Anomalous diffusion equation modeled by the joint use of domain boundary element method and analytical derived solution based on green equation

Resumo

In some situations, the mathematical formulation of the diffusion phenomenon might be described
through a differential equation, which takes into account complementary and different effects with respect to the
physical processes simulated with the support of the Fick ́s equation, which is usually adopted to represent the
diffusion process. In particular, diffusion applied to spatial-temporal retention problems with bimodal mass
transmission is highlighted. To better understand this physical phenomenon, the proper use of the analytical
Green function (GF) was investigated. The formulation employs the steady-state fundamental solution. In
addition to the basic integral equation, another one is required, due to the fourth-order differential operator
introduced in the differential equation of the problem evaluated. The domain discretization employs linear cells.
The first order time derivative is approximated by a backward finite difference scheme. Two examples are
presented. Numerical results are compared with analytical solutions showing good agreement between them;
such framework provides a novel perspective for the use of the combined approach here developed to assess the
behavior of physical phenomena better described by the fourth order analytical equation based on Green
Function. In this work, the Domain Boundary Element Method (D-BEM) is explored to model that anomalous
diffusion process taking into consideration that we were also able to originally develop the Green analytical
solutions for the fourth order diffusion equation. Such combination of approaches proves to establish a new
conceptual reference in this area.