High-Order Accurate Local Schemes for Fractional Differential Equations

High-order methods inspired by the multi-step Adams methods are proposed for systems of fractional differential equations. The schemes are based on an expansion in a weighted space. To obtain the schemes this expansion is terminated after terms. We study the local truncation error and its behavior with respect to the step-size h and P. Building on this analysis, we develop an error indicator based on the Milne device. Methods with fixed and variable step-size are tested numerically on a number of problems, including problems with known solutions, and a fractional version on the Van der Pol equation.