Title
Numerical Analysis of Fractional Differential Equations of Caputo type

Instructors
Luísa Morgado (University of Trás-os-Montes e Alto Douro)
Magda Rebelo (New University of Lisbon)

Abstract
In this course we focus on numerical methods for differential equations of noninteger order, in the Caputo sense.
The course is divided in two main parts.
In the first one, after a brief review of the basic definitions of fractional derivatives we start with the theory of initial value problems of Caputo type. We emphasize the equivalence of such problems with singular Volterra integral equations, and we present and discuss several numerical schemes to solve them. Then terminal boundary value problems are considered and the equivalence between them and singular Fredholm integral equations is established. Once again some numerical methods are presented. In both cases, we consider ordinary single-term equations, and we highlight the main issues arising in the numerical approximation of this kind of problems, which are mainly related to the non-locality of the fractional differential operator, and to the existence of singular solutions. Later, we discuss how these ideas can be extended to solve multi-term or partial differential problems.
The second part is devoted to the numerical analysis of distributed-order ordinary and partial differential equations. The distributed-order derivative is a linear operator, defined as a weighted integral of different differentiation orders over a certain range. Thus, distributed-order differential equations can be regarded as natural generalizations of differential equations of integer and fractional order. The available literature on this type of derivatives is still scarce and only very few studies were devoted to the numerical approximation of such models. We will present the ones for Caputo type derivatives.