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September  2012, 11(5): 1839-1857. doi: 10.3934/cpaa.2012.11.1839

Collocation methods for differential equations with piecewise linear delays

1. 

School of Mathematical Sciences, Heilongjiang University, Harbin, Heilongjiang, China

2. 

Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong SAR, China

Received  March 2011 Revised  September 2011 Published  March 2012

After analyzing the regularity of solutions to delay differential equations (DDEs) with piecewise continuous (linear) non-vanishing delays, we describe collocation schemes using continuous piecewise polynomials for their numerical solution. We show that for carefully designed meshes these collocation solutions exhibit optimal orders of global and local superconvergence analogous to the ones for DDEs with constant delays. Numerical experiments illustrate the theoretical superconvergence results.
Citation: Hui Liang, Hermann Brunner. Collocation methods for differential equations with piecewise linear delays. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1839-1857. doi: 10.3934/cpaa.2012.11.1839
References:
[1]

A. Bellen, One-step collocation for delay differential equations, J. Comput. Appl. Math., 10 (1984), 275-283. doi: 10.1016/0377-0427(84)90039-6.

[2]

A. Bellen, S. Maset, M. Zennaro and N. Guglielmi, Recent trends in the numerical solution of retarded functional differential equations, Acta Numer., 18 (2009), 1-110. doi: 10.1017/S0962492906390010.

[3]

A. Bellen and M. Zennaro, "Numerical Methods for Delay Differential Equations," Clarendon Press, Oxford, 2003. doi: 10.1093/acprof:oso/9780198506546.001.0001.

[4]

R. Bellman and K. L. Cooke, "Differential-Difference Equations," Academic Press, New York-London, 1963. doi: 10.1063/1.3050672.

[5]

H. Brunner, The numerical solution of neutral Volterra integro-differential equations with delay arguments, Ann. Numer. Math., 1 (1994), 309-322.

[6]

H. Brunner, "Collocation Methods for Volterra Integral and Related Functional Differential Equations," Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511543234.

[7]

H. Brunner and S. Maset, Time transformations for delay differential equations, Discrete Contin. Dyn. Syst., 25 (2009), 751-775. doi: 10.3934/dcds.2009.25.751.

[8]

H. Brunner and W. K. Zhang, Primary discontinuities in solutions for delay integro-differential equations, Methods Appl. Anal., 6 (1999), 525-533.

[9]

K. L. Cooke and J. Wiener, A survey of differential equations with piecewise continuous arguments, in "Delay Differential Equations and Dynamical Systems (Claremont, CA, 1990)" (eds. S. Busenberg and M. Martelli), 1-15, Lecture Notes Math., 1475, Springer-Verlag, Berlin, (1991). doi: 10.1007/BFb0083475.

[10]

L. E. El'sgol'ts and S. B. Norkin, "Introduction to the Theory and Application of Differential Equations with Deviating Arguments," Academic Press, New York, 1973.

[11]

N. Guglielmi and E. Hairer, Computing breaking points in implicit delay differential equations, Adv. Comput. Math., 29 (2008), 229-247. doi: 10.1007/s10444-007-9044-5.

[12]

I. Györi, F. Hartung and J. Turi, Numerical approximations for a class of differential equations with time- and state-dependent delays, Appl. Math. Lett., 8 (1995), 19-24. doi: 10.1016/0893-9659(95)00079-6.

[13]

I. Györi and F. Hartung, On numerical approximation using differential equations with piecewise-constant arguments, Period. Math. Hungar., 56 (2008), 55-69. doi: 10.1007/s10998-008-5055-5.

[14]

J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional Differential Equations," Springer-Verlag, New York, 1993.

[15]

V. Kolmanovskii and A. Myshkis, "Applied Theory of Functional-Differential Equations," Kluwer, Dordrecht, 1992.

[16]

H. Liang, M. Z. Liu and W. J. Lv, Stability of $\theta $-schemes in the numerical solution of a partial differential equation with piecewise continuous arguments, Period. Appl. Math. Lett., 23 (2010), 198-206. doi: 10.1016/j.aml.2009.09.012.

[17]

M. Z. Liu, M. H. Song and Z. W. Yang, Stability of Runge-Kutta methods in the numerical solution of equation $u'(t)=au(t)+a_0u([t])$, J. Comput. Appl. Math., 166, (2004), 361-370. doi: 10.1016/j.cam.2003.04.002.

[18]

Z. W. Yang, M. Z. Liu and M. H. Song, Stability of Runge-Kutta methods in the numerical solution of equation $u'(t) = au(t)+a_0 u([t])+a_1 u([t-1])$, Appl. Math. Comput., 162 (2005), 37-50. doi: 10.1016/j.amc.2003.12.081.

show all references

References:
[1]

A. Bellen, One-step collocation for delay differential equations, J. Comput. Appl. Math., 10 (1984), 275-283. doi: 10.1016/0377-0427(84)90039-6.

[2]

A. Bellen, S. Maset, M. Zennaro and N. Guglielmi, Recent trends in the numerical solution of retarded functional differential equations, Acta Numer., 18 (2009), 1-110. doi: 10.1017/S0962492906390010.

[3]

A. Bellen and M. Zennaro, "Numerical Methods for Delay Differential Equations," Clarendon Press, Oxford, 2003. doi: 10.1093/acprof:oso/9780198506546.001.0001.

[4]

R. Bellman and K. L. Cooke, "Differential-Difference Equations," Academic Press, New York-London, 1963. doi: 10.1063/1.3050672.

[5]

H. Brunner, The numerical solution of neutral Volterra integro-differential equations with delay arguments, Ann. Numer. Math., 1 (1994), 309-322.

[6]

H. Brunner, "Collocation Methods for Volterra Integral and Related Functional Differential Equations," Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511543234.

[7]

H. Brunner and S. Maset, Time transformations for delay differential equations, Discrete Contin. Dyn. Syst., 25 (2009), 751-775. doi: 10.3934/dcds.2009.25.751.

[8]

H. Brunner and W. K. Zhang, Primary discontinuities in solutions for delay integro-differential equations, Methods Appl. Anal., 6 (1999), 525-533.

[9]

K. L. Cooke and J. Wiener, A survey of differential equations with piecewise continuous arguments, in "Delay Differential Equations and Dynamical Systems (Claremont, CA, 1990)" (eds. S. Busenberg and M. Martelli), 1-15, Lecture Notes Math., 1475, Springer-Verlag, Berlin, (1991). doi: 10.1007/BFb0083475.

[10]

L. E. El'sgol'ts and S. B. Norkin, "Introduction to the Theory and Application of Differential Equations with Deviating Arguments," Academic Press, New York, 1973.

[11]

N. Guglielmi and E. Hairer, Computing breaking points in implicit delay differential equations, Adv. Comput. Math., 29 (2008), 229-247. doi: 10.1007/s10444-007-9044-5.

[12]

I. Györi, F. Hartung and J. Turi, Numerical approximations for a class of differential equations with time- and state-dependent delays, Appl. Math. Lett., 8 (1995), 19-24. doi: 10.1016/0893-9659(95)00079-6.

[13]

I. Györi and F. Hartung, On numerical approximation using differential equations with piecewise-constant arguments, Period. Math. Hungar., 56 (2008), 55-69. doi: 10.1007/s10998-008-5055-5.

[14]

J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional Differential Equations," Springer-Verlag, New York, 1993.

[15]

V. Kolmanovskii and A. Myshkis, "Applied Theory of Functional-Differential Equations," Kluwer, Dordrecht, 1992.

[16]

H. Liang, M. Z. Liu and W. J. Lv, Stability of $\theta $-schemes in the numerical solution of a partial differential equation with piecewise continuous arguments, Period. Appl. Math. Lett., 23 (2010), 198-206. doi: 10.1016/j.aml.2009.09.012.

[17]

M. Z. Liu, M. H. Song and Z. W. Yang, Stability of Runge-Kutta methods in the numerical solution of equation $u'(t)=au(t)+a_0u([t])$, J. Comput. Appl. Math., 166, (2004), 361-370. doi: 10.1016/j.cam.2003.04.002.

[18]

Z. W. Yang, M. Z. Liu and M. H. Song, Stability of Runge-Kutta methods in the numerical solution of equation $u'(t) = au(t)+a_0 u([t])+a_1 u([t-1])$, Appl. Math. Comput., 162 (2005), 37-50. doi: 10.1016/j.amc.2003.12.081.

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