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)
On the attainable order of collocation methods for the neutral functional-differential equations with proportional delays. (English) Zbl 0955.65098

The author extends the recent results of H. Brunner [On the discretization of differential and Volterra integral equations with variable delay, BIT 37, 1-12 (1997; Zbl 0873.65126)] for the y ' (t)=by(qt),y(0)=1, and the delay Volterra integral equations y(t)=1+ 0 t by(qs)ds with proportional delay qt,0<q1, to the neutral functional-differential equation (NFDE):

y ' (t)=ay(t)+ i=1 b i y(q i t)+ i=1 c i y ' (p i t),y(0)=1,

and the delay Volterra integro-differential equation (DVIDE):

y(t)=1+ 0 t ay(s)ds+ i=1 0 t b i y(q i s)ds+ i=1 0 t c i y ' (p i τ)dτ

with proportional delays p i t and q i t,0<p i ,q i 1, and complex numbers a,b i and c i · He analyzes the attainable order of m-stage implicit (collocation-based) Runge-Kutta methods at the first mesh point t=h for the collocation solution v(t) of the NFDE and the ‘iterated collocation solution u it (t)’ of the DVIDE to the solution y(t), and investigates the existence of the collocation polynomials M m (t) of v(th) or M ^ m (t) of u it (th), t[0,1], such that the rational approximant v(h) or u it (h) is the (m,m)-Padé approximant to y(h) and satisfies |v(h)-y(h)|=O(h 2m+1 )· If they exist, then the conditions on M m (t) and M ^ m (t), respectively, actually are given.

MSC:
65R20Integral equations (numerical methods)
45D05Volterra integral equations
34K28Numerical approximation of solutions of functional-differential equations
45G10Nonsingular nonlinear integral equations
65L05Initial value problems for ODE (numerical methods)
34K06Linear functional-differential equations