A SPECTRAL METHOD FOR PANTOGRAPH-TYPE DELAY DIFFERENTIAL EQUATIONS AND ITS CONVERGENCE ANALYSIS *

We propose a novel numerical approach for delay differential equations with vanishing proportional delays based on spectral methods. A Legendre-collocation method is employed to obtain highly accurate numerical approximations to the exact solution. It is proved theoretically and demonstrated numerically that the proposed method converges exponentially provided that the data in the given pantograph delay differential equation are smooth.

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

[2]  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.

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

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

[5]  Giuseppe Mastroianni,et al.  Optimal systems of nodes for Lagrange interpolation on bounded intervals. A survey , 2001 .

[6]  A. Bellen.  Preservation of superconvergence in the numerical integration of delay differential equations with proportional delay , 2002 .

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

[8]  Jie Shen,et al.  Spectral and High-Order Methods with Applications , 2006 .

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

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

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

[12]  Xiang Xu,et al.  Accuracy Enhancement Using Spectral Postprocessing for Differential Equations and Integral Equations , 2008 .

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

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

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