SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1736-1758, August 2024.
Abstract. We consider nonlinear delay differential and renewal equations with infinite delay. We extend the work of Gyllenberg et al. [Appl. Math. Comput., 333 (2018), pp. 490–505] by introducing a unifying abstract framework, and we derive a finite-dimensional approximating system via pseudospectral discretization. For renewal equations, we consider a reformulation in the space of absolutely continuous functions via integration. We prove the one-to-one correspondence of equilibria between the original equation and its approximation, and that linearization and discretization commute. Our most important result is the proof of convergence of the characteristic roots of the pseudospectral approximation of the linear(ized) equations when the collocation nodes are chosen as the family of scaled zeros or extrema of Laguerre polynomials. This ensures that the finite-dimensional system correctly reproduces the stability properties of the original linear equation if the dimension of the approximation is large enough. The result is illustrated with several numerical tests, which also demonstrate the effectiveness of the approach for the bifurcation analysis of equilibria of nonlinear equations. The new approach used to prove convergence also provides the exact location of the spectrum of the differentiation matrices for the Laguerre zeros and extrema, adding new insights into properties that are important in the numerical solution of differential equations by pseudospectral methods.

中文翻译：

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无限延迟方程：抽象框架中数值稳定性和分岔的伪谱离散化


《SIAM 数值分析杂志》，第 62 卷，第 4 期，第 1736-1758 页，2024 年 8 月。

抽象的。我们考虑具有无限延迟的非线性时滞微分和更新方程。我们扩展了 Gyllenberg 等人的工作。 [应用。数学。 Comput., 333 (2018), pp. 490–505]通过引入统一的抽象框架，我们通过伪谱离散化推导出有限维逼近系统。对于更新方程，我们考虑通过积分在绝对连续函数空间中重新表述。我们证明了原方程与其近似方程之间的平衡一一对应，以及线性化和离散化可交换。我们最重要的结果是当配置节点被选为拉盖尔多项式的缩放零点或极值族时，线性（化）方程的伪谱近似的特征根的收敛性证明。如果近似的维数足够大，这可以确保有限维系统正确地再现原始线性方程的稳定性特性。通过多次数值试验验证了结果，也证明了该方法用于非线性方程平衡分岔分析的有效性。用于证明收敛性的新方法还提供了拉盖尔零点和极值的微分矩阵谱的精确位置，为通过伪谱方法数值求解微分方程的重要性质添加了新的见解。