Spectral methods for pantograph-type differential and integral equations with multiple delays

Abstract

We analyze the convergence properties of the spectral method when used to approximate smooth solutions of delay differential or integral equations with two or more vanishing delays. It is shown that for the pantograph-type functional equations the spectral methods yield the familiar exponential order of convergence. Various numerical examples are used to illustrate these results.

Keywords

Delay differential equation / Volterra functional integral equation / multiple vanishing delays / Legendre spectral method / convergence analysis

References

[1] Ali I, Brunner H, Tang T. A spectral method for pantograph-type delay differential equations and its convergence analysis

[2] Bellen A., Zennaro M. Numerical Methods for Delay differentials Equations, 2003, Oxford: Oxford University Press

[3] Brunner H. Collocation Methods for Volterra Integral and Related Functional Equations, 2004, Cambridge: Cambridge University Press.

[4] Canuto C., Hussaini M., Quarteroni A., Zang T. A. Spectral Methods: Fundamentals in Single Domains, 2006, Berlin: Springer.

[5] Derfel G. A., Vogl F. On the asymptotics of solutions of a class of linear functional-differential equations. Europ J Appl Math, 1996, 7: 511-518.

[6] Guglielmi N. Short proofs and a counterexample for analytical and numerical stability of delay equations with infinite memory. IMA J Numer Anal, 2006, 26: 60-77.

[7] Guglielmi N., Hairer E. Implementing Radau IIA methods for stiff delay differential equations. Computing, 2001, 67: 1-12.

[8] Iserles A. On the generalized pantograph functional-differential equation. Europ J Appl Math, 1993, 4: 1-38.

[9] Iserles A., Liu Y. -K. On pantograph integro-differential equations. J Integral Equations Appl, 1994, 6: 213-237.

[10] Ishiwata E. On the attainable order of collocation methods for the neutral functionaldi differential equations with proportional delays. Computing, 2000, 64: 207-222.

[11] Li D., Liu M. Z. Asymptotic stability of numerical solution of pantograph delay differential equations. J Harbin Inst Tech, 1999, 31: 57-59.

[12] Liu M. Z., Li D. Properties of analytic solution and numerical solution of multipantograph equation. Appl Math Comput, 2004, 155: 853-871.

[13] Mastroianni G., Occorsio D. Optimal systems of nodes for Lagrange interpolation on bounded intervals: A survey. J Comput Appl Math, 2001, 134: 325-341.

[14] Qiu L., Mitsui T., Kuang J. X. The numerical stability of the θ-method for delay differential equations with many variable delays. J Comput Math, 1999, 17: 523-532.

[15] Shen J., Tang T. Spectral and High-Order Methods with Applications, 2006, Beijing: Science Press.

[16] Tang T., Xu X., Cheng J. On spectral methods for Volterra type integral equations and the convergence analysis. J Comput Math, 2008, 26: 825-837.

[17] Zhao J., Xu Y., Qiao Y. The attainable order of the collocation method for doublepantograph delay differential equation. Numer Math J Chinese Univ, 2005, 27: 297-308.