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首页> 外文学位 >Numerical solution of retarded and neutral delay differential equations using continuous Runge-Kutta methods.
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Numerical solution of retarded and neutral delay differential equations using continuous Runge-Kutta methods.

机译：使用连续Runge-Kutta方法的延迟和中立延迟微分方程的数值解。

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摘要

A delay differential equation (DDE) can provide us with a realistic model of many phenomena arising in applied mathematics. For example, a DDE can be used for the modeling of population dynamics, the spread of infectious diseases, and two-body problems of electrodynamics. In this thesis, we present a numerical method for solving DDEs and analysis for the numerical solution.;We have developed the method by adapting recently developed techniques for initial value ordinary differential equations (continuous Runge-Kutta formulas, defect error control, and an automatic handling technique for derivative discontinuities) and developing a new approach (that is based on an iterative scheme determined by extrapolation or a special interpolant) to handle vanishing delays. This results in a robust variable stepsize method which can be applied to problems with state dependent delays. This approach can also be applied to any continuous Runge-Kutta formula.;We determine convergence properties for the numerical solution associated with our method. We first analyze such properties for retarded type DDEs and then the analysis is extended to the case of neutral type DDEs. The main theoretical result we establish is that the global error of the numerical solution is bounded by a multiple of the prescribed tolerance. Our analysis can also be applied to other numerical methods based on Runge-Kutta formulas for DDEs.;We have developed an experimental Fortran code as a modified version of DVERK based on a 6th order continuous Runge-Kutta formula. An implementation of our method is described and numerical results for various kinds of DDEs are presented. Numerical comparisons with two other existing codes, ARCHI and DRKLAG, are considered and some advantages of our approach are identified.
机译：延迟微分方程（DDE）可以为我们提供应用数学中出现的许多现象的真实模型。例如，DDE可以用于人口动力学，传染病传播和电动力学两体问题的建模。在本文中，我们提出了一种求解DDE的数值方法，并对数值解进行了分析。;我们通过将最新开发的技术应用于初值常微分方程（连续Runge-Kutta公式，缺陷误差控制和自动处理微分不连续性的技术），并开发一种新方法（基于由外推法或特殊内插法确定的迭代方案）来处理消失的延迟。这导致了一种鲁棒的可变步长调整方法，该方法可以应用于状态依赖的延迟问题。这种方法也可以应用于任何连续的Runge-Kutta公式。;我们确定与我们的方法有关的数值解的收敛性。我们首先分析延迟型DDE的这种特性，然后将分析扩展到中性型DDE的情况。我们建立的主要理论结果是，数值解的整体误差受规定公差的倍数限制。我们的分析还可以应用于基于DDE的Runge-Kutta公式的其他数值方法。我们已经基于6阶连续Runge-Kutta公式开发了实验性的Fortran代码，作为DVERK的修改版本。描述了我们方法的实现，并给出了各种DDE的数值结果。考虑了与其他两个现有代码（ARCHI和DRKLAG）的数值比较，并确定了我们方法的一些优势。

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  • 作者

    Hayashi, Hiroshi.;

  • 作者单位

    University of Toronto (Canada).;

  • 授予单位 University of Toronto (Canada).;
  • 学科 Engineering Mechanical.;Computer Science.
  • 学位 Ph.D.
  • 年度 1996
  • 页码 92 p.
  • 总页数 92
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

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