Numerical Solutions of Fractional Differential Equations

In this thesis, we focus on numerical solutions of general linear multi-term fractional differential equations (FDEs) with fractional derivatives defined in the Caputo sense. Multi-term fractional order differential equations are involving both ordinary and fractional derivative operators. Numerical methods plays very crucial role for solving fractional differential equations, since analytical solutions are not always possible for solving them. Memory trait of fractional calculus is one of the main reason for difficulty of developing analytical techniques for such a equations. Therefore, there has been considerable interest in solving FDEs numerically in recent years and many powerful schemes have been developed. Essentially, most of the developed methods are modified from original versions for classical differential equations and applied to FDEs. In this study, we introduce a numerical technique based on the fractional Taylor vector and we construct fractional Taylor operational matrix of fractional integration to solve multi-term FDEs. The main characteristic of this technique is to reduce the given IVP of fractional order to a system of algebraic equations by employing the fractional Taylor operational matrix of fractional integration. Finally, this set of algebraic equations can be solved easily and efficiently for unknown coefficients by using computer programming. Consequently, by using these coefficients, the approximate solution of the given problem can be obtained. Some numerical examples are presented to demonstrate the accuracy and applicability of given method. The approximate solutions obtained by use of given technique are compared with numerical results of some other methods in literature and exact solutions of given problems. From these results, we can conclude that the presented technique is efficient and applicable for solving high order multi-term fractional order differential equations numerically. Keywords: numerical solutions, fractional Taylor vector,fractional differential equations, spectral method, Caputo fractional derivative, Riemann-Liouville fractional integral, operational matrices.