Second Advanced School on Integral Equations and Applications
Lisbon, May 18-20, 2017
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.