A novel adaptive procedure for solving fractional differential equations

In this paper, a novel adaptive procedure for step size selection for fractional differential equations is presented. The new adaptive approach is based on the implementation of a single numerical method and uses two numerical approximations, obtained at two successive steps, to advance the computation. We define a step size selection function that allows to adapt the size of the step according to the behaviour of solution. The new approach is easy to implement and leads to a low computational cost compared to classic step doubling procedure. The reported numerical results are satisfactory and show that our adaptive approach attains more accurate results than the results obtained on uniform grids, and results as good as the step doubling procedure but with very low implementation and computational effort.