@incollection{Diethelm, author = {Diethelm, Kai}, title = {General theory of Caputo-type fractional differential equations}, series = {Handbook of Fractional Calculus with Applications, Vol. 2: Fractional Differential Equations}, booktitle = {Handbook of Fractional Calculus with Applications, Vol. 2: Fractional Differential Equations}, editor = {Kochubei, Anatoli and Luchko, Yuri}, publisher = {De Gruyter}, address = {Berlin}, doi = {10.1515/9783110571660-001}, pages = {1 -- 20}, abstract = {This article describes the fundamentals of the theory of ordinary fractional differential equations of Caputo's type. Starting from the existence and uniqueness of solutions and the well-posedness in general, the flow of topics continues via a derivation of explicit solution formulas for certain important classes of problems and the discussion of their smoothness properties to the stability properties of these solutions. The main focus is on initial value problems, but terminal value problems are briefly considered as well. In addition to dealing with standard single-order problems, the presentation also contains a short discussion of multiterm equations and multiorder systems.}, language = {en} } @incollection{Diethelm, author = {Diethelm, Kai}, title = {Fundamental approaches for the numerical handling of fractional operators and time-fractional differential equations}, series = {Handbook of Fractional Calculus with Applications, Vol. 3: Numerical Methods}, booktitle = {Handbook of Fractional Calculus with Applications, Vol. 3: Numerical Methods}, editor = {Karniadakis, George Em}, publisher = {De Gruyter}, address = {Berlin}, doi = {10.1515/9783110571684-001}, pages = {1 -- 22}, abstract = {This article describes fundamental approaches for the numerical handling of problems arising in fractional calculus. This includes, in particular, methods for approximately computing fractional integrals and fractional derivatives, where the emphasis is placed on Caputo operators, as well as solvers for the associated differential and integral equations.}, language = {en} }