Efficient Numerical Methods for Fractional Differential Equations and their Analytical Background

Abstract

Efficient numerical methods for fractional differential equations and their theoretical background are presented. A historical review introduces and motivates the field of fractional calculus. Analytical results on classical calculus as well as special functions and integral transforms are repeated for completeness. Known analytical results on non-integer order differentiation and integrations are presented and corrected and extended where needed. On those results several numerical methods for the solution of fractional differential equations are based. These methods are described and compared to each other in detail. Special attention is paid to the question of applicability of higher oder methods and in connection the practical implementation of such methods is analyzed. Different ways of improvements of the presented numerical methods are given. Numerical calculations confirm the results which were deduced theoretically. Moreover, some of the presented methods are generalized to deal with partial differential equations of fractional order. Finally a problem of physics/chemistry is presented and some of the presented numerical methods are applied.