Stability of the Discretized Pantograph Differential Equation

Abstract

Abstract. In this paper we study discretizations of the general pantograph equation y'(t) = ay(t) + by(6(t)) + cy'(<t>(t)), f>0, y(0)=y0, where a , b , c , and yo are complex numbers and where 9 and <¡> are strictly increasing functions on the nonnegative reals with 0(0) = <^>(0) = 0 and 8(t) < t, 4>(t) < t for positive /. 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 {yn/^o 's DOunded or that it tends to zero algebraically, as a negative power of n . X. IntroductionIn this article we shall consider, in the most general form, the generalized pantograph equationwhere a, b, c, and yo are complex and where 8 and <p are sufficiently smooth "delay" functions on the nonnegative reals which increase strictly monotonically and satisfy 0(0) = 0(0) = 0 and 6(t) < t, <p(t) < t for positive t. Thus, (1.1) is a differential equation which couples function values and derivatives on distinct time levels. When c vanishes, (1.1) is called a delay equation and the term cy'((j>(t)) is called the "neutral" term. Our principal focus in this paper is the case of proportional delays 8(t) = qt and 4>(t) = pt, where q and p are between 0 and 1. The equation (1.1) can also be considered for vector-valued y and matrices a, b, c, but we will dispense with this generalization here. There are many applications for the generalized pantograph equation. Here we only mention applications in number theory (Mahler [15]), in electrodynamics (Fox et al. [9]) and the collection of current by the pantograph of an electric locomotive (whence its name; cf. Ockendon and Tayler [18]), and in nonlinear dynamical systems (Derfel [6]). A more comprehensive list features in Iserles [12].Delay differential equations with constant delays, i.e., c = 0 and 6(t) = t-x, where one also prescribes y's values on (-t, 0), have been investigated extensively in the past (see, for instance, Bellman and Cooke [1,2] and Hale [10]).

A taylor collocation method for solving high-order linear pantograph equations with linear functional argument

“…j , and M r for α 1 = 0, β = 0 are defined in relations (9) and (13), respectively. Table I shows the solutions of Eq.…”

“…In particular, they turn out to be fundamental when ordinary differential equation (ODE)-based model fail. These equations arise in industrial applications [6,7] and in studies based on biology, economy, control, and electro-dynamic [8,9]. On the other hand, many methods based on Taylor polynomials have been given to find approximate solutions of the differential-difference and integro differential-difference equations [10][11][12][13][14][15][16].…”

About the existence of solutions for a hybrid nonlinear generalized fractional pantograph equation

“…The name pantograph originated from the work [5] on the collection of current by the pantograph head of an electric locomotive. There are a great number of papers devoted to the qualitative properties and numerical solutions of these equations (see, for example, [6][7][8][9][10]). …”

Stability analysis of $\theta$ -methods for neutral functional-differential equations

“…Equation (1.1) has been used as a test model for numerical methods by Buhmann and Iserles [3,4,5] and Buhmann, Iserles and Nørsett [6]. The long time dynamical behaviour of numerical solutions is the main issue in these papers.…”

“…These two methods can be viewed as a linear multi-step method and a linear one-leg method respectively. We will not discuss those methods used in [3,4,5,6], though similar to θ-methods with θ = 1/2, as we believe that θ-methods are more standard and more general. In Sect.…”