In this work, we consider and analyze explicit exponential Runge–Kutta methods for solving semilinear time-fractional integro-differential equation, which involves two nonlocal terms in time. Firstly, the temporal Runge–Kutta discretizations follow the idea of exponential integrators. Subsequently, we utilize the spectral Galerkin method to introduce a fully discrete scheme. Then, we mainly focus on discussing the one-stage and two-stage methods for solving the proposed semilinear problem. Based on special abstract settings, we perform the convergence analysis for the proposed two different stage methods. In this process, we heavily use estimates about the operator family {S̃(t)}, and in combination with Lipschitz continuous condition. Finally, some numerical experiments confirm theoretical results. Meanwhile, applying this scheme to the related linear problem yields high-order convergence, highlighting the advantages of explicit exponential Runge–Kutta methods.

中文翻译：

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用于半线性时间分数积分微分方程的显式指数 Runge-Kutta 方法


在这项工作中，我们考虑并分析了用于求解半线性时间分数积分微分方程的显式指数 Runge-Kutta 方法，该方程涉及两个非局部时间项。首先，时间 Runge-Kutta 离散化遵循指数积分器的思想。随后，我们利用光谱 Galerkin 方法引入了一种完全离散的方案。然后，我们主要讨论解决所提出的半线性问题的一阶段和两阶段方法。基于特殊的抽象设置，我们对提出的两种不同的阶段方法进行收敛分析。在此过程中，我们大量使用有关算子族 {S̃（t）} 的估计值，并结合 Lipschitz 连续条件。最后，一些数值实验证实了理论结果。同时，将此方案应用于相关的线性问题会产生高阶收敛，突出了显式指数 Runge-Kutta 方法的优势。