High-order accurate local schemes for fractional differential equations (Q514460)

Various different approaches for the numerical solution of fractional-order differential equations are known today. Most of these methods can be interpreted as generalizations of classical and well-known schemes for first-order differential equations. A particularly frequently occurring example is the classical Adams-Bashforth-Moulton method for which various different ``fractionalizations'' are known. The paper under review adds another completely new algorithm to this collection. The idea is based on a carefully chosen expansion of the unknown solution's fractional derivative. This expansion is constructed in terms of an orthogonal basis of a suitable function space. A convergence analysis is provided, and a Milne-type error indicator is constructed. Based on the latter, the authors then develop and test a stepsize control strategy.