Summary
Operational quadrature rules are applied to problems in numerical integration and the numerical solution of integral equations: singular integrals (power and logarithmic singularities, finite part integrals), multiple timescale convolution, Volterra integral equations, Wiener-Hopf integral equations. Frequency domain conditions, which determine, the stability of such equations, can be carried over to the discretization.
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References
Ciarlet, P. G.: The Finite Element Method for Elliptic Problems. Amsterdam: North Holland 1978
Corduneanu, C.: Integral Equations and Stability of Feedback Systems. New York: Academic Press 1973
Dahlquist, G.: A special stability problem for linear multistep methods. BIT3, 27–43 (1963)
Desoer, C.A., Vidyasagar, M.: Feedback Systems: Input-Output Properties. New York: Academic Press 1975
Doetsch, G.: Handbuch der Laplace-Transformation, Vol. II. Basel: Birkhäuser 1955
Eggermont, P.P.B.: Uniform error estimates of Galerkin methods for, monotone Abel-Nolterra integral equations on the half-line. Report, Univ. of Delaware, Newark (1987)
Gelbin, D.: Measuring absorption, rates from an aqueous solution. AICHE J. (to appear) orGienger, G.: On convolution quadratures and their application to Fredholm integral equations. Dissertation, Univ. Heidelberg (1987)
Gochberg, I.Z., Feldman, I.A.: Faltungsgleichungen und Projektionsverfahren zu ihrer Lösung. Stuttgart: Birkhäuser 1974
Golub, G.H., Van Loan, C.F.: Matrix Computations. Oxford: North Oxford Academic 1983
Hairer, E., Lubich, Ch., Schlichte, M.: Fast numerical solution of weakly singular Volterra integral equations. Report, Université de Genève (1986) J. Comput. Appl. Math. (to appear)
Hairer, E., Maass, P.: Numerical methods for singular nonlinear integro-differential equations. Appl. Numer. Math.3, 243–256 (1987)
Hannsgen, K.B., Herdman, T.L., Stech, H.W., Wheeler, R.L. (eds.): Volterra and Functional Differential Equations New York: Marcel Dekker 1982
Henrici, P.: Fast Fourier methods in computational complex analysis. SIAM Rev.21, 487–527 (1979)
Hoppensteadt, F.C.: An algorithm for approximate solutions to weakly filtered synchronous control systems and nonlinear renewal processes. SIAM J. Appl. Math.43, 834–843 (1983)
Krein, M.G.: Integral equations on a half-line with kernel depending upon the difference of the arguments. Uspekhi Mat. Nauk13, 3–120 (1958) (engl.: AMS Translations22, 163–288 (1962))
Londen, S.O., Staffans, O.J. (eds.): Volterra Equations. Springer LNM, Vol. 737 (1979)
Lopez-Castillo, J.M., Jay-Gerin, J.P., Tannous, C.: Dynamics of electron delocalization: an exact treatment. Europhys. Lett. (to appear)
Lubich, C.: Fractional linear multistep methods for Abel-Volterra integral equations of the second kind. Math. Comput.45, 463–469 (1985)
Lubich, C.: Fractional linear multistep methods for Abel-Volterra integral equations of the first kind. IMA J. Numer. Anal.7, 97–106 (1987)
Lubich, C.: Convolution Quadrature and Discretized Operational Calculus. I. Numer. Math.52, 129–145 (1988)
Sanz-Serna, J.M.: A numerical method for a partial integro-differetial equation. SIAM. J. Numer. Anal. (to appear)
Sloan, I.H., Spence, A.: Projection methods for integral equations on the half-line. IMA J. Numer. Anal.6, 153–172 (1986)
Stenger, F.: The approximate solution of Wiener-Hopf integral equations. J. Math. Anal. Appl.37, 687–724 (1972)
Talbot, A.: The accurate numerical inversion of Laplace transforms. J. Inst. Math. Appl.23, 97–120 (1979)
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This is Part II to the article with the same title (Part I), which was published in Volume 52, No. 2, pp. 129–145 (1988)
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Lubich, C. Convolution quadrature and discretized operational calculus. II. Numer. Math. 52, 413–425 (1988). https://doi.org/10.1007/BF01462237
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DOI: https://doi.org/10.1007/BF01462237