A self-singularity-capturing scheme for fractional differential equations

Abstract

We develop a two-stage computational framework for robust and accurate time-integration of multi-term linear/nonlinear fractional differential equations. In the first stage, we formulate a self-singularity-capturing scheme, given available/observable data for diminutive time, experimentally obtained or sampled from an approximate numerical solution utilizing a fine grid nearby the initial time. 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. In the second stage, we utilize the multi-singular behaviour 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 . We show that this novel framework can control the error even in the presence of strong multi-singularities.