Volume 11, Issue 1
A Meshless and Parallelizable Method for Differential Equations with Time-Delay

Numer. Math. Theor. Meth. Appl., 11 (2018), pp. 92-127.

Published online: 2018-11

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• Abstract

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 $∆t < τ$ 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.

• Keywords

Delay differential equations, meshless/parallel computatio, contour integral, Filon-Clenshaw-Curtis quadrature.

65M15, 65D05, 65D30

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@Article{NMTMA-11-92, 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 = {

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 $∆t < τ$ 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.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2018.m1636}, url = {http://global-sci.org/intro/article_detail/nmtma/10645.html} }
TY - JOUR T1 - A Meshless and Parallelizable Method for Differential Equations with Time-Delay JO - Numerical Mathematics: Theory, Methods and Applications VL - 1 SP - 92 EP - 127 PY - 2018 DA - 2018/11 SN - 11 DO - http://doi.org/10.4208/nmtma.2018.m1636 UR - https://global-sci.org/intro/article_detail/nmtma/10645.html KW - Delay differential equations, meshless/parallel computatio, contour integral, Filon-Clenshaw-Curtis quadrature. AB -

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 $∆t < τ$ 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.

Shulin Wu & Chengming Huang. (2020). A Meshless and Parallelizable Method for Differential Equations with Time-Delay. Numerical Mathematics: Theory, Methods and Applications. 11 (1). 92-127. doi:10.4208/nmtma.2018.m1636
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