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Stability of the discretized pantograph differential equation


Authors: Martin Buhmann and Arieh Iserles
Journal: Math. Comp. 60 (1993), 575-589
MSC: Primary 65L20; Secondary 34K20
DOI: https://doi.org/10.1090/S0025-5718-1993-1176707-2
MathSciNet review: 1176707
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Abstract: In this paper we study discretizations of the general pantograph equation

$\displaystyle y'(t) = ay(t) + by(\theta (t)) + cy'(\phi (t)),\quad t \geq 0,\quad y(0) = {y_0},$

, where a, b, c, and $ {y_0}$ are complex numbers and where $ \theta $ and $ \phi $ are strictly increasing functions on the nonnegative reals with $ \theta (0) = \phi (0) = 0$ and $ \theta (t) < t, \phi (t) < t$ for positive t. Our purpose is an analysis of the stability of the numerical solution with trapezoidal rule discretizations, and we will identify conditions on a, b, c and the stepsize which imply that the solution sequence $ \{ {y_n}\} _{n=0}^\infty $ is bounded or that it tends to zero algebraically, as a negative power of n.

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DOI: https://doi.org/10.1090/S0025-5718-1993-1176707-2
Article copyright: © Copyright 1993 American Mathematical Society

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