Modification and error analysis of an adaptive collocation method for solving delay differential equations

[1] I. Ali, H. Brunner, and T. Tang, A spectral method for pantograph–type delay differential equations and its convergence analysis, J. Comput. Math. 27 (2009), 254–265.

[2] U.M. Ascher and L.R. Petzold, The numerical solution of delay-differential algebraic equations of retarded and neutral type, SIAM J. Numer. Anal. 32 (1995), 1635–1657.

[3] C.T.H. Baker and C.A.H. Paul, Discontinuous solutions of neutral delay differential equations, Appl. Numer. Math. 56 (2006), 284–304.

[4] M.V. Barbarossa, K.P. Hadeler, and C. Kuttler, State-dependent neutral delay equations from population dynamics, J. Math. Bio. 69 (2014), no. 4, 1027–1056.

[5] M.V. Barbarossa, C. Kuttler, and J. Zinsl, Delay equations modeling the effects of phase-specific drugs and immunotherapy on proliferating tumor cells, Math. Biosc. Eng. 9 (2012), no. 2, 241–257.

[6] A. Bellen and N. Guglielmi, Solving neutral delay differential equations with state-dependent delays, J. Comput. Appl. Math. 229 (2009), 350–362.

[7] H. Brunner, H. Xie, and R. Zhang, Analysis of collocation solutions for a class of functional equations with vanishing delays, IMA J. Numer. Anal. 31 (2011), no. 2, 698–718.

[8] A. Bellen, Preservation of superconvergence in the numerical integration of delay differential equations with proportional delay, IMA J. Numer. Anal. 22 (2002), no. 4, 529-536.

[9] A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations, Oxford University Press, Oxford, 2003.

[10] C. Canuto, M.Y. Hussaini, A. Quarteroni, and T.A. Zang, Spectral Methods: Fundamentals in Single Domains, Springer–Verlag, Berlin, 2006.

[11] M.J. Crawley, Natural Enemies: The Population Biology of Predators, Parasites and Disease, Blackwell Scientific Publications, Oxford, 1992.

[12] R.D. Driver, Existence and continuous dependence of solutions of a neutral functional-differential equation, Rational, Mech. Anal. 19 (1965), no. 2, 149–186.

[13] L.E. Elsgolts and S.B. Norkin, Introduction to the Theory and Application of Differential Equations with Deviating Arguments, Academic Press, New York, 1973.

[14] W.H. Enright, Software for ordinary and delay differential equations: Accurate discrete approximate solutions are not enough, Appl. Numer. Math. 56 (2006), no. 3-4, 459–471.

[15] W.H. Enright and H. Hayashi, Convergence analysis of the solution of retarded and neutral delay differential equations by continuous numerical methods, SIAM J. Numer. Anal. 35 (1998), no. 2, 572–585.

[16] L.J. Grimm, Existence and continuous dependence for a class of nonlinear neutral differential equations, Proc. Amer. Math. Soc. 29 (1971), no. 3, 467–473.

[17] N. Guglielmi and E. Hairer, Implementing Radau IIA methods for stiff delay differential equations, Computing 67 (2001), 1–12.

[18] H. Hayashi, Numerical solution of retarded and neutral delay differential equations using continuous Runge-Kutta methods, Ph.D. thesis, Department of Computer Science, University of Toronto, ON, Canada, 1996.

[19] K. Ito, H.T. Tran, and A. Manitius, A fully-discrete spectral method for delay-differential equations, SIAM J. Numer. Anal. 28 (1991), 1121–1140.

[20] Z. Jackiewicz, Existence and uniqueness of solutions of neutral delay-differential equations with state dependent delays, Funk. Ekvac. 30 (1987), 9–17.

[21] Z. Jackiewicz and E. Lo, Numerical solution of neutral functional differential equations by Adams methods in divided difference form, J. Comput. Appl. Math. 189 (2006), 592–605.

[22] J.C. Jimenez, L.M. Pedroso, F. Carbonell, and V. Hernandez, Local linearization method for numerical integration of delay differential equations, SIAM J. Numer. Anal. 44 (2006), no. 6, 2584–2609.

[23] H. Kocak and A. Yildirim, Series solution for a delay differential equation arising in electrodynamics, Int. J. Numer. Method. Biomed. Engin. 25 (2009), no. 11, 1084–1096.

[24] V.B. Kolmanovski and A. Myshkis, Introduction to the Theory and Applications of Functional-Differential Equations, Kluwer Academic, Dordrecht, 1999.

[25] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Mathematics in Science and Engineering, vol. 191, Academic Press, Boston, 1993.

[26] Y. Kuang and A. Feldstein, Boundedness of solutions of a nonlinear nonautonomous neutral delay equation, J. Math. Anal. Appl. 156 (1991), 293–304.

[27] Y. Liu, Numerical solution of implicit neutral functional-differential equations, SIAM J. Numer. Anal. 36 (1999), 516–528.

[28] M. Maleki, I. Hashim, and S. Abbasbandy, Solution of time-varying delay systems using an adaptive collocation method, Appl. Math. Comput. 219 (2012), 1434–1448.

[29] M. Maleki and I. Hashim, Adaptive pseudospectral methods for solving constrained linear and nonlinear time-delay optimal control problems, J. Franklin. Inst. 351 (2014), 811–839.

[30] M. Mohr, M.V. Barbarossa, and C. Kuttler, Predator-prey interactions, age structures and delay equations, Math. Model. Nat. Phenom. 9 (2014), no. 1, 92–107.

[31] K.W. Neves and S. Thompson, Software for the numerical solution of system of functional differential equations with state-dependent delays, Appl. Numer. Math. 9 (1992), 385–401.

[32] H.J. Oberle and H.J. Pesch, Numerical treatment of delay differential Equations by Hermite interpolation, Numer. Math. 37 (1981), 235–255.

[33] C.A.H. Paul, Developing a delay differential equation solver, Appl. Numer. Math. 9 (1992), 403–414.

[34] Z. Tan and C. Zhang, Implicit–explicit one-leg methods for nonlinear stiff neutral equations, Appl. Math. Comput. 335 (2018), 196–210.

[35] Z. Wang and B. Guo, Legendre-Gauss–Radau collocation method for solving initial value problems of first order ordinary differential equations, J. Sci. Comput. 52 (2012), no. 1, 226-225.

[36] Z.Q. Wang and L.L. Wang, A Legendre-Gauss collocation method for nonlinear delay differential equations, Discrete Contin. Dyn. Syst. Ser. B 13 (2010), 685–708.

[37] H. ZivariPiran and H. Enright, An efficient unified approach for the numerical solution of delay differential equations, Numer. Algor. 53 (2010), no. 2-3, 397–417.