Numerical Solutions of Fractional Differential Equations

Fractional analysis has almost the same history as classical calculus. Fractional analysis did not attract enough attention for a long time. However, in recent decades, fractional analysis and fractional differential equations become very popular because of its powerful applications. A large number of new differential models that involve fractional calculus are developed. For most fractional differential equations we can not provide methods to compute the exact solutions analytically. Therefore it is necessary to revert to numerical methods. The structure of this thesis is arranged in the following way. We begin by recalling some classical facts from calculus. Partically, we recall definition and some properties of gamma, beta and Mittag-Leffler function. Then, in Chapter 3, we introduce the fundamental concepts and definitions of fractional calculus. This includes, in particular, some basic results concerning Riemann–Liouville differentiation and integration, and basic properties of Caputo derivative. In Chapter 4 we discuss fractional variant of the classical second-order Adams–Bashforth–Moulton method. It has been introduced by K. Diethelm, A.D. Freed, and discussed in book by K. Diethelm. Keywords: R-L Fractional Derivative, Caputo Fractional Derivative, Adams-Bashforth- Moulton Method, Fractional Differential Equations