TY - JOUR

T1 - A super-attainable order in collocation methods for differential equations with proportional delay

AU - Ishiwata, Emiko

AU - Muroya, Yoshiaki

AU - BRUNNER, Hermann

N1 - Funding Information:
This work was partially supported by Grant-in-Aid for Young Scientists (B), No. 14740086 of the Japanese Ministry of Education, Science, Sports and Culture, and Waseda University Grant for Special Research Projects 2005A-077, 2006B-167 and Scientific Research (c), No. 16540207 of Japan Society for the Promotion of Science.

PY - 2008/4/15

Y1 - 2008/4/15

N2 - We are concerned with the pantograph differential equation y′ (t) = ay (t) + by (qt) + f (t), t > 0, y (0) = y0, with proportional delay qt, 0 < q < 1. In the literatures, it is known that if we choose some proper m collocation points for m ≥ 2, then collocation leads to a superconvergence result of order p* = 2 m + 1 at the first mesh point t = h. In this paper, in such collocation solution to the above equation, we show that there are cases for some 0 < q < 1 such that the attainable order at the first mesh point t = h, becomes a super-attainable order, just O (h2 m + 2). Numerical experiments of such 0 < q < 1 are also presented.

AB - We are concerned with the pantograph differential equation y′ (t) = ay (t) + by (qt) + f (t), t > 0, y (0) = y0, with proportional delay qt, 0 < q < 1. In the literatures, it is known that if we choose some proper m collocation points for m ≥ 2, then collocation leads to a superconvergence result of order p* = 2 m + 1 at the first mesh point t = h. In this paper, in such collocation solution to the above equation, we show that there are cases for some 0 < q < 1 such that the attainable order at the first mesh point t = h, becomes a super-attainable order, just O (h2 m + 2). Numerical experiments of such 0 < q < 1 are also presented.

KW - Collocation methods

KW - Delay differential equations

KW - Proportional delay

KW - Super-attainable order

UR - http://www.scopus.com/inward/record.url?scp=40149095247&partnerID=8YFLogxK

U2 - 10.1016/j.amc.2007.08.078

DO - 10.1016/j.amc.2007.08.078

M3 - Article

AN - SCOPUS:40149095247

SN - 0096-3003

VL - 198

SP - 227

EP - 236

JO - Applied Mathematics and Computation

JF - Applied Mathematics and Computation

IS - 1

ER -