zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
High-order nonlinear initial-value problems countably determined. (English) Zbl 1169.65071
An iterative approach to approximate global solutions of high order initial value problems (IVPs): $$x^{(m)}(t)= f(t, x(t), x'(t), \dots x^{(m-1)}(t) ), \quad t \in [t_0, t_0+T]\tag1$$ with $ x^{(j)}(t_0)= x_0^j,$ $j=0, \dots ,m-1, (1)$ is proposed. The approach extends previous results of the same authors for first order non linear IVPs [Nonlinear Anal., Theory Methods Appl. 63, No. 1 (A), 97--105 (2005; Zbl 1097.34005)]. Under continuity and uniform Lipschitz conditions on the nonlinear function $f$, the IVP (1) is equivalent to an integral equation $ x(t) = T x(t)$ with some integral operator $T$ and the problem reduces to the computation of a fixed point of $T$ in a Banach space. For the approximation of such a fixed point a suitable Faber-Schauder basis $ ( \Gamma_j )_{j \ge 0}$ of piecewise linear functions associated to a sequence of nodes $ (t_j)_{j \ge 0} $ with $ t_1 = t_0+T$ dense in $[t_0, t_0+T]$ is proposed such that $ ( \varphi_k = \sum_{i \ge 0} \lambda_k^i \; \Gamma_i )$ with $ \lambda_k^i$ appropriately chosen tends to the exact solution. Several convergence results for an iterative to the solution as well as error estimates are proposed and finally the technique is applied to an IVP second order test problem to show the convergence of the method depending on the number of the nodes $t_j$ and the number of iterations.

65L05Initial value problems for ODE (numerical methods)
34A34Nonlinear ODE and systems, general
65L20Stability and convergence of numerical methods for ODE
65L70Error bounds (numerical methods for ODE)
Full Text: DOI
[1] Berenguer, M. I.; Fortes, M. A.; Guillem, A. I. Garralda; Galán, M. Ruiz: Linear Volterra integro-differential equation and Schauder bases, Appl. math. Comput. 159, 495-507 (2004) · Zbl 1068.65143 · doi:10.1016/j.amc.2003.08.132
[2] Castro, E.; Gámez, D.; Guillem, A. I. Garralda; Galán, M. Ruiz: High order linear initial-value problems and Schauder bases, Appl. math. Model. 31, 2629-2638 (2007) · Zbl 1137.65046 · doi:10.1016/j.apm.2006.10.013
[3] Gámez, D.; Guillem, A. I. Garralda; Galán, M. Ruiz: Nonlinear initial-value problems and Schauder bases, Nonlinear anal. TMA 63, 97-105 (2005) · Zbl 1097.34005 · doi:10.1016/j.na.2005.05.005
[4] Jamenson, G. J. O.: Topology and normed spaces, (1974)
[5] Megginson, R. E.: An introduction to Banach space theory, (1998) · Zbl 0910.46008
[6] Palomares, A.; Galán, M. Ruiz: Isomorphisms, Schauder bases in Banach spaces and numerical solution of integral and differential equations, Numer. funct. Anal. optim. 26, 129-137 (2005) · Zbl 1079.65067 · doi:10.1081/NFA-200051625
[7] Semadeni, Z.: Schauder bases in Banach spaces of continuous functions, (1982) · Zbl 0478.46014