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Numerical integration of a differential-difference equation with a decreasing time-lag

Authors:

R. E. Bellman,

J. D. Buell,

R. E. KalabaAuthors Info & Claims

Communications of the ACM, Volume 8, Issue 4

Pages 227 - 228

https://doi.org/10.1145/363831.364879

Published: 01 April 1965 Publication History

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Abstract

Systems in which variable time-lags are present are of common occurrence in biology. Variable flow rates are a common cause of these variable lags. At present no extensive body of knowledge exists concerning the effects which these variable lags can cause. Shown here is a method of reducing some differential-difference equations to ordinary differential equations which can then be studied numerically with ease. Subsequent study will deal with situations in which multiple-lags and lags dependent on the solution itself are present.

References

[1]

BELLMAN, R. E., JACQUEZ, J., AND KALABA, R. E. Mathematical models of chemotherapy. Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Vol. IV, Biology and Problems of Health. U. of California Press, Berkeley, 1961, 57-66.

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[2]

--. From chemotherapy to computers to trajectories. In Mathematical Problems in the Biological Sciences. R. E. Bellman (Ed.), Amer. Math. Soc., Providence, R. I., 1962, 225- 231.

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[3]

--, KALABA, R. E., AND KOTKIN, B. Differential approximation applied to the solution of convolution equations. Math. Comput. 18, 87 (July 1964), 487-491.

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Baker CPaul CWillé D(2023)Issues in the numerical solution of evolutionary delay differential equationsAdvances in Computational Mathematics10.1007/BF029886253:1(171-196)Online publication date: 22-Mar-2023
https://dl.acm.org/doi/10.1007/BF02988625
Farhloul MFortin M(2023)A mixed finite element for the stokes problem using quadrilateral elementsAdvances in Computational Mathematics10.1007/BF024319983:1-2(101-113)Online publication date: 22-Mar-2023
https://dl.acm.org/doi/10.1007/BF02431998
Isaia V(2021)A Fixed Point Approach to Simulation of Functional Differential Equations with a Delayed ArgumentJournal of Dynamics and Differential Equations10.1007/s10884-021-10081-735:2(1047-1082)Online publication date: 7-Oct-2021
https://doi.org/10.1007/s10884-021-10081-7
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Index Terms

Numerical integration of a differential-difference equation with a decreasing time-lag
1. Mathematics of computing
  1. Mathematical analysis
    1. Differential equations
      1. Ordinary differential equations
      2. Partial differential equations

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Communications of the ACM Volume 8, Issue 4

April 1965

63 pages

ISSN:0001-0782

EISSN:1557-7317

DOI:10.1145/363831

Editor:
C. C. Gotlieb
Univ. of Toronto, Toronto, Ont., Canada

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 April 1965

Published in CACM Volume 8, Issue 4

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Baker CPaul CWillé D(2023)Issues in the numerical solution of evolutionary delay differential equationsAdvances in Computational Mathematics10.1007/BF029886253:1(171-196)Online publication date: 22-Mar-2023
https://dl.acm.org/doi/10.1007/BF02988625
Farhloul MFortin M(2023)A mixed finite element for the stokes problem using quadrilateral elementsAdvances in Computational Mathematics10.1007/BF024319983:1-2(101-113)Online publication date: 22-Mar-2023
https://dl.acm.org/doi/10.1007/BF02431998
Isaia V(2021)A Fixed Point Approach to Simulation of Functional Differential Equations with a Delayed ArgumentJournal of Dynamics and Differential Equations10.1007/s10884-021-10081-735:2(1047-1082)Online publication date: 7-Oct-2021
https://doi.org/10.1007/s10884-021-10081-7
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Kaushik ASharma M(2008)Numerical analysis of a mathematical model for propagation of an electrical pulse in a neuronNumerical Methods for Partial Differential Equations10.1002/num.2030124:4(1055-1079)Online publication date: 24-Mar-2008
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Kaushik A(2007)Pointwise uniformly convergent numerical treatment for the non-stationary Burger-Huxley equation using grid equidistributionInternational Journal of Computer Mathematics10.1080/0020716070131450584:10(1527-1546)Online publication date: 1-Oct-2007
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Kaushik A(2007)Nonstandard perturbation approximation and travelling wave solutions of nonlinear reaction diffusion equationsNumerical Methods for Partial Differential Equations10.1002/num.2024424:1(217-238)Online publication date: 19-Jun-2007
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Baker CPaul CWillé D(1995)Issues in the numerical solution of evolutionary delay differential equationsAdvances in Computational Mathematics10.1007/BF030283703:3(171-196)Online publication date: 1-Apr-1995
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Jackiewicz Z(1987)Variable-step variable-order algorithm for the numerical solution of neutral functional differential equationsApplied Numerical Mathematics10.1016/0168-9274(87)90036-53:4(317-329)Online publication date: 1-Aug-1987
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