Boundary value problems for partial differential equations with piecewise constant delay (Q1175438)

This note continues the earlier work of the author and others in an attempt to extend the theory of functional differential equations with continuous argument to differential equations with discontinuous argument deviations. The study of ordinary differential equations with piecewise constant argument (EPCA) was initiated by the author [Lect. Notes Pure Appl. Math. 90, 547-552 (1984; Zbl 0531.34059)], \textit{S. M. Shah} and the author [Int. J. Math. Math. Sci. 6, 671-703 (1983; Zbl 0543.34067)] and \textit{K. L. Cooke} and the author [J. Math. Anal. Appl. 99, 265-297 (1984; Zbl 0557.34059)]. These equations represent a hybrid of continuous and discrete dynamical systems and combine properties of both differential and difference equations. They include as particular cases loaded and impulse equations, hence their importance in control theory and in certain biomedical models. Continuity of a solution at a point joining any two consecutive intervals implies recursion relations for the values of the solution at such points. Therefore, EPCA are intrinsically closer to difference rather than differential equations. Here boundary value problems for some linear EPCA in partial derivatives are considered and the behavior of their solutions studied. The results are also extended to equations with positive definite operators in Hilbert spaces. This topic is of considerable theoretical, computational and applied interest, because it creates the possibility of approximating some complicated problems of mathematical physics by simpler EPCA.