High-order collocation methods for singular Volterra functional equations of neutral type

Applied Numerical Mathematics

Volumn 57, Issue 5-7 SPEC. ISS., 2007, Pages 533-548

  Brunner, Hermann ^a  

^a MEMORIAL UNIVERSITY OF NEWFOUNDLAND   (Canada)

Author keywords

Collocation methods; Neutral equations; Optimal order of convergence; Variable delays; Volterra functional integro differential equations; Weakly singular kernels

Indexed keywords

COMPUTATION THEORY; CONVERGENCE OF NUMERICAL METHODS; DIFFERENTIAL EQUATIONS; INTEGRAL EQUATIONS;

COLLOCATION METHODS; NEUTRAL EQUATIONS; VARIABLE DELAYS; VOLTERRA FUNCTIONAL INTEGRO-DIFFERENTIAL EQUATIONS; WEAKLY SINGULAR KERNELS;

FUNCTION EVALUATION;

EID: 34147094895     PISSN: 01689274     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.apnum.2006.07.006     Document Type: Article

References (46)

1. Baker C.T.H. Numerical analysis of Volterra functional and integral equations. In: Duff I.S., and Watson G.A. (Eds). The State of the Art in Numerical Analysis. York, 1996 (1997), Clarendon Press, Oxford 193-222
2. Bélair J. Population models with state-dependent delays. In: Arino O., Axelrod D.E., and Kimmel M. (Eds). Mathematical Population Dynamics (1991), Marcel Dekker, New York 165-176
3. Bellen A. One-step collocation for delay differential equations. J. Comput. Appl. Math. 10 (1984) 275-283
4. Bellen A. Constrained mesh methods for functional-differential equations. In: Meinardus G., and Nürnberger G. (Eds). Delay Equations, Approximation and Application. Internat. Ser. Numer. Math. vol. 74 (1985), Birkhäuser Verlag, Basel, Boston 52-70
5. Bellen A., and Zennaro M. Numerical Methods for Delay Differential Equations (2003), Oxford University Press, Oxford
6. Bocharov G.A., and Rihan F.A. Numerical modelling in biosciences using delay differential equations. J. Comput. Appl. Math. 125 (2000) 183-199
7. Bownds J.M., Cushing J.M., and Schutte R. Existence, uniqueness, and extendibility of solutions to Volterra integral systems with multiple variable delays. Funkcial. Ekvac. 19 (1976) 101-111
8. Brauer F., and van den Driessche P. Some directions for mathematical epidemiology. In: Ruan S., Wolkowicz G.S.K., and Wu J. (Eds). Dynamical Systems and Their Applications in Biology. Cape Breton, 2001. Fields Institute Communications vol. 36 (2003), American Mathematical Society, Providence, RI 95-112
9. Brunner H. The numerical solution of neutral Volterra integro-differential equations with delay arguments. Ann. Numer. Math. 1 (1994) 309-322
10. Brunner H. The discretization of Volterra functional integral equations with proportional delays. In: Elaydi S., Ladas G., Wu J., and Zou X. (Eds). Difference and Differential Equations. Changsha, 2002. Fields Institute Communications vol. 42 (2004), American Mathematical Society, Providence, RI 3-27
11. Brunner H. The numerical analysis of functional integral and integro-differential equations of Volterra type. Acta Numerica 00 (2004) 55-145
12. Brunner H. Collocation Methods for Volterra Integral and Related Functional Equations. Cambridge Monographs on Applied and Computational Mathematics vol. 15 (2004), Cambridge University Press, Cambridge
13. Brunner H. The numerical solution of weakly singular Volterra functional integro-differential equations with variable delays. Comm. Pure Appl. Anal. 5 (2006) 261-276
14. Brunner H., and Hu Q.-Y. Optimal superconvergence orders of iterated collocation solutions for Volterra integral equations with vanishing delays. SIAM J. Numer. Anal. 43 (2005) 1934-1949
15. Brunner H., and Ma J.-T. On the regularity of solutions to neutral-type Volterra functional-differential equations with weakly singular kernels. J. Integral Equations Appl. 18 (2006)
16. Brunner H., Pedas A., and Vainikko G. The piecewise polynomial collocation method for nonlinear weakly singular Volterra equations. Math. Comp. 68 (1999) 1079-1095
17. Brunner H., Pedas A., and Vainikko G. Piecewise polynomial collocation methods for linear Volterra integro-differential equations with weakly singular kernels. SIAM J. Numer. Anal. 39 (2001) 957-982
18. Brunner H., and Vermiglio R. Stability of solutions of neutral functional integro-differential equations and their discretizations. Computing 71 (2003) 229-245
19. Busenberg S., and Cooke K.L. The effect of integral conditions in certain equations modelling epidemics and population growth. J. Math. Biol. 10 (1980) 13-32
20. Cahlon B., and Nachman L.J. Numerical solutions of Volterra integral equations with a solution dependent delay. J. Math. Anal. Appl. 112 (1985) 541-562
21. Cañada A., and Zertiti A. Systems of nonlinear delay integral equations modelling population growth in a periodic environment. Comment. Math. Univ. Carolin. 35 (1994) 633-644
22. Cao Y., Herdman T.L., and Xu Y. A hybrid collocation method for Volterra integral equations with weakly singular kernels. SIAM J. Numer. Anal. 41 (2003) 364-381
23. Cooke K.L. An epidemic equation with immigration. Math. Biosci. 29 (1976) 135-158
24. Cushing J.M. Integro-Differential Equations and Delay Models in Population Dynamics. Lecture Notes in Biomath. vol. 20 (1977), Springer-Verlag, Berlin, Heidelberg, New York
25. Feldstein A., and Neves K.W. High order methods for state-dependent delay differential equations with nonsmooth solutions. SIAM J. Numer. Anal. 21 (1984) 844-863
26. Györi I., Hartung F., and Turi J. Numerical approximations for a class of differential equations with time- and state-dependent delays. Appl. Math. Lett. 8 (1995) 19-24
27. Hale J.K., and Verduyn Lunel S.M. Introduction to Functional Differential Equations (1993), Springer-Verlag, New York
28. Hartung F., Herdman T.L., and Turi J. On existence, uniqueness and numerical approximation for neutral equations with state-dependent delay. Appl. Numer. Math. 24 (1997) 393-409
29. Herdman T.L., and Turi J. An application of finite Hilbert transforms in the derivation of a state space model for an aeroelastic system. J. Integral Equations Appl. 3 (1991) 271-287
30. Herdman T.L., and Turi J. On the solutions of a class of integral equations arising in unsteady aerodynamics. In: Elaydi S. (Ed). Differential Equations. Colorado Springs, 1989. Lecture Notes in Pure and Appl. Math. vol. 127 (1991), Marcel Dekker, New York 241-248
31. Hethcote H.W., and Tudor D.W. Integral equation models for endemic infectious diseases. J. Math. Biol. 9 (1980) 37-47
32. Ito K., and Turi J. Numerical methods for a class of singular integro-differential equations based on semigroup approximation. SIAM J. Numer. Anal. 28 (1991) 1698-1722
33. Kangro R., and Parts I. Superconvergence in the maximum norm of a class of piecewise polynomial collocation methods for solving linear weakly singular Volterra integro-differential equations. J. Integral Equations Appl. 15 (2003) 403-427
34. Liu Y. Runge-Kutta-collocation methods for systems of functional-differential and functional equations. Adv. Comput. Math. 11 (1999) 315-329
35. I. Parts, Piecewise Polynomial Collocation Methods for Solving Weakly Singular Integro-Differential Equations, PhD thesis, Diss. Math. Univ. Tartuensis, University of Tartu, 2005
36. Parts I., and Pedas A. Collocation approximations for weakly singular Volterra integro-differential equations. Math. Model. Anal. 8 (2003) 315-328
37. Tang T. Superconvergence of numerical solutions to weakly singular Volterra integro-differential equations. Numer. Math. 61 (1992) 373-382
38. Tavernini L. One-step methods for the numerical solution of Volterra functional differential equations. SIAM J. Numer. Anal. 8 (1971) 786-795
39. Tavernini L. The approximate solution of Volterra differential systems with state dependent time lags. SIAM J. Numer. Anal. 15 (1978) 1039-1052
40. Vermiglio R., and Torelli L. A stable numerical approach for implicit non-linear neutral delay differential equations. BIT 43 (2003) 195-215
41. V. Volterra, Variazioni e fluttuazioni del numero d'individui in specie animali conviventi, in: Memorie del R. Comitato talassografico italiano, CXXXI, 1927; also Volterra [44], vol. V, pp. 1-111
42. Volterra V. Leçons sur la théorie mathématique de la lutte pour la vie (1931), Gauthier-Villars, Paris also:
43. Volterra V. Leçons sur la théorie mathématique de la lutte pour la vie (1990), Éditions Jacques Gabay, Sceaux
44. Volterra V. The general equations of biological strife in the case of historical actions. Proc. Edinburgh Math. Soc. (2) 6 (1939) 4-10
45. Volterra V. Opere Matematiche, vol. I-V (1956), Accademia Nazionale dei Lincei, Roma
46. Willé D.R., and Baker C.T.H. The tracking of derivative discontinuities in systems of delay differential equations. Appl. Numer. Math. 9 (1992) 209-222