American Institute of Mathematical Sciences

Advanced
May  2010, 13(3): 685-708. doi: 10.3934/dcdsb.2010.13.685

A Legendre-Gauss collocation method for nonlinear delay differential equations

Zhong-Qing Wang 1, and  Li-Lian Wang 2,

1. 

Department of Mathematics, Shanghai Normal University, Shanghai, 200234
2. 

Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 637371

Received  November 2008 Revised  October 2009 Published  February 2010

In this paper, we introduce an efficient Legendre-Gauss collocation method for solving nonlinear delay differential equations with variable delay. We analyze the convergence of the single-step and multi-domain versions of the proposed method, and show that the scheme enjoys high order accuracy and can be implemented in a stable and efficient manner. We also make numerical comparison with other methods.
Keywords: nonlinear delay differential equations, spectral accuracy., Legendre-Gauss collocation methods.
Mathematics Subject Classification: Primary: 5L60, 65L05, 65Q05, 41A1.
Citation: Zhong-Qing Wang, Li-Lian Wang. A Legendre-Gauss collocation method for nonlinear delay differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 685-708. doi: 10.3934/dcdsb.2010.13.685
[1]

Lijun Yi, Zhongqing Wang. Legendre spectral collocation method for second-order nonlinear ordinary/partial differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 299-322. doi: 10.3934/dcdsb.2014.19.299
[2]

Jie Tang, Ziqing Xie, Zhimin Zhang. The long time behavior of a spectral collocation method for delay differential equations of pantograph type. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 797-819. doi: 10.3934/dcdsb.2013.18.797
[3]

Can Huang, Zhimin Zhang. The spectral collocation method for stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 667-679. doi: 10.3934/dcdsb.2013.18.667
[4]

Hui Liang, Hermann Brunner. Collocation methods for differential equations with piecewise linear delays. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1839-1857. doi: 10.3934/cpaa.2012.11.1839
[5]

Angelamaria Cardone, Dajana Conte, Beatrice Paternoster. Two-step collocation methods for fractional differential equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2709-2725. doi: 10.3934/dcdsb.2018088
[6]

Ben-Yu Guo, Zhong-Qing Wang. A spectral collocation method for solving initial value problems of first order ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1029-1054. doi: 10.3934/dcdsb.2010.14.1029
[7]

Wilhelm Schlag. Spectral theory and nonlinear partial differential equations: A survey. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 703-723. doi: 10.3934/dcds.2006.15.703
[8]

Mickaël D. Chekroun, Michael Ghil, Honghu Liu, Shouhong Wang. Low-dimensional Galerkin approximations of nonlinear delay differential equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4133-4177. doi: 10.3934/dcds.2016.36.4133
[9]

Seyedeh Marzieh Ghavidel, Wolfgang M. Ruess. Flow invariance for nonautonomous nonlinear partial differential delay equations. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2351-2369. doi: 10.3934/cpaa.2012.11.2351
[10]

Saroj Panigrahi, Rakhee Basu. Oscillation results for second order nonlinear neutral differential equations with delay. Conference Publications, 2015, 2015 (special) : 906-912. doi: 10.3934/proc.2015.0906
[11]

Cemil Tunç. Stability, boundedness and uniform boundedness of solutions of nonlinear delay differential equations. Conference Publications, 2011, 2011 (Special) : 1395-1403. doi: 10.3934/proc.2011.2011.1395
[12]

Anatoli F. Ivanov, Musa A. Mammadov. Global asymptotic stability in a class of nonlinear differential delay equations. Conference Publications, 2011, 2011 (Special) : 727-736. doi: 10.3934/proc.2011.2011.727
[13]

H.Thomas Banks, Danielle Robbins, Karyn L. Sutton. Theoretical foundations for traditional and generalized sensitivity functions for nonlinear delay differential equations. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1301-1333. doi: 10.3934/mbe.2013.10.1301
[14]

István Győri, Ferenc Hartung, Nahed A. Mohamady. Boundedness of positive solutions of a system of nonlinear delay differential equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 809-836. doi: 10.3934/dcdsb.2018044
[15]

Zhenyu Lu, Junhao Hu, Xuerong Mao. Stabilisation by delay feedback control for highly nonlinear hybrid stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4099-4116. doi: 10.3934/dcdsb.2019052
[16]

Tian Zhang, Huabin Chen, Chenggui Yuan, Tomás Caraballo. On the asymptotic behavior of highly nonlinear hybrid stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5355-5375. doi: 10.3934/dcdsb.2019062
[17]

Giuseppina di Blasio, Filomena Feo, Maria Rosaria Posteraro. Existence results for nonlinear elliptic equations related to Gauss measure in a limit case. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1497-1506. doi: 10.3934/cpaa.2008.7.1497
[18]

Qiumei Huang, Xiuxiu Xu, Hermann Brunner. Continuous Galerkin methods on quasi-geometric meshes for delay differential equations of pantograph type. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5423-5443. doi: 10.3934/dcds.2016039
[19]

Jie Shen, Li-Lian Wang. Laguerre and composite Legendre-Laguerre Dual-Petrov-Galerkin methods for third-order equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1381-1402. doi: 10.3934/dcdsb.2006.6.1381
[20]

Pham Huu Anh Ngoc. Stability of nonlinear differential systems with delay. Evolution Equations & Control Theory, 2015, 4 (4) : 493-505. doi: 10.3934/eect.2015.4.493

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (16)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences