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In this paper we present a comparison of numerical methods for the solution of single term fractional differential equations. We review five available methods and use a graphical technique to compare their relative merits. We conclude by giving recommendations on the choice of efficient methods for any given single term fractional differential equation.

1. Introduction. Over the past decade the development of numerical methods for the solution of equations containing derivatives and integrals of non-integer order has become a hot topic. There have been several simple algorithms published for producing approximate solutions for fractional differential equations. Despite the interest these algorithms have generated among mathematical modellers and applied scientists from several fields there seems to have been no systematic comparison of their relative merits from an applications viewpoint to date. Much theoretical work has been completed already, but as the recent paper [9] indicates, there can be a considerable gap between methods that perform well in theory and those whose practical implementations are effective. For existing theoretical work we refer to the recent works by authors such as Diethelm, Ford, Luchko and Podlubny (see [7],[10],[11],[12],[14],[21],[23].)