Green's functions for the fourth-order diffusion equation

Distinct physical processes are involved in the representation of diffusion phenomenon. On the other side, the more commonly employed differential equation for diffusion in the reviewed literature does not provide neither complete nor necessarily adequate mathematical representation of such processes what ends up to still be an ongoing unsolved challenge. In particular, this complex task of mathematical modeling is more visible in when, for instance, a set of diffusive particles are characterized by the simultaneous existence of two states of energy, possessing both a deceleration and temporary retention of particles. The fundamental hypothesis considered in this paper is that there is a predominant flux, which is the one represented by the potential defined by means of the Fick's law. In addition, after a short period, other possible excitation microstates begin to exist, leading to the modeling of such physical phenomena using the fourth-order anomalous diffusion equation. Given the exposed framework, the present work contributes with the development of the analytical solution of a partial differential equation (PDE) that represents the diffusion process caused by a pulse of local and instantaneous energy. Green's function is obtained through the solution of a specific PDE associated with the equation governing the original problem and in the same domain with the support of Fourier and Laplace transforms. These expressions were then validated using the finite difference numerical method. The successful proposed approach allowed us to produce completely new results for the diffusion and propagation processes, also taking into account retention processes or retardation effects, propagation disorders, acceleration or deceleration, or even fluid stagnation.