# zbMATH — the first resource for mathematics

Continuous Runge-Kutta methods for neutral Volterra integro-differential equations with delay. (English) Zbl 0876.65089
Explicit and implicit continuous Runge-Kutta methods (i.e., Runge-Kutta methods with dense output) are used for the numerical treatment of neutral Volterra integro-differential equations with delay. The order of convergence is investigated and it is shown that the arising nonlinear systems can be solved by fixed point iteration as long as the considered problem is non-stiff. A few numerical experiments are included.
Reviewer: E.Hairer (Genève)

##### MSC:
 65R20 Numerical methods for integral equations 45J05 Integro-ordinary differential equations
RODAS
Full Text:
##### References:
 [1] Baker, C.T.H.; Paul, C.A.H., Parallel continuous Runge-Kutta methods and vanishing lag delay differential equations, Adv. comput. math., 1, 367-394, (1993) · Zbl 0824.65055 [2] Bellen, A.; Jackiewicz, Z.; Zennaro, M., Stability analysis of one-step methods for neutral delay-differential equations, Numer. math., 52, 605-619, (1988) · Zbl 0644.65049 [3] Brunner, H., The numerical solutions of neutral Volterra integro-differential equations with delay arguments, () · Zbl 0828.65146 [4] Brunner, H.; van der Houwen, P.J., The numerical solution of Volterra equations, () · Zbl 0611.65092 [5] Enright, W.H., The relative efficiency of alternative defect control schemes for high order continuous Runge-Kutta formulas, SIAM J. numer. anal., 30, 1419-1445, (1993) · Zbl 0787.65046 [6] W.H. Enright and H. Hayashi, Convergence analysis for the numerical solutions of retarded and neutral type delay differential equations by continuous numerical methods, SIAM J. Numer. Anal., to appear. · Zbl 0914.65084 [7] Hairer, E.; Nørsett, S.P.; Wanner, G., Solving ordinary differential equations I. nonstiff problems, (1993), Springer Berlin · Zbl 0789.65048 [8] Hairer, E.; Wanner, G., Solving ordinary differential equations II. stiff and differential-algebraic problems, (1991), Springer Berlin · Zbl 0729.65051 [9] Hayashi, H., Numerical solution of retarded and neutral delay differential equations using continuous Runge-Kutta methods, () [10] Jackiewicz, Z., One-step methods of any order for neutral functional differential equations, SIAM J. numer. anal., 21, 486-511, (1984) · Zbl 0562.65056 [11] Jackiewicz, Z.; Lo, E., The numerical integration of neutral functional-differential equations by fully implicit one-step methods, Z. angew. math. mech., 75, 207-221, (1995) · Zbl 0830.65079 [12] Kamont, Z.; Kwapisz, M., On the Cauchy problem for differential-delay equations in a Banach space, Math. nachr., 74, 173-190, (1976) · Zbl 0288.34069 [13] Vermiglio, R., Natural continuous extensions of Runge-Kutta methods for Volterra integro-differential equations, Numer. math., 53, 439-458, (1988) · Zbl 0629.65145
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.