Exponential Integrators for Fractional Differential Equations

Exponential integrators are a well-established class of effective methods for the numerical integration of systems of differential equations of large dimension, especially in the presence of stiffness. Problems of this kind are common in several applications and usually come from semi-discretization of partial differential equations. The main idea behind exponential integrators is to solve in an exact way the stiff term of the problem and hence apply a time-step integration to the non-stiff term, thus to allow to use less expensive explicit schemes and obtain, at the same time, good stability properties. Recently, the investigation of exponential integrators has been generalized to differential equations of fractional order (FDEs). In this context exponential integrators turn out to be very effective since they overcome some of the typical weak points of numerical methods for FDEs: indeed, it is possible to derive methods with excellent stability properties and, when working with linear problems, to surmount some limitations in accuracy and order of convergence which represent a severe constriction for common methods for FDEs. In this Chapter we illustrate the main aspects concerning the derivation of some families of exponential integrators for FDEs and we study the convergence properties. Moreover, we address their numerical implementation with special attention to some peculiar aspects as, for example, the evaluation of Mittag-Leffler functions, with scalar and/or matrix arguments; this is a central feature in the implementation of exponential integrators for fractional order problems. The application to some time- fractional partial differential equations is also presented by means of some numerical examples.