ON THE SOLUTIONS OF SOME PARTIAL DIFFERENTIAL EQUATIONS WITH LAGGING ARGUMENTS

Abstract

A fundamental problem in science is the problem of obtaining a mathematical model of a physical system or process. Usually a scientific theory developed by the theoretician postulating a mathematical model, based perhaps on some physical laws; and this model reasonably represents the process under consideration.It is then possible to make a prediction from the model, or design controls for the process so that it functions in some desirable manner. By far the largest class of mathematical models are those given in terms of differential equations . our aim will be to examine for a certain type of differential equations, "Differential Equations with lagging arguments II ,• specially equations with constant operators and equations of scalar coefficients, the formula of the solution and its

exponential growth.

- Around 1956, z. r.AfCHLfCKII (11], studied the "Differential- Equation with lagging arguments " of the form dy ---- - A y(t - a) = x(t) (1) dt with initial conditions Q$ t <

y(t) = e ts o

where A is a scalar, is the zero element in Banach space.Around 1 1967, with a completely different approach z. I.Rf£HL!CKII again studied "Differential Equation with lagging arguments " [12) in its general form which is given by