Fractional Calculus and Symbolic Solution of Fractional Differential Equations

Abstract

The aim of this article is to present a practical introduction to fractional calculus. Fractional calculus is an old mathematical subject concerned with fractional derivatives. Fractional derivatives used in this paper are restricted to the Riemann-Liouville type. Based on the Riemann-Liouville calculus, we formulate fractional differential equations. Fractional differential equations are applied to models in relaxation and diffusion problems. Fractional calculus is used to formulate and to solve different physical models allowing a continuous transition from relaxation to oscillation phenomena. An application to an anomalous diffusion process demonstrates that the method used is also useful for more than one independent variable. Based on the theory of fractional derivatives and linear transformation theory, we demonstrate how symbolic calculations on a computer can be used to support practical calculations. The symbolic program FractionalCalculus based on Mathematica is used to demonstrate the solution of fractional differential equations step by step. The key method applied is linear transformation theory in connection with generalized functions.