Numerical resolution of fractional partial differential equations using the Legendre's pseudo-spectral method and the Adams method

Abstract

The study of fractional partial differential equations is a topic of great importance in various areas of science and engineering. This is due to wide range of applications that can be developed based on this type of model. From a mathematical perspective, the fractional order characterizing each differential contribution can be interpreted as an additional parameter that can be adjusted for a given application. Solving such fractional models analytically or numerically fractional models constitutes a complex task, as the majority of models are inherently non-linear. In this context, the present work aims to extend the Legendre pseudo-spectral method to fractional context. Furthermore, to associate the proposed approach to traditional fractional Adams Method. The first approach is employed to transform the original model into a set of time-fractional ordinary differential equations, which are integrated considering the fractional Adams Method. To evaluate the proposed methodology, some case studies are solved and its numerical results are compared with analytical solutions. The results obtained demonstrate that the proposed methodology is an interesting strategy to solve this type of problem.