An Automated Singularity-Capturing Scheme for Fractional Differential Equations

Abstract

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 multiterm 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-singularitycapturing 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 O(∆t 3−α ). 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.

Data-Driven Failure Prediction in Brittle Materials: A Phase-Field Based Machine Learning Framework

“…Future works will focus on addressing the noted issues. Besides, the focus of future studies will be on the evaluation of the developed framework with the inclusion of plasticity/visco-elasto-plasticity [45,46,50,48] in the damage and fatigue phase-field model, combined with efficient and stable long time integration schemes [47,56].…”

A Thermodynamically Consistent Fractional Visco-Elasto-Plastic Model with Memory-Dependent Damage for Anomalous Materials

“…• The developed discretization recovered the limit Hookean β → 0 and Newtonian β → 1 cases for the free-energy. In the presence of single-to multi-singularities, the accuracy of the developed scheme can improve through a variant of a self-singularity-capturing approach [53] for all fractional operators. Nevertheless, non-smooth loading/unloading conditions pose additional challenges to develop high-order schemes for the model.…”

“…With particular arrangements of SB and standard elements, fractional models were applied, e.g., to describe the far from equilibrium power-law dynamics of multifractional visco-elastic [23,26,[37][38][39][40], distributed visco-elastic [17] and visco-elastoplastic [25,50,51,54,59] complex materials. Concurrently, significant advances in numerical methods allowed numerical solutions to time-and space-fractional partial differential equations (FPDEs) for smooth/non-smooth solutions, such as finitedifference (FD) schemes [32,34], fractional Adams methods [16,60], implicit-explicit (IMEX) schemes [11,63], spectral methods [44,45], fractional subgrid-scale modeling [43], fractional sensitivity equations [29], operator-based uncertainty quantification [28] and self-singularity-capturing approaches [53].…”

“…Database: ε ve and 2N − 1 coefficients b Compute ∆ε ve n+1 using(53), and form ∆ε ve * n+1 using(56). Compute the FFT F(∆ε ve * n+1 ).4: Compute c (β E ) n+1 using (55), using the known b (β E ) coefficients.…”

Fast IMEX Time Integration of Nonlinear Stiff Fractional Differential Equations

“…It is known that time-fractional operators possess power-law kernels with a singularity nearby the initial time, which produces non-smooth solutions that deteriorate the accuracy of many existing numerical schemes. In order to handle such problem, Lubich [28] introduced the so-called correction method, which was later applied to a series of direct/multi-step schemes for linear/nonlinear FDEs [6,[59][60][61], and also employed in a self-singularity-capturing approach by Suzuki and Zayernouri [47]. In the aforementioned works, the correct determination of singularity powers leads to global high accuracy of the numerical schemes.…”