The Modified Local Green's Function Method for the solution of the anoma- lous diffusion equation

Resumo

The field of fractional calculus is currently gaining momentum in mathematics and has extensive practical applications across several scientific and engineering problems, particularly those that involve nonlocality. This study aims to enhance the discourse surrounding the utilization of numerical techniques for solving problems governed by partial differential equations with fractional time derivatives. To this end, the present work employs an enriched formulation of the Modified Local Green's Function Method (MLGFM) to solve the anomalous diffusion equation in two dimensions. The anomalous, or fractional, diffusion equation presents a time-derivative of

non-integer order, in the interval (0,1]. When the order of the time-derivative is equal to 1, the classical diffusion equation is recovered, which means that it can be treated as the simplest case of the fractional diffusion equation. To represent the fractional time derivative, the Caputo representation is chosen based on the authors’ previous work. The MLGFM is an integral method hybrid of the Finite Element Method (FEM) and the Boundary Element Method (BEM). The method uses the FEM to create discrete projections of the Green's functions and use them as fundamental solutions in BEM formulation. The MLGFM presents high convergence for the potential in the domain, inherited from the FEM, and for the normal flux in the boundary, inherited from the BEM. This paper proposes a trigonometric enrichment based on the Generalized Finite Element technique for the solution of the anomalous diffusion equation. The results are compared with analytical solutions available in the literature.