@Article{NMTMA-11-1,
author = {},
title = {A Meshless and Parallelizable Method for Differential Equations with Time-Delay},
journal = {Numerical Mathematics: Theory, Methods and Applications},
year = {2018},
volume = {11},
number = {1},
pages = {92--127},
abstract = {<p style="text-align: justify;">Numerical computation plays an important role in the study of differential
equations with time-delay, because a simple and explicit analytic solution is usually un
available. Time-stepping methods based on discretizing the temporal derivative with
some step-size $∆t$ are the main tools for this task. To get accurate numerical solutions,
in many cases it is necessary to require $∆t &lt; τ$ and this will be a rather unwelcome
restriction when $τ$, the quantity of time-delay, is small. In this paper, we propose a
method for a class of time-delay problems, which is completely meshless. The idea
lies in representing the solution by its Laplace inverse transform along a carefully designed contour in the complex plane and then approximating the contour integral by
the Filon-Clenshaw-Curtis (FCC) quadrature in a few fast growing subintervals. The
computations of the solution for all time points of interest are naturally parallelizable
and for each time point the implementations of the FCC quadrature in all subintervals
are also parallelizable. For each time point and each subinterval, the FCC quadrature
can be implemented by fast Fourier transform. Numerical results are given to check the
efficiency of the proposed method.</p>},
issn = {2079-7338},
doi = {https://doi.org/10.4208/nmtma.2018.m1636},
url = {https://global-sci.com/article/90445/a-meshless-and-parallelizable-method-for-differential-equations-with-time-delay}
}