Efficient algorithms for solving the fractional ordinary differential equations

Fractional calculus and fractional differential equations are popular in describing anomalous diffusion, ground water flow and transport, and the price fluctuation in finance, etc. Some numerical methods are developed to solve the fractional ordinary differential equations. However, for most of these methods it seems that we always have to make a trade-off between efficiency and accuracy because of the non-local properties of fractional operators. In other words, for ensuring the accuracy, usually the computation cost is hard to accept; on the other hand if the computation cost is reduced then the accuracy is greatly lost. Based on the idea of equidistributing meshes, this paper designs efficient numerical schemes, which have linearly increasing computation cost with time t but not losing the accuracy at the same time. Error estimates for the proposed schemes are performed; and the numerical examples demonstrate the efficacy of our algorithms.