We propose a family of compactly supported parametric interaction functions in the general Cucker-Smale flocking dynamics such that the mean-field macroscopic system of mass and momentum balance equations with non-local damping terms can be converted from a system of partial integro-differential equations to an augmented system of partial differential equations in a compact set. We treat the interaction functions as Green's functions for an operator corresponding to a semi-linear Poisson equation and compute the density and momentum in a translating reference frame, i.e. one that is taken in reference to the flock's centroid. This allows us to consider the dynamics in a fixed, flock-centered compact set without loss of generality. We approach the computation of the non-local damping using the standard finite difference treatment of the chosen differential operator, resulting in a tridiagonal system which can be solved quickly.

中文翻译：

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Cucker-Smale模型中的半线性Poisson介导的植绒

我们提出了一个在一般的Cucker-Smale植绒动力学中的一组紧密支持的参数相互作用函数，这样，就可以从偏积分微分方程组中转换具有非局部阻尼项的质量和动量平衡方程的平均场宏观系统紧集中的偏微分方程的扩充系统。对于与半线性泊松方程相对应的算子，我们将相互作用函数视为格林函数，并在平移参考系（即参考鸡群质心的参考系）中计算密度和动量。这使我们可以考虑以固定的，以羊群为中心的紧凑集合中的动力学，而不会失去一般性。