Métodos de Diferenças Finitas para a Equação de Difusão Fracionária

Abstract

There is an extensive range of formulations involving the term fractional derivative, and this number continues to increase. This is due to the fact that fractional derivative is a complex and comprehensive mathematical concept that can be used to describe a wide range of phenomena. Considering the growing number of definitions, this work presents a study using numerical methods with six fractional derivatives: Riemann-Liouville, Caputo, Chen, Katugampola, Caputo-Fabrizio, and Atangana-Baleanu. These derivatives are applied to the fractional diffusion equation in one dimension with a constant coefficient, where the fractional temporal variation is simulated in the diffusion term. The selection of these operators was based on the classification carried out by Teodoro in his doctoral thesis, which, based on the criterion proposed by Ortigueira and Machado in 2015 − composed of five properties that help us conclude when a specific operator is a fractional derivative − verified whether these or many other operators can be considered fractional derivatives, according to the aforementioned criterion. The numerical approach adopted is the finite difference method, employing schemes based on the classical progressive and regressive Euler methods. The numerical analysis of stability and convergence is carried out using the forward and backward Euler-type method with the fractional derivative according to Riemann-Liouville. The imbalance generated by the fractional derivative in the fractional diffusion equation is corrected by inserting the dimensional correction parameter T into the model. The central contribution of this work was the development of numerical methods and the construction of respective codes for fractional-order diffusion. Numerical experiments are presented, displaying results with the aim of confirming theoretical conclusions, verifying the convergence rate, making comparisons between different approaches, and understanding the importance of the validity of the properties of the Ortigueira and Machado criterion for modeling. Algorithms, along with their corresponding Python language codes, are available.