An Automated Singularity-Capturing Scheme for Fractional Differential Equations

Solutions to fractional models inherently exhibit non-smooth behavior, which significantly deteriorates the accuracy and therefore efficiency of existing numerical methods. We develop a two-stage data-infused computational framework for accurate time-integration of single- and multi-term fractional differential equations. In the first stage, we formulate a self-singularity-capturing scheme, given available/observable data for diminutive time. In this approach, the fractional differential equation provides the necessary knowledge/insight on how the hidden singularity can bridge between the initial and the subsequent short-time solution data. We develop a new self-singularity-capturing finite-difference algorithm for automatic determination of the underlying power-law singularities nearby the initial data, employing gradient descent optimization. In the second stage, we can utilize the multi-singular behavior of solution in a variety of numerical methods, without resorting to making any ad-hoc/uneducated guesses for the solution singularities. Particularly, we employed an implicit finite-difference method, where the captured singularities, in the first stage, are taken into account through some Lubich-like correction terms, leading to an accuracy of order \(\mathcal{O}(\Delta t^{3-\alpha})\). Our computational results demonstrate that the developed framework can either fully capture or successfully control the solution error in the time-integration of fractional differential equations, especially in the presence of strong multi-singularities.