Response of a diffusion-wave system subjected to deterministic and stochastic fields

This paper presents a general analytical technique for a Fractional Diffusion-Wave (FDW) system subjected to a non-homogeneous field, which can be deterministic or stochastic. In this formulation, the response of the system is written as a linear combination of the eigenfunctions which are identified using the method of separation of variables. The properties of the eigenfunctions are used to reduce the FDW equation defined in space-time domain into a set of infinite Fractional Differential Equations (FDEs) defined in the time domain only. A Laplace transform technique is used to obtain the fractional Green's function and a Duhamel integral type expression for the system's response. For a stochastic analysis, the input field is treated as a random process with specified mean and correlation functions. An expectation operator is applied on a set of expressions to obtain the stochastic characteristics of the system. Several special cases are discussed and numerical results are presented to show the deterministic and stochastic response of FDW systems. When the order of the fractional derivative of the equation is < 1, 1, > 1, and 2 the equation represents a model of a sub-diffusion, a diffusion, a diffusion-wave propagation, and a wave propagation system, respectively. Thus the formulation provides a unified approach for stochastic analysis of all these systems. The approach can be equally applied to all those systems for which the existence of eigenmodes is guaranteed.