On the Convergence of Waveform Relaxation Methods for Differential-Functional Systems of Equations

Abstract

In this paper the convergence of a waveform relaxation method applied to an initial value problem for the Volterra functional-differential system is discussed. It is shown that the method is convergent under the assumption that the splitting function satisfies only the one side Lipschitz condition with respect to some arguments and the Lipschitz condition with respect to the others. The conditions given in the paper also guarantee the existence and uniqueness of the solution to the initial problem discussed in the paper. The convergence of the perturbed continuous time waveform relaxation method is also discussed. ᮊ

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

“…The paper contains results which generalize the results of papers [1][2][3][4]. It complements the results of papers [5,6], where neutral differential-functional systems were not discussed and the results of paper [7], where only the convergence of WR method was discussed without paying any attention to error estimates.…”

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]). …”

Remarks on evaluation of spectral radius of operators arising in WR methods for linear systems of ODEs

“…However, the method was born much earlier, see, for instance, the papers [10][11][12] as well as a very general formulation due to Ważewski [27,28] and Kwapisz [13,14]. A number of papers discuss in details the convergence and error estimates of various WR methods (see, for example, [1][2][3][4]7,[17][18][19][20]25,29] and many others).…”