The numerical solution of differential equations of fractional order is a notoriously difficult and complex matter, mainly due to the nonlocality of the operators and the nonsmoothness of the exact solutions. The former leads to a very high arithmetic complexity, and the latter imposes severe challenges regarding the construction of numerical methods with a high convergence order. Baffet and Hesthaven address these issues with an unconventional combination of new and known ideas.

In particular, they decompose the operators arising in the problem into two parts. One part describes the history of the process under consideration but does not contain any components that give rise to nonsmoothness. This feature allows the use of kernel compression techniques to provide a fast and accurate approximation of this part. The second component is then strictly local but contains the singularities, so that it can be handled by suitable generalizations of classical numerical integration schemes for singular integrands.

Although a full theoretical analysis is not available yet, the numerical examples given in the paper indicate that the combination of these two ingredients leads to a very efficient overall approach. As Baffet and Hesthaven demonstrate, it is also possible to add a stepsize control procedure to the algorithm, thus allowing the use of an adaptively chosen grid, which should improve the efficiency even more.

Very many papers describing numerical methods for computationally solving differential equations of fractional order have been published in recent years. Since it contains really novel ideas that are likely to have a lasting impact on the development of the field, Baffet and Hesthaven’s text is one of the few works that are actually worth reading for members of the scientific computing and mathematical modeling communities who have an interest in solving such equations.