The aim of this paper is to construct and analyze explicit exponential Runge-Kutta methods for the temporal discretization of linear and semilinear integro-differential equations. By expanding the errors of the numerical method in terms of the solution, we derive order conditions that form the basis of our error bounds for integro-differential equations. The order conditions are further used for constructing numerical methods. The convergence analysis is performed in a Hilbert space setting, where the smoothing effect of the resolvent family is heavily used. For the linear case, we derive the order conditions for general order $p$ and prove convergence of order $p$, whenever these conditions are satisfied. In the semilinear case, we consider in addition spatial discretization by a spectral Galerkin method, and we require locally Lipschitz continuous nonlinearities. We derive the order conditions for orders one and two, construct methods satisfying these conditions and prove their convergence. Finally, some numerical experiments illustrating our theoretical results are given.

翻译：本文的目的是为了构建和分析线性和半线性内分泌异方程的时间分解的显性龙格-库塔方法。 通过扩大溶液中数字方法的误差,我们得出构成我们内分解方程误差基础的顺序条件。 命令条件被进一步用于构建数字方法。 趋同分析在Hilbert空间环境中进行, 大量使用固态家族的平滑效果。 对于线性案例, 我们得出一般顺序的顺序条件 $p$, 并证明在满足这些条件时, 单价 $p$ 的趋同。 在半线性案例中, 我们考虑用光谱加列金法来补充空间分解, 我们要求本地的Lipschitz连续的非线性。 我们为一级和二级命令得出顺序的顺序条件, 构建满足这些条件的方法, 并证明这些条件的趋同性。 最后, 提供了一些说明我们理论结果的数字实验。