Mittag–Leffler time differencing for fractional oscillation models with nonlinear force

In this paper, we develop Mittag-Leffler time differencing (MLTD) schemes for solving nonlinear fractional oscillation differential equations of order α∈(1,2) with both smooth and nonsmooth forcing terms. The proposed multistep schemes are designed to achieve uniform optimal convergence rates on graded meshes, which are particularly crucial for addressing the challenges posed by nonsmooth forcing terms. Partial fraction decompositions of rational approximations for the Mittag-Leffler function are utilized to enhance computational efficiency. Numerical examples, including oscillatory problems for both scalar equations and systems, are provided to demonstrate the robustness and efficiency of the proposed schemes for both smooth and nonsmooth cases, validating the theoretical results. This study highlights the adaptability of exponential time differencing technique for nonlinear fractional oscillation problems, offering a reliable and efficient framework for real-world applications involving fractional dynamics with varying levels of forcing term regularity.