Solving ODEs and DDEs with residual control. We first consider the numerical integration of ordinary differential equations (ODEs) with Runge-Kutta methods that have continuous extensions. For some methods of this kind we develop robust and inexpensive estimates of both the local error and the size of the residual. We then develop an effective program, ddesd, to solve delay differential equations (DDEs) with time- and state-dependent delays. To get reliable results for these difficult problems, the code estimates and controls the size of the residual. The user interface of dies’ makes it easy to formulate and solve DDEs, even those with complications like event location and restarts.

References in zbMATH (referenced in 20 articles , 1 standard article )

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  1. Bocchi, Edoardo: On the return to equilibrium problem for axisymmetric floating structures in shallow water (2020)
  2. Bica, Alexandru Mihai: Initial value problems with retarded argument solved by iterated quadratic splines (2016)
  3. Calzada, Juan A.; Obaya, Rafael; Sanz, Ana M.: Continuous separation for monotone skew-product semiflows: from theoretical to numerical results (2015)
  4. Khasi, M.; Ghoreishi, F.; Hadizadeh, M.: Numerical analysis of a high order method for state-dependent delay integral equations (2014)
  5. Lenz, Simon M.; Schlöder, Johannes P.; Bock, H. Georg: Numerical computation of derivatives in systems of delay differential equations (2014)
  6. Botchev, M. A.: A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. (2013)
  7. Trif, Damian: Direct operatorial tau method for pantograph-type equations (2012)
  8. Yagoub, Hemza; Nguyen-Ba, Truong; Vaillancourt, Rémi: Variable-step variable-order 3-stage Hermite-Birkhoff-Obrechkoff DDE solver of order 4 to 14 (2011)
  9. Enright, W. H.; Muir, P. H.: New interpolants for asymptotically correct defect control of BVODEs (2010)
  10. Enright, W. H.; Yan, Li: The reliability/cost trade-off for a class of ODE solvers (2010)
  11. Foley, Catherine; Mackey, Michael C.: Mathematical model for G-CSF administration after chemotherapy (2009)
  12. Shampine, L. F.: Dissipative approximations to neutral DDEs (2008)
  13. Li, Chengqing; Li, Shujun; Alvarez, Gonzalo; Chen, Guanrong; Lo, Kwok-Tung: Cryptanalysis of two chaotic encryption schemes based on circular bit shift and XOR operations (2007)
  14. Shampine, L. F.: Design of software for ODEs (2007)
  15. Feldstein, Alan; Neves, Kenneth W.; Thompson, Skip: Sharpness results for state dependent delay differential equations: An overview (2006)
  16. Ilie, Silvana; Corless, Robert M.; Reid, Greg: Numerical solutions of index-1 differential algebraic equations can be computed in polynomial time (2006)
  17. Shampine, L. F.; Gahinet, P.: Delay-differential-algebraic equations in control theory (2006)
  18. Thompson, S.; Shampine, L. F.: A friendly Fortran DDE solver (2006)
  19. Shampine, L. F.: Error estimation and control for ODEs (2005)
  20. Shampine, L. F.: Solving ODEs and DDEs with residual control (2005)