Implicit-Explicit difference schemes for nonlinear fractional differential equations with non-smooth solutions

TL;DR

This paper introduces second-order implicit-explicit schemes for nonlinear fractional differential equations with non-smooth solutions, ensuring accuracy, stability, and efficiency through correction terms and rigorous analysis.

Contribution

The paper develops and proves the convergence and stability of novel second-order IMEX schemes tailored for nonlinear fractional differential equations with non-smooth solutions.

Findings

Schemes achieve uniform second-order accuracy.

Numerical examples confirm effectiveness for complex systems.

Methods are flexible and computationally efficient.

Abstract

We propose second-order implicit-explicit (IMEX) time-stepping schemes for nonlinear fractional differential equations with fractional order $0<\beta<1$. From the known structure of the non-smooth solution and by introducing corresponding correction terms, we can obtain uniformly second-order accuracy from these schemes. We prove the convergence and linear stability of the proposed schemes. Numerical examples illustrate the flexibility and efficiency of the IMEX schemes and show that they are effective for nonlinear and multi-rate fractional differential systems as well as multi-term fractional differential systems with non-smooth solutions.