Métodos numéricos para solução de equações diferenciais segundo a derivada de caputo

Abstract

This thesis deals with the subject of non-integer order calculus mainly numerical methods or solving differential equations and dynamic systems analysis, according to the definition of the Caputo derivative.In this work, a brief historical review on the subject will be given in a general covering the period from the first discussion up to the 1970s. Due to the large number of papers published from the 1970s, it was decided to concentrate in a review for the areas of numerical methods and dynamic systems from the 1970s until the present day. As a theoretical framework, an analytical introductory background of a Riemann-Liouville(used in the definition of Caputo), Caputo and Günwald-Letnikov (used to obtain numerical methods) derivatives will be given. As well as a review on the existence and uniqueness of differential equations for Riemann-Liouville and Caputo derivatives. The main results of this thesis are: i) a counter example to the theorem of existence and uniqueness for differential equations derived according to the Riemann-Liouville found inthe literature, and will propose a reformulation of the theorem; ii) a hybrid method for solving Caputos differential equations; iii) method with concentrated hereditary memory in the initial condition (MHCCI), also used in Caputos differential equations, but only in the cases in which the system does not explicitly depend on time. A preliminary analysisof the stability of fixed points for Caputos differential equations will be shown using the MHCCI method. The results suggest a generalization of the indirect method of Lyapunov.