Iterative techniques with computer realization for the initial value problem for Caputo fractional differential equations

The main aim of the paper is to suggest some algorithms and to use them in an appropriate computer environment to solve approximately the initial value problem for scalar nonlinear Caputo fractional differential equations on a finite interval. The iterative schemes are based on appropriately defined lower and upper solutions to the given problem. A number of different cases depending on the type of lower and upper solutions are studied and various schemes for constructing successive approximations are provided. In some of the algorithms we do not use Mittag-Leffler functions and as a result the practical application of the algorithms are easier. Also we suggest two algorithms generalizing the monotone-iterative techniques and using Mittag-Leffler functions and in a practical problem computer software is build and used.