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The approximate solution of initial-value problems for general Volterra integro-differential equations. (English) Zbl 0631.65141
We study the application of certain polynomial spline collocation methods to Volterra integrodifferential equations of order r where the rth derivative of the unknown solution occurs also in the kernel function of the equation. The analysis focuses on the questions of the attainable order of discrete (local) convergence and the numerical implementation of these methods. The performance of the resulting implicit Runge-Kutta- Nyström type methods is illustrated by numerical examples.

MSC:
65R20 Numerical methods for integral equations
45J05 Integro-ordinary differential equations
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[1] Aguilar, M., Brunner, H.: Colocation methods for second-order Volterra integro-differential equations (to appear in Appl. Numer. Math.). · Zbl 0651.65098
[2] Brunner, H.: Implicit Runge-Kutta methods of optimal order for Volterra integro-differential equations. Math. Comp.42, 95–109 (1984). · Zbl 0543.65092 · doi:10.1090/S0025-5718-1984-0725986-6
[3] Brunner, H.: High-order methods for the numerical solution of Volterra integro-differential equations. J. Comput. Appl. Math.15, 301–309 (1986). · Zbl 0634.65143 · doi:10.1016/0377-0427(86)90221-9
[4] Brunner, H., van der Houwen, P. J.: The Numerical Solution of Volterra Equations. Amsterdam-New York: North-Holland 1986. · Zbl 0611.65092
[5] Burton, T. A.: Volterra Integral and Differential Equations. New York: Academic Press 1983. · Zbl 0515.45001
[6] Denisenko, N. V., Yanovich, L. A.: One-step numerical methods for the solution of systems of linear Volterra integrodifferential equations of the second kind. Differential Equations19, 650–662 (1983).
[7] Goldfine, A.: Taylor series methods for the solution of Volterra integral and integro-differential equations. Math. Comp.31, 691–707 (1977). · Zbl 0372.65054 · doi:10.1090/S0025-5718-1977-0440970-4
[8] Hrusa, W. J., Nohel, J. A.: Global existence and asymptotics in one-dimensional nonlinear viscoelasticity. In: Trends and Applications of Pure Mathematics to Mechanics (P. G. Ciarlet and M. Roseau, eds.), pp. 165–187). Berlin-Heidelberg-New York: Springer (Lecture Notes in Phys., Vol. 195) 1984. · Zbl 0543.73043
[9] MacCamy, R. C.: An integro-differential equation with application in heat flow. Quart. Appl. Math.35, 1–19 (1977). · Zbl 0351.45018
[10] MacCamy, R. C.: A model for one-dimensional, nonlinear viscoelasticity. Quart. Appl. Math.35, 21–33 (1977). · Zbl 0355.73041
[11] Pouzet, P.: Etude, en vue de leur approximation numérique, des solutions d’équations intégrales et intégrodifférentielles de type Volterra pour des problèmes de conditions initiales. Thèse, Université de Strasbourg 1962.
[12] Pouzet, P.: Systemes différentiels, équations intégrales et intégrodifférentielles. In: Procédures ALGOL en Analyse Numérique I, pp. 203–208. Paris: Centre National de la Recherche Scientifique 1967.
[13] Volterra, V.: Theory of Functionals and of Integral and Integro-Differential Equations, pp. 147–149. New York: Dover Publ. 1959. · Zbl 0086.10402
[14] Yanovich, L. A.: A fourth-order numerical method for solving systems of linear integrodifferential equations of Volterra type (Russian). Dokl. Akad. Nauk BSSR28, 293–296 (1984). · Zbl 0544.65097
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