TY - JOUR
T1 - Optimal Superconvergence Results for Delay Integro‐Differential Equations of Pantograph Type
AU - Brunner, Hermann
AU - Hu, Qiya
N1 - Funding information:
Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL, A1C 5S7 Canada ([email protected]). This author’s research was supported by the Natural Sciences and Engineering Research Council of Canada (Discovery grant 9406).
*LSEC and Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China (hqy@lsec. cc.ac.cn). This author’s research was supported by the Natural Science Foundation of China G10371129, the Key Project of the Natural Science Foundation of China G10531080, and the National Basic Research Program of China 2005CB321702.
Publisher copyright:
Copyright © 2007 Society for Industrial and Applied Mathematics
PY - 2007/5/4
Y1 - 2007/5/4
N2 - We analyze the optimal (global and local) orders of superconvergence of collocation solutions Uh on uniform meshes Ih for delay Volterra integro-differential equations with proportional delay functions given by θ(t) = qt (0 < q < 1, t € [0, T]). In particular, we show that if Uh is a continuous piecewise polynomial of degree m > 2, and if collocation is at the Gauss ( Legendre) points, then the (optimal) order of local superconvergence on Ih is p* = m + 2. It turns out that the same order p* holds for nonlinear (strictly increasing) delay functions vanishing at t = 0. However, on judiciously chosen geometric meshes, collocation at the Gauss points yields the order 2m - εN, where εN → 0 as the number N of mesh points tends to infinity. Optimal local superconvergence results for the pantograph delay differential equation are obtained as special cases of our general analysis.
AB - We analyze the optimal (global and local) orders of superconvergence of collocation solutions Uh on uniform meshes Ih for delay Volterra integro-differential equations with proportional delay functions given by θ(t) = qt (0 < q < 1, t € [0, T]). In particular, we show that if Uh is a continuous piecewise polynomial of degree m > 2, and if collocation is at the Gauss ( Legendre) points, then the (optimal) order of local superconvergence on Ih is p* = m + 2. It turns out that the same order p* holds for nonlinear (strictly increasing) delay functions vanishing at t = 0. However, on judiciously chosen geometric meshes, collocation at the Gauss points yields the order 2m - εN, where εN → 0 as the number N of mesh points tends to infinity. Optimal local superconvergence results for the pantograph delay differential equation are obtained as special cases of our general analysis.
KW - Collocation solutions
KW - Optimal order of superconvergence
KW - Pantograph equation
KW - Proportional delays
KW - Vanishing delays
KW - Volterra integro-differential equation
UR - http://www.scopus.com/inward/record.url?scp=46949097714&partnerID=8YFLogxK
U2 - 10.1137/060660357
DO - 10.1137/060660357
M3 - Journal article
AN - SCOPUS:46949097714
SN - 0036-1429
VL - 45
SP - 986
EP - 1004
JO - SIAM Journal on Numerical Analysis
JF - SIAM Journal on Numerical Analysis
IS - 3
ER -