Analytic and numerical exponential asymptotic stability of nonlinear impulsive differential equations

Stability Analysis of Analytical and Numerical Solutions to Nonlinear Delay Differential Equations with Variable Impulses

“…., the transformation function (2) can be reduced to ( ) = (1/(1 + )) { / }−1 . Then (4) is the same as the equivalent equations proposed in [9,13]. Proof.…”

Convergence analysis of three parareal solvers for impulsive differential equations

“…From (10), z = ∆T λ and Euler (z) = 1/(1 + z) is the stability function of the implicit Euler method applied to the regular ODE, α = 0 in (8). For a general RK method used as the propagator with the small step-size ∆t, the function f (z/J) can be derived similarly,…”

“…However, for many IDEs, even in the purely linear case, an analytic solution of simple form is often unavailable. In face of this situation, numerical computation is a natural choice [8][9][10][11] . The goal of this study is to study the parareal algorithm proposed by Lions et al 12 for a class of representative IDEs:…”

Asymptotical stability of Runge–Kutta methods for nonlinear impulsive differential equations

“…In recent years, the stability of numerical methods for IDEs has attracted more and more attention (see [11,12,15,17,22,29] etc.). Stability of Runge-Kutta methods with the constant stepsize for scalar linear IDEs has been studied by [17].…”

“…Stability of the exact and numerical solutions of nonlinear IDEs has been studied by the Lyapunov method in [11]. Stability of Runge-Kutta methods for a special kind of nonlinear IDEs has been investigated by the properties of the differential equations without impulsive perturbations in [15]. Stability and asymptotic stability of implicit Euler method for stiff IDEs in Banach space has been studied by [22].…”