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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 & 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.  Google Scholar [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.  Google Scholar [3] A. Bellen and M. Zennaro, "Numerical Methods for Delay Differential Equations," Clarendon Press, Oxford, 2003. doi: 10.1093/acprof:oso/9780198506546.001.0001.  Google Scholar [4] R. Bellman and K. L. Cooke, "Differential-Difference Equations," Academic Press, New York-London, 1963. doi: 10.1063/1.3050672.  Google Scholar [5] H. Brunner, The numerical solution of neutral Volterra integro-differential equations with delay arguments, Ann. Numer. Math., 1 (1994), 309-322.  Google Scholar [6] H. Brunner, "Collocation Methods for Volterra Integral and Related Functional Differential Equations," Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511543234.  Google Scholar [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.  Google Scholar [8] H. Brunner and W. K. Zhang, Primary discontinuities in solutions for delay integro-differential equations, Methods Appl. Anal., 6 (1999), 525-533.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional Differential Equations," Springer-Verlag, New York, 1993.  Google Scholar [15] V. Kolmanovskii and A. Myshkis, "Applied Theory of Functional-Differential Equations," Kluwer, Dordrecht, 1992.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [3] A. Bellen and M. Zennaro, "Numerical Methods for Delay Differential Equations," Clarendon Press, Oxford, 2003. doi: 10.1093/acprof:oso/9780198506546.001.0001.  Google Scholar [4] R. Bellman and K. L. Cooke, "Differential-Difference Equations," Academic Press, New York-London, 1963. doi: 10.1063/1.3050672.  Google Scholar [5] H. Brunner, The numerical solution of neutral Volterra integro-differential equations with delay arguments, Ann. Numer. Math., 1 (1994), 309-322.  Google Scholar [6] H. Brunner, "Collocation Methods for Volterra Integral and Related Functional Differential Equations," Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511543234.  Google Scholar [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.  Google Scholar [8] H. Brunner and W. K. Zhang, Primary discontinuities in solutions for delay integro-differential equations, Methods Appl. Anal., 6 (1999), 525-533.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional Differential Equations," Springer-Verlag, New York, 1993.  Google Scholar [15] V. Kolmanovskii and A. Myshkis, "Applied Theory of Functional-Differential Equations," Kluwer, Dordrecht, 1992.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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