zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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.

MSC:
65R20Integral equations (numerical methods)
45J05Integro-ordinary differential equations
Software:
RODAS
WorldCat.org
Full Text: DOI
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. Proceedings SCADE ’93 (1993) · Zbl 0847.65087
[4] Brunner, H.; Van Der Houwen, P. J.: The numerical solution of Volterra equations. CWI monographs 3 (1986) · 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) · Zbl 0789.65048
[8] Hairer, E.; Wanner, G.: Solving ordinary differential equations II. Stiff and differential-algebraic problems. (1991) · Zbl 0729.65051
[9] Hayashi, H.: Numerical solution of retarded and neutral delay differential equations using continuous Runge-Kutta methods. Ph.d. thesis (1996)
[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