Runge-Kutta theory for Volterra and Abel integral equations of the second kind
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Abstract:
The present paper develops the local theory of general Runge-Kutta methods for a broad class of weakly singular and regular Volterra integral equations of the second kind. Further, the smoothness properties of the exact solutions of such equations are investigated.References
-
M. Abramowitz & I. A. Stegun (Editors), Handbook of Mathematical Functions, Dover, New York, 1964.
- Uwe an der Heiden, Analysis of neural networks, Lecture Notes in Biomathematics, vol. 35, Springer-Verlag, Berlin-New York, 1980. MR 617008
- H. Brunner, E. Hairer, and S. P. Nørsett, Runge-Kutta theory for Volterra integral equations of the second kind, Math. Comp. 39 (1982), no. 159, 147–163. MR 658219, DOI 10.1090/S0025-5718-1982-0658219-8
- Hermann Brunner and Syvert P. Nørsett, Superconvergence of collocation methods for Volterra and Abel integral equations of the second kind, Numer. Math. 36 (1980/81), no. 4, 347–358. MR 614853, DOI 10.1007/BF01395951 H. Brunner & H. J. J. te Riele, Volterra-Type Integral Equations of the Second Kind with Non-Smooth Solutions: High-Order Methods Based on Collocation Techniques, Report NW 118, Mathematisch Centrum, Amsterdam, 1982. J. C. Butcher, "Implicit Runge-Kutta and related methods," in Modern Numerical Methods for Ordinary Differential Equations (G. Hall and J. M. Watt, eds.), Clarendon Press, Oxford, 1976, pp. 136-151.
- Frank de Hoog and Richard Weiss, High order methods for a class of Volterra integral equations with weakly singular kernels, SIAM J. Numer. Anal. 11 (1974), 1166–1180. MR 368458, DOI 10.1137/0711088
- E. Hairer and G. Wanner, On the Butcher group and general multi-value methods, Computing (Arch. Elektron. Rechnen) 13 (1974), no. 1, 1–15 (English, with German summary). MR 403225, DOI 10.1007/bf02268387
- Richard K. Miller and Alan Feldstein, Smoothness of solutions of Volterra integral equations with weakly singular kernels, SIAM J. Math. Anal. 2 (1971), 242–258. MR 287258, DOI 10.1137/0502022
- S. P. Nørsett and G. Wanner, The real-pole sandwich for rational approximations and oscillation equations, BIT 19 (1979), no. 1, 79–94. MR 530118, DOI 10.1007/BF01931224
- H. Oulès, Résolution numérique d’une équation intégrale singulière, Rev. Française Traitement Information Chiffres 7 (1964), 117–124 (French). MR 172478
- P. Pouzet, Étude en vue de leur traitement numérique des équations intégrales de type Volterra, Rev. Française Traitement Information Chiffres 6 (1963), 79–112 (French). MR 152152 H. J. J. te Riele, Collocation Methods for Weakly Singular Second Kind Volterra Integral Equations with Non-Smooth Solution, Report NW 115, Mathematisch Centrum, Amsterdam, 1981.
- Brooks Ferebee, The tangent approximation to one-sided Brownian exit densities, Z. Wahrsch. Verw. Gebiete 61 (1982), no. 3, 309–326. MR 679677, DOI 10.1007/BF00539832
Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Math. Comp. 41 (1983), 87-102
- MSC: Primary 65R20
- DOI: https://doi.org/10.1090/S0025-5718-1983-0701626-6
- MathSciNet review: 701626