Error analysis of arbitrarily high-order stepping schemes for fractional integro-differential equations with weakly singular kernels.

We propose high-order computational schemes for solving nonlinear fractional integro-differential equations (FIDEs) that are commonly used to model systems with memory or long-term behavior. From the known structure of the smooth solution, we show that the solutions of such FIDEs are equivalent to those of Volterra integral equations (VIEs). The fractional integral appearing in the integral form of the resulting VIE is then split into a history term and a local term. Subsequently, we develop an efficient strategy that utilizes a combination of a kernel compression scheme and an integral deferred correction (IDC) scheme to obtain a high-order solution. The kernel compression scheme reduces the costs in approximating the history term, while the IDC scheme approximates VIEs on short intervals to obtain the local information. Error analysis demonstrates high-order accuracy of the proposed schemes, and numerical examples illustrate their effectiveness, particularly for nonlinear FIDEs. The results suggest that the proposed scheme provides accurate solutions even for large time steps, making it a valuable tool for researchers and engineers working on systems with memory or long-term behavior.