Formation of Fractional Derivative in Time due to Propagation of Free Green’s Function in Spatial Stochastic Disorder Field for Transport Phenomena

Abstract

In this paper, we develop perturbation method combined with Feynman’s diagram and graphic approach for evolution of average transport equation in disordered media. The disorder is a field, with a spatial gradient, and evolve basic average transport mechanism; via ‘perturbative technique’; where ‘disorder free Green’s function’ propagates through several realizations of spatial disorder, via fixed point equation; resulting in fractional derivative in time; as additional term representing spatial heterogeneity. For solving the singular integrals, arising out of infinite series via graphical technique in Fourier-Laplace space; Jordon’s lemma of complex analysis is used; which returns operator in Fourier-Laplace space; and inverting the same we get fractional derivatives in time domain for evolution of average; which is regarded as self-energy of disordered transport.