TY - JOUR
T1 - Analysis of collocation solutions for a class of functional equations with vanishing delays
AU - BRUNNER, Hermann
AU - Xie, Hehu
AU - Zhang, Ran
N1 - Funding Information:
Natural Sciences and Engineering Research Council of Canada (DG A-9406); Hong Kong Research Grants Council; National Nature Science Foundation of China (10801062); 985 program of Jilin University; K. C. Wong Education Foundation in Hong Kong.
PY - 2011/4
Y1 - 2011/4
N2 - We study the existence, uniqueness and regularity properties of solutions for the functional equation y(t) = b(t)y(θ(t)) + f(t), t ∈ [0, T], where the delay function θ(t) vanishes at t = 0. Functional equations corresponding to the linear delay function θ(t) = qt (0 < q < 1) represent an important special case. We then analyse the optimal order of convergence of piecewise polynomial collocation approximations to solutions of these functional equations. The theoretical results are illustrated by extensive numerical examples.
AB - We study the existence, uniqueness and regularity properties of solutions for the functional equation y(t) = b(t)y(θ(t)) + f(t), t ∈ [0, T], where the delay function θ(t) vanishes at t = 0. Functional equations corresponding to the linear delay function θ(t) = qt (0 < q < 1) represent an important special case. We then analyse the optimal order of convergence of piecewise polynomial collocation approximations to solutions of these functional equations. The theoretical results are illustrated by extensive numerical examples.
KW - collocation solutions
KW - functional equation with vanishing delay
KW - integro-functional equation
KW - optimal order of convergence
KW - q-difference equation
KW - uniqueness and regularity of solution
UR - http://www.scopus.com/inward/record.url?scp=79953676594&partnerID=8YFLogxK
U2 - 10.1093/imanum/drp051
DO - 10.1093/imanum/drp051
M3 - Journal article
AN - SCOPUS:79953676594
SN - 0272-4979
VL - 31
SP - 698
EP - 718
JO - IMA Journal of Numerical Analysis
JF - IMA Journal of Numerical Analysis
IS - 2
ER -