The Collocation Method in the Numerical Solution of Boundary Value Problems for Neutral Functional Differential Equations. Part I: Convergence Results

We consider the numerical solution of boundary value problems for general neutral functional differential equations by the collocation method. The collocation method can be applied in two versions: the finite element method and the spectral element method. We give convergence results for the collocation method deduced by the convergence theory developed in [S. Maset, Numer. Math., (2015), pp. 1--31] for a general discretization of an abstract reformulation of the problems. Such convergence results are then applied in Part II [S. Maset, SIAM J. Numer. Anal., 53 (2015), pp. 2794--2821] of this paper to boundary values problems for a particular type of neutral functional differential equations, namely, differential equations with deviating arguments

On the stability, boundedness, and square integrability of solutions of third order neutral delay differential equations

In this paper, sufficient conditions are established for the stability, boundedness and square integrability of solutions for some non-linear neutral delay differential equations of third order. Lyapunov’s direct method is used to obtain the results

Discontinuous almost periodic type functions, almost automorphy of solutions of differential equations with discontinuous delay and applications

In this work, using discontinuous almost periodic type functions, exponential dichotomy and the notion of Bi-almost automorphicity we give sufficient conditions to obtain a unique almost automorphic solution of a quasilinear system of differential equations with piecewise constant arguments. Finally, an application to the Lasota–Wazewska model with piecewise constant delayed argument is given