Partial differential equations with piecewise constant delay

Joseph Wiener,Lokenath Debnath

doi:10.1155/s0161171291000674

Abstract

The influence of certain discontinuous delays on the behavior of solutions to partial differential equations is studied. In Section 2, the initial value problems (IVP) are discussed for differential equations with piecewise constant argument (EPCA) in partial derivatives. A class of loaded partial differential equations that arise in solving certain inverse problems is studied in some detail in Section 3. Section 4 is devoted to obtain the solutions of IVP for linear partial differential equations with piecewise constant delay by using integral transforms. Finally, an abstract Cauchy problem is discussed.

Highlights

Functional differential equations (FDE) with delay provide a mathematical model for a physical or biological system in which the rate of change of the system depends upon iis past history
This paper continues our earlier work [1,2,3,4,5] in an attempt to extend this theory to differential equations with discontinuous argument deviations
Ordinary differential equations with arguments having intervals of constancy have been studied. Such 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, their importance in control theory and in certain biomedical models

Summary

INTRODUCTION

Functional differential equations (FDE) with delay provide a mathematical model for a physical or biological system in which the rate of change of the system depends upon iis past history. A class of loaded equations that arise in solving certain inverse problems is explored within the general framework of differential equations with piecewise constant delay. It has been shown in [6] that partial differential equations (PDE) with piecewise constant time naturally arise in the process of approximating PDE by simpler EPCA. The above method may be used to solve IVP for PDE of any order in with piecewise constant delay or systems of such equations. In the latter case, P and Q in (2.6) are square matrices of linear differential operators and u(x,O is a vector function. A Uo(s)e’, k k(P(is),tl),B Q(is)U(s, ti) 1-1 multiply by Qj(is) each of the equations

The equation

Consider the initial value problem