Stability of the discretized pantograph differential equation

Source abstract

In this paper we study discretizations of the general pantograph equation $$y ′ ( t ) = a y ( t ) + b y ( θ ( t ) ) + c y ′ ( ϕ ( t ) ) , t ≥ 0 , y ( 0 ) = y 0 , y’(t) = ay(t) + by(\theta (t)) + cy’(\phi (t)),\quad t \geq 0,\quad y(0) = {y_0},$$ , where a, b, c , and y 0 {y_0} are complex numbers and where θ \theta and ϕ \phi are strictly increasing functions on the nonnegative reals with θ ( 0 ) = ϕ ( 0 ) = 0 \theta (0) = \phi (0) = 0 and θ ( t ) > t , ϕ ( t ) > t \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 ∞ \{ {y_n}\} _{n=0}^\infty is bounded or that it tends to zero algebraically, as a negative power of n .