In this paper, we propose a general approach for approximate simulation and analysis of delay differential equations (DDEs) with distributed time delays based on methods for ordinary differential equations (ODEs). The key innovation is that we 1) approximate the kernel by the probability density function of an Erlang mixture and 2) use the linear chain trick to transform the approximate DDEs to ODEs. Furthermore, we prove that an approximation with infinitely many terms converges for continuous and bounded kernels and for specific choices of the coefficients. We compare the steady states of the original DDEs and their stability criteria to those of the approximate system of ODEs, and we propose an approach based on bisection and least-squares estimation for determining optimal parameter values in the approximation. Finally, we present numerical examples that demonstrate the accuracy and convergence rate obtained with the optimal parameters and the efficacy of the proposed approach for bifurcation analysis and Monte Carlo simulation. The numerical examples involve a modified logistic equation and a point reactor kinetics model of a molten salt nuclear fission reactor.

翻译：本文提出了一种基于常微分方程方法对具有分布时滞的时滞微分方程进行近似仿真与分析的一般性框架。核心创新点在于：1）采用Erlang混合分布的概率密度函数逼近核函数；2）运用线性链技巧将近似时滞微分方程转化为常微分方程组。进一步地，我们证明了在核函数连续有界且系数满足特定选择条件下，具有无穷多项的逼近形式具有收敛性。通过比较原时滞微分方程与近似常微分方程组的稳态解及其稳定性判据，我们提出了一种基于二分法和最小二乘估计的优化参数确定方法。最后，通过数值算例展示了采用优化参数所获得的精度与收敛速度，并验证了所提方法在分岔分析和蒙特卡洛仿真中的有效性。数值算例涉及修正的逻辑斯蒂方程和熔盐核裂变反应堆的点堆动力学模型。