Stability of the discretized pantograph differential equation

Buhmann, Martin; Iserles, Arieh

doi:10.1090/S0025-5718-1993-1176707-2

Stability of the discretized pantograph differential equation
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Math. Comp. 60 (1993), 575-589 Request permission

Abstract:

In this paper we study discretizations of the general pantograph equation \[ y’(t) = ay(t) + by(\theta (t)) + cy’(\phi (t)),\quad t \geq 0,\quad y(0) = {y_0},\], 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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