunner,et al.  A SPECTRAL METHOD FOR PANTOGRAPH-TYPE DELAY DIFFERENTIAL EQUATIONS AND ITS CONVERGENCE ANALYSIS * , 2009 .

[8]  A. Bellen,et al.  Numerical methods for delay differential equations , 2003 .

[9]  Qiya Hu,et al.  Optimal Superconvergence Results for Delay Integro-Differential Equations of Pantograph Type , 2007, SIAM J. Numer. Anal..

[10]  Arieh Iserles,et al.  On nonlinear delay differential equations , 1994 .

[11]  Xiaojun Zhou,et al.  Spectral Petrov-Galerkin Methods for the Second Kind Volterra Type Integro-Differential Equations , 2011 .

[12]  T. A. Zang,et al.  Spectral Methods: Fundamentals in Single Domains , 2010 .

[13]  L. Fox,et al.  On a Functional Differential Equation , 1971 .

[14]  Sigal Gottlieb,et al.  Spectral Methods , 2019, Numerical Methods for Diffusion Phenomena in Building Physics.

[15]  Zhongqing Wang,et al.  A LEGENDRE-GAUSS COLLOCATION METHOD FOR NONLINEAR DELAY DIFFERENTIAL EQUATIONS , 2010 .

[16]  Hermann Brunner,et al.  Current work and open problems in the numerical analysis of Volterra functional equations with vanishing delays , 2009 .

[17]  Hermann Brunner,et al.  Superconvergence in Collocation Methods on Quasi-Graded Meshes for Functional Differential Equations with Vanishing Delays , 2006 .

[18]  Ishtiaq Ali,et al.  Spectral methods for pantograph-type differential and integral equations with multiple delays , 2009 .

[19]  Ivan P. Gavrilyuk,et al.  Collocation methods for Volterra integral and related functional equations , 2006, Math. Comput..

[20]  Jack Carr,et al.  13.—The Functional Differential Equation y′(x) = ay(λx) + by(x) , 1976, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[21]  Ben-Yu Guo,et al.  Legendre--Gauss collocation method for initial value problems of second order ordinary differential equations , 2009 .