Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Uniform error estimates of operational quadrature methods for nonlinear convolution equations on the half-line

Authors: P. P. B. Eggermont and Ch. Lubich
Journal: Math. Comp. 56 (1991), 149-176
MSC: Primary 65R20; Secondary 45D05, 47H15, 47H17
MathSciNet review: 1052091
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study uniform error estimates of operational quadrature methods for nonlinear convolution equations on the half-line. Equations of this kind arise in control engineering and diffusion problems. The essential ingredients are the stability of the operational quadrature method in an $ {L^2}$ setting, which is inherited from the continuous equation by its very construction, and a theorem that says that the behavior of the linearized equations is the same in all $ {L^p}$ spaces $ (1 \leq p \leq \infty )$.

References [Enhancements On Off] (What's this?)

  • [1] B. A. Barnes, The spectrum of integral operators on Lebesgue spaces, J. Operator Theory 18 (1987), 115-132. MR 912815 (89i:46065)
  • [2] H. Brunner and P. J. van der Houwen, The numerical solution of Volterra equations, North-Holland, Amsterdam, 1986. MR 871871 (88g:65136)
  • [3] J. R. Cannon, The one-dimensional heat equation, Addison-Wesley, Reading, MA, 1984. MR 747979 (86b:35073)
  • [4] C. Corduneanu, Integral equations and stability of feedback systems, Academic Press, New York, 1973. MR 0358245 (50:10710)
  • [5] G. Dahlquist, A special stability problem for linear multistep methods, BIT 3 (1963), 27-43. MR 0170477 (30:715)
  • [6] C. A. Desoer and M. Vidyasagar, Feedback systems: Input-output properties, Academic Press, New York, 1975. MR 0490289 (58:9636)
  • [7] P. P. B. Eggermont, Uniform error estimates of Galerkin methods for monotone Abel-Volterra integral equations on the half-line, Math. Comp. 53 (1989), 157-189. MR 969485 (90h:65216)
  • [8] R. Ghez, A primer of diffusion problems, Wiley, New York, 1988. MR 945142 (89m:35001)
  • [9] E. Hairer, Ch. Lubich and M. Schlichte, Fast numerical solution of weakly singular Volterra integral equations, J. Comput. Appl. Math. 23 (1988), 87-98. MR 952066 (89g:65166)
  • [10] I. I. Kolodner, Equations of Hammerstein in Hilbert space, J. Math. Mech. 13 (1964), 701-750. MR 0171184 (30:1415)
  • [11] P. Linz, Analytical and numerical methods for Volterra equations, SIAM, Philadelphia, PA, 1985. MR 796318 (86m:65163)
  • [12] A. S. Lodge, M. Renardy and J. A. Nohel, eds., Viscoelasticity and rheology, Academic Press, Orlando, 1985. MR 830195 (87a:76007)
  • [13] S. O. Londen, On some nonintegrable Volterra kernels with integrable resolvents including some applications to Riesz potentials, J. Integral Equations 10 (1985), 241-289. MR 831246 (87f:45017)
  • [14] Ch. Lubich, Convolution quadrature and discretized operational calculus. I and II, Numer. Math. 52 (1988), 129-145 and 413-425. MR 923707 (89g:65018)
  • [15] R. E. A. C. Paley and N. Wiener, Fourier transforms in the complex domain, Amer. Math. Soc. Colloq. Publ., vol. 19, Amer. Math. Soc., Providence, RI, 1934. MR 1451142 (98a:01023)
  • [16] A. C. Pipkin, Lectures on viscoelasticity theory, Springer-Verlag, New York, 1972.
  • [17] I. W. Sandberg, A frequency domain condition for stability of feedback systems containing a single time-varying nonlinear element, Bell Sys. Tech. J. 43 (1964), 1581-1608. MR 0175684 (30:5868)
  • [18] E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces. Princeton Univ. Press, Princeton, NJ, 1971. MR 0304972 (46:4102)
  • [19] G. Zames, On the input-output stability of nonlinear time-varying feedback systems. I and II, IEEE Trans. Automat. Control AC-11 (1966), 228-238 and 465-477.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65R20, 45D05, 47H15, 47H17

Retrieve articles in all journals with MSC: 65R20, 45D05, 47H15, 47H17

Additional Information

Article copyright: © Copyright 1991 American Mathematical Society

American Mathematical Society