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A new error analysis for a cubic spline approximate solution of a class of Volterra integro-differential equations

Authors: Joseph A. Guzek and Gene A. Kemper
Journal: Math. Comp. 27 (1973), 563-570
MSC: Primary 65R05
MathSciNet review: 0337044
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Abstract: In this paper a third-order numerical method is considered which utilizes a twice continuously differentiable third degree spline to approximate the solution of \[ \begin {array}{*{20}{c}} {\dot x(t) = F\left ( {t,x(t),\int _a^t {K(t,u,x(u))\;du} } \right ),} \hfill \\ {x(a) = {x_0},} \hfill \\ \end {array} \] at discrete points in the interval [a, b]. The error analysis uses a technique usually associated with linear multistep methods.

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

    R. C. Buck, Advanced Calculus, 2nd ed., McGraw-Hill, New York, 1965. MR 42 #431. J. A. Guzek & G. A. Kemper, A Cubic Spline Approximate Solution of a Class of Integro-Differential Equations, Proc. Conf. Numerical Mathematics, University of Manitoba, October 1971.
  • Peter Henrici, Discrete variable methods in ordinary differential equations, John Wiley & Sons, Inc., New York-London, 1962. MR 0135729
  • H.-S. Hung, Application of Linear Spline Functions to the Numerical Solution of Volterra Integral Equations of the Second Kind, University of Wisconsin Comput. Sci. Tech. Rep. No. 27, 1968. H.-S. Hung, The Numerical Solution of Differential and Integral Equations by Spline Functions, Math. Res. Center Tech. Rep. No. 1053, Mathematics Research Center, University of Wisconsin, Madison, Wis., 1970.
  • Gene A. Kemper, Linear multistep methods for a class of functional differential equations, Numer. Math. 19 (1972), 361–372. MR 317561, DOI
  • Peter Linz, Linear multistep methods for Volterra integro-differential equations, J. Assoc. Comput. Mach. 16 (1969), 295–301. MR 239786, DOI

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Keywords: Volterra integro-differential equations, spline approximation
Article copyright: © Copyright 1973 American Mathematical Society