##
**Current work and open problems in the numerical analysis of Volterra functional equations with vanishing delays.**
*(English)*
Zbl 1396.65161

Summary: The aims of this paper are (i) to present a survey of recent advances in the analysis of superconvergence of collocation solutions for linear Volterra-type functional integral and integro-differential equations with delay functions \(\theta(t)\) vanishing at the initial point of the interval of integration (with \(\theta(t) = qt\) \((0 < q < 1\), \(t \geq 0\)) being an important special case), and (ii) to point, by means of a list of open problems, to areas in the numerical analysis of such Volterra functional equations where more research needs to be carried out.

### MSC:

65R20 | Numerical methods for integral equations |

### Keywords:

Volterra functional integral and integro-differential equation; vanishing delay; pantograph equation; collocation solution; optimal order of superconvergence
Full Text:
DOI

### References:

[1] | Ali I, Brunner H, Tang T. A spectral method for pantograph-type delay differential equations and its convergence analysis. J Comput Math (in press) · Zbl 1212.65308 |

[2] | Ali I, Brunner H, Tang T. Spectral methods for pantograph differential and integral equations with multiple delays (to appear) · Zbl 1396.65107 |

[3] | Andreoli G. Sulle equazioni integrali. Rend Circ Mat Palermo, 1914, 37: 76–112 · JFM 45.0539.03 |

[4] | Bellen A. Preservation of superconvergence in the numerical integration of delay differential equations with proportional delay. IMA J Numer Anal, 2002, 22: 529–536 · Zbl 1031.65089 |

[5] | Bellen A, Brunner H, Maset S, Torelli L. Superconvergence in collocation methods on quasi-graded meshes for functional differential equations with vanishing delays. BIT, 2006, 46: 229–247 · Zbl 1109.65112 |

[6] | Bellen A, Guglielmi N, Torelli L. Asymptotic stability properties of {\(\theta\)}-methods for the pantograph equation. Appl Numer Math, 1997, 24: 275–293 · Zbl 0878.65064 |

[7] | Bellen A, Zennaro M. Numerical Methods for Delay Differential Equations. Oxford: Oxford University Press, 2003 · Zbl 1038.65058 |

[8] | Brunner H. On the discretization of differential and Volterra integral equations with variable delay. BIT, 1997, 37: 1–12 · Zbl 0873.65126 |

[9] | Brunner H. The numerical analysis of functional integral and integro-differential equations of Volterra type. Acta Numerica, 2004, 55–145 · Zbl 1118.65391 |

[10] | Brunner H. Collocation Methods for Volterra Integral and Related Functional Equations. Cambridge Monographs on Applied and Computational Mathematics, Vol 15. Cambridge: Cambridge University Press, 2004 · Zbl 1059.65122 |

[11] | Brunner H. Recent advances in the numerical analysis of Volterra functional differential equations with variable delays. J Comput Appl Math, 2008 (in press) |

[12] | Brunner H. On the regularity of solutions for Volterra functional equations with weakly singular kernels and vanishing delays (to appear) |

[13] | Brunner H. Collocation methods for pantograph-type Volterra functional equations with multiple delays. Comput Methods Appl Math, 2008 (in press) · Zbl 1153.65123 |

[14] | Brunner H, Hu Q -Y. Superconvergence of iterated collocation solutions for Volterra integral equations with variable delays. SIAM J Numer Anal, 2005, 43: 1934–1949 · Zbl 1103.65136 |

[15] | Brunner H, Hu Q -Y. Optimal superconvergence results for delay integro-differential equations of pantograph type. SIAM J Numer Anal, 2007, 45: 986–1004 · Zbl 1144.65083 |

[16] | Brunner H, Maset S. Time transformations for delay differential equations. Discrete Contin Dyn Syst (in press) · Zbl 1187.34093 |

[17] | Brunner H, Maset S. Time transformations for state-dependent delay differential equations. Preprint, 2008 · Zbl 1194.34135 |

[18] | Brunner H, Pedas A, Vainikko G. The piecewise polynomial collocation method for nonlinear weakly singular Volterra equations. Math Comp, 1999, 68: 1079–1095 · Zbl 0941.65136 |

[19] | Brunner H, Pedas A, Vainikko G. Piecewise polynomial collocation methods for linear Volterra integro-differential equations with weakly singular kernels. SIAM J Numer Anal, 2001, 39: 957–982 · Zbl 0998.65134 |

[20] | Buhmann M D, Iserles A. Numerical analysis of functional equations with a variable delay. In: Griffiths D F, Watson G A, eds. Numerical Analysis (Dundee 1991). Pitman Res Notes Math Ser, 260. Harlow: Longman Scientific & Technical, 1992, 17–33 · Zbl 0795.65048 |

[21] | Buhmann M D, Iserles A. On the dynamics of a discretized neutral equation. IMA J Numer Anal, 1992, 12: 339–363 · Zbl 0759.65056 |

[22] | Buhmann M D, Iserles A. Stability of the discretized pantograph differential equation. Math Comp, 1993, 60: 575–589 · Zbl 0774.34057 |

[23] | Buhmann M, Iserles A, Nørsett S P. Runge-Kutta methods for neutral differential equations. In: Agarwal R P, ed. Contributions in Numerical Mathematics (Singapore 1993). River Edge: World Scientific Publ, 1993, 85–98 · Zbl 0834.65061 |

[24] | Carvalho L A V, Cooke K L. Collapsible backward continuation and numerical approximations in a functional differential equation. J Differential Equations, 1998, 143: 96–109 · Zbl 0911.34063 |

[25] | Li G Chambers. Some properties of the functional equation {\(\phi\)}(x) = f(x)+ 0 {\(\lambda\)}x g(x, y, {\(\phi\)}(y))dy. Internat J Math Math Sci, 1990, 14: 27–44 |

[26] | Denisov A M, Korovin S V. On Volterra’s integral equation of the first kind. Moscow Univ Comput Math Cybernet, 1992, 3: 19–24 · Zbl 0785.45001 |

[27] | Denisov A M, Lorenzi A. On a special Volterra integral equation of the first kind. Boll Un Mat Ital B (7), 1995, 9: 443–457 · Zbl 0846.45001 |

[28] | Denisov A M, Lorenzi A. Existence results and regularization techniques for severely ill-posed integrofunctional equations. Boll Un Mat Ital B (7), 1997, 11: 713–732 · Zbl 0882.45013 |

[29] | Feldstein A, Iserles A, Levin D. Embedding of delay equations into an infinitedimensional ODE system. J Differential Equations, 1995, 117: 127–150 · Zbl 0817.34045 |

[30] | Feldstein A, Liu Y K. On neutral functional-differential equations with variable time delays. Math Proc Cambridge Phil Soc, 1998, 124: 371–384 · Zbl 0913.34067 |

[31] | Fox L, Mayers D F, Ockendon J R, Tayler A B. On a functional differential equation. J Inst Math Appl, 1971, 8: 271–307 · Zbl 0251.34045 |

[32] | Frederickson P O. Dirichlet solutions for certain functional differential equations. In: Urabe M, ed. Japan-United States Seminar on Ordinary Differential and Functional Equations (Kyoto 1971). Lecture Notes in Math, Vol 243. Berlin-Heidelberg: Springer-Verlag, 1971, 249–251 |

[33] | Frederickson P O. Global solutions to certain nonlinear functional differential equations. J Math Anal Appl, 1971, 33: 355–358 · Zbl 0203.15001 |

[34] | Gan S Q. Exact and discretized dissipativity of the pantograph equation. J Comput Math, 2007, 25: 81–88 · Zbl 1142.65375 |

[35] | Guglielmi N. Short proofs and a counterexample for analytical and numerical stability of delay equations with infinite memory. IMA J Numer Anal, 2006, 26: 60–77 · Zbl 1118.65090 |

[36] | Guglielmi N, Zennaro M. Stability of one-leg {\(\theta\)}-methods for the variable coefficient pantograph equation on the quasi-geometric mesh. IMA J Numer Anal, 2003, 23: 421–438 · Zbl 1055.65094 |

[37] | Huang C M, Vandewalle S. Discretized stability and error growth of the nonautonomous pantograph equation. SIAM J Numer Anal, 2005, 42: 2020–2042 · Zbl 1080.65068 |

[38] | Iserles A. On the generalized pantograph functional differential equation. Europ J Appl Math, 1993, 4: 1–38 · Zbl 0767.34054 |

[39] | Iserles A. Numerical analysis of delay differential equations with variable delays. Ann Numer Math, 1994, 1: 133–152 · Zbl 0828.65083 |

[40] | Iserles A. On nonlinear delay-differential equations. Trans Amer Math Soc, 1994, 344: 441–477 · Zbl 0804.34065 |

[41] | Iserles A. Beyond the classical theory of computational ordinary differential equations. In: Duff I S, Watson G A, eds. The State of the Art in Numerical Analysis (York 1996). Oxford: Clarendon Press, 1997, 171–192 · Zbl 0886.65073 |

[42] | Iserles A, Liu Y K. On pantograph integro-differential equations. J Integral Equations Appl, 1994, 6: 213–237 · Zbl 0816.45005 |

[43] | Iserles A, Terjéki J. Stability and asymptotic stability of functional-differential equations. J London Math Soc (2), 1995, 51: 559–572 · Zbl 0832.34080 |

[44] | Ishiwata E. On the attainable order of collocation methods for the neutral functionaldifferential equations with proportional delays. Computing, 2000, 64: 207–222 · Zbl 0955.65098 |

[45] | Ishiwata E, Muroya Y. Rational approximation method for delay differential equations with proportional delay. Appl Math Comput, 2007, 187: 741–747 · Zbl 1117.65105 |

[46] | Jackiewicz Z. Asymptotic stability analysis of {\(\theta\)}-methods for functional differential equations. Numer Math, 1984, 43: 389–396 · Zbl 0557.65047 |

[47] | Kato T, McLeod J B. The functional-differential equation y’(x) = ay({\(\lambda\)}x) + by(x). Bull Amer Math Soc, 1971, 77: 891–937 · Zbl 0236.34064 |

[48] | Koto T. Stability of Runge-Kutta methods for the generalized pantograph equation. Numer Math, 1999, 84: 233–247 · Zbl 0943.65091 |

[49] | Lalesco T. Sur l’équation de Volterra. J de Math (6), 1908, 4: 309–317 · JFM 39.0417.01 |

[50] | Lalesco T. Sur une équation intégrale du type Volterra. C R Acad Sci Paris, 1911, 152: 579–580 · JFM 42.0379.02 |

[51] | Li D, Liu M Z. Asymptotic stability of numerical solution of pantograph delay differential equations. J Harbin Inst Tech, 1999, 31: 57–59 (in Chinese) · Zbl 0970.65548 |

[52] | Li D, Liu M Z. The properties of exact solution of multi-pantograph delay differential equation. J Harbin Inst Tech, 2000, 32: 1–3 (in Chinese) · Zbl 1087.34536 |

[53] | Liang J, Liu M Z. Stability of numerical solutions to pantograph delay systems. J Harbin Inst Tech, 1996, 28: 21–26 (in Chinese) · Zbl 0970.65550 |

[54] | Liang J, Liu M Z. Numerical stability of {\(\theta\)}-methods for pantograph delay differential equations. J Numer Methods Comput Appl, 1996, 12: 271–278 (in Chinese) · Zbl 0852.73037 |

[55] | Liang J, Qiu S, Liu M Z. The stability of {\(\theta\)}-methods for pantograph delay differential equations. Numer Math J Chinese Univ (Engl Ser), 1996, 5: 80–85 · Zbl 0869.65052 |

[56] | Liu M Z, Li D. Properties of analytic solution and numerical solution of multipantograph equation. Appl Math Comput, 2004, 155: 853–871 · Zbl 1059.65060 |

[57] | Liu M Z, Wang Z, Hu G. Asymptotic stability of numerical methods with constant stepsize for pantograph equations. BIT, 2005, 45: 743–759 · Zbl 1095.65075 |

[58] | Liu M Z, Yang Z W, Xu Y. The stability of modified Runge-Kutta methods for the pantograph equation. Math Comp, 2006, 75: 1201–1215 · Zbl 1094.65075 |

[59] | Liu Y K. Stability analysis of {\(\theta\)}-methods for neutral functional-differential equations. Numer Math, 1995, 70: 473–485 · Zbl 0824.65081 |

[60] | Liu Y K. The linear q-difference equation y(x) = ay(qx) + f(x). Appl Math Lett, 1995, 8: 15–18 · Zbl 0820.65091 |

[61] | Liu Y K. On {\(\theta\)}-methods for delay differential equations with infinite lag. J Comput Appl Math, 1996, 71: 177–190 · Zbl 0853.65076 |

[62] | Liu Y K. Asymptotic behaviour of functional-differential equations with proportional time delays. Europ J Appl Math, 1996, 7: 11–30 · Zbl 0856.34078 |

[63] | Liu Y K. Numerical investigation of the pantograph equation. Appl Numer Math, 1997, 24: 309–317 · Zbl 0878.65065 |

[64] | Ma S F, Yang Z W, Liu M Z. H {\(\alpha\)}-stability of modified Runge-Kutta methods for nonlinear neutral pantograph equations. J Math Anal Appl, 2007, 335: 1128–1142 · Zbl 1125.65076 |

[65] | Morris G R, Feldstein A, Bowen E W. The Phragmén-Lindelöf principle and a class of functional differential equations. In: Weiss L, ed. Ordinary Differential Equations (Washington, DC, 1971). New York: Academic Press, 1972, 513–540 |

[66] | Mureşan V. On a class of Volterra integral equations with deviating argument. Studia Univ Babeş-Bolyai Math, 1999, XLIV: 47–54 |

[67] | Muroya Y, Ishiwata E, Brunner H. On the attainable order of collocation methods for pantograph integro-differential equations. J Comput Appl Math, 2003, 152: 347–366 · Zbl 1023.65146 |

[68] | Ockendon J R, Tayler A B. The dynamics of a current collection system for an electric locomotive. Proc Roy Soc London Ser A, 1971, 322: 447–468 |

[69] | Pukhnacheva T P. A functional equation with contracting argument. Siberian Math J, 1990, 31: 365–367 · Zbl 0723.45003 |

[70] | Qiu L, Mitsui T, Kuang J X. The numerical stability of the {\(\theta\)}-method for delay differential equations with many variable delays. J Comput Math, 1999, 17: 523–532 · Zbl 0942.65087 |

[71] | Si J G, Cheng S S. Analytic solutions of a functional differential equation with proportional delays. Bull Korean Math Soc, 2002, 39: 225–236 · Zbl 1021.34052 |

[72] | Takama N, Muroya Y, Ishiwata E. On the attainable order of collocation methods for the delay differential equations with proportional delay. BIT, 2000, 40: 374–394 · Zbl 0965.65101 |

[73] | Terjéki J. Representation of the solutions to linear pantograph equations. Acta Sci Math (Szeged), 1995, 60: 705–713 · Zbl 0833.34059 |

[74] | Volterra V. Sopra alcune questioni di inversione di integrali definite. Ann Mat Pura Appl, 1897, 25: 139–178 · JFM 28.0366.02 |

[75] | Volterra V. Leçcons sur les équations intégrales. Paris: Gauthier-Villars, 1913 (VFIEs with proportional delays as limits of integration are treated on pp. 92–101) |

[76] | Xu Y, Zhao J, Liu M. H-stability of Runge-Kutta methods with variable stepsize for systems of pantograph equations. J Comput Math, 2004, 22: 727–734 · Zbl 1059.65068 |

[77] | Yu Y, Li S. Stability analysis of Runge-Kutta methods for nonlinear systems of pantograph equations. J Comput Math, 2005, 23: 351–356 · Zbl 1081.65078 |

[78] | Zhang C, Sun G. The discrete dynamics of nonlinear infinite-delay differential equations. Appl Math Lett, 2002, 15: 521–526 · Zbl 1001.65091 |

[79] | Zhang C, Sun G. Boundedness and asymptotic stability of multistep methods for pantograph equations. J Comput Math, 2004, 22: 447–456 · Zbl 1054.65083 |

[80] | Zhao J J, Cao W R, Liu M Z. Asymptotic stability of Runge-Kutta methods for the pantograph equations. J Comput Math, 2004, 22: 523–534 · Zbl 1065.65102 |

[81] | Zhao J J, Xu Y, Qiao Y. The attainable order of the collocation method for double-pantograph delay differential equation. Numer Math J Chinese Univ, 2005, 27: 297–308 (in Chinese) · Zbl 1100.65061 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.