New methods for the study and resolution of equations involving fractional operators and their applications

Resumo

The main topic of research in this dissertation is Mathematical Analysis and, more specifically, Fractional Calculus. We provide new techniques to study, and sometimes solve, fractional integral equations and fractional differential equations. The first two chapters of this thesis provide the introductory notions to be used in the rest of the research. In the third chapter, we provide an original algorithm to find the unique solution of a linear fractional integral equation of constant coefficients. In the fourth chapter, we apply the previously obtained results to fractional differential equations. In particular, we see their implications in the problem of imposing natural initial conditions in such a way that the existence and uniqueness of solution is guaranteed. In the fifth chapter, we develop some applications based on the aforementioned ideas. In the sixth chapter, we discuss conclusions and possible lines of future work.