Transitional processes in linear stochastic parabolic and hyperbolic systems with constant delays

A scheme combining the classical method of steps with expansion of the state space (MSSSE) was earlier proposed for an analysis of systems of stochastic ordinary differential equations with one constant time delay (SODDEs). This two-stage scheme is adapted for the analysis of new models described by stochastic partial differential equations (SPDEs) with delays (SPDDEs). The modified scheme together with a usage of the generalized Fokker–Planck– Kolmogorov equation (FPK Eq.) for the probability density functional makes it possible to construct a chain of SPDEs without delays. We exploit this chain to obtain new sequence of PDEs for calculating the first moment functions (fields) of the solution on successive time intervals. Some results of symbolic and numeric calculations for parabolic and hyperbolic equations that carried out in the environment of the mathematical package Mathematica, are presented.