Delay dependent estimates for waveform relaxation methods for neutral differential-functional systems

Convergence analysis of waveform relaxation methods for neutral differential-functional systems

“…2 and 3 that, since the function K (t) in these two examples satisfies the condition K 1 (0) + K 2 (0) < 1, the WR method (4.3) is convergent. However, such convergence results cannot be guaranteed by [11] and [6], since max 0≤t≤2 K 1 (t) + K 2 (t) > 1 in both Examples 1 and 2. For the superlinear convergence aspect, from Tables 1 and 2, we find that compared with Example 2, 11 additional iterations with non-acceleration WR method are needed by Example 1 to achieve the error tolerance 10 −13 .…”

“…In [6], the convergence conditions are β(t) ≤ t and max 0≤t≤T (K 1 (t) + K 2 (t)) < 1. Our new conditions are β(t) < t and K 1 (0) + K 2 (0) < 1.…”

“…Since the other convergence conditions given in [6] depend on the initial approximation y 0 , we omit them at this moment.…”

“…Recently, articles [6,11] give some consideration of WR methods for Volterra functional-differential systems of the form y (t) = f (t, y(·), y (·)), t ∈ I = [0, T ], y(t) = g(t), t ∈ I τ = [−τ, 0], (1.1) where the function f : I × C g (I, R n ) × C g (I, R n ) → R n and g : [−τ, 0] → R n is a given initial function, which is continuous with its first-order derivative and satisfies the consistency condition g (0) = f (t, y(·), y (·))| t=0 . Here, we denote by C g (I, R n ) the class of continuous functions defined on I with values in R n which are equal to g for t ∈ [−τ, 0].…”

Waveform relaxation methods for fractional functional differential equations

“…And it is suitable excellently for parallel computation. For these virtues, the method has been applied to solve ordinary differential equations, differential-algebraic equations, functional differential equations, and partial differential equations (for example, see [4,5,6,14,15,16,32]). …”