标题： 关于具有无限多个导数的线性微分方程 显示英文标题

标题： On linear differential equations with infinitely many derivatives

摘要： 具有无限多导数的微分方程，有时也被称为“非局部”微分方程，在现代物理的各个分支中频繁出现，如弦理论、引力和宇宙学。本文的目标是展示如何正确解释和求解这类方程，特别关注基于Borel变换的求解方法。这种方法是对以前方法的广泛推广（N. Barnaby和N. Kamran，无限多导数的动力学：初值问题。{\em J. 高能物理}2008年第02期，论文008，40页；P. Górka，H. Prado和E.G. Reyes，通过拉普拉斯变换的功能演算和具有无限多导数的方程。{\em 杂志 of 数学物理}51（2010），103512；P. Górka，H. Prado和E.G. Reyes，具有无限多导数的常微分方程的初值问题。{\em 经典和量子引力}29（2012），065017）。特别是我们重新考虑了广义的初值问题，并驳斥了现代文献中的各种猜想。我们用具体的例子说明了可能出现的各种现象，并且还讨论了该理论的有效实现。 显示英文摘要

摘要： Differential equations with infinitely many derivatives, sometimes also referred to as ``nonlocal'' differential equations, appear frequently in branches of modern physics such as string theory, gravitation and cosmology. The goal of this paper is to show how to properly interpret and solve such equations, with a special focus on a solution method based on the Borel transform. This method is a far-reaching generalization of previous approaches (N. Barnaby and N. Kamran, Dynamics with infinitely many derivatives: the initial value problem. {\em J. High Energy Physics} 2008 no. 02, Paper 008, 40 pp.; P. G\'orka, H. Prado and E.G. Reyes, Functional calculus via Laplace transform and equations with infinitely many derivatives. {\em Journal of Mathematical Physics} 51 (2010), 103512; P. G\'orka, H. Prado and E.G. Reyes, The initial value problem for ordinary equations with infinitely many derivatives. {\em Classical and Quantum Gravity} 29 (2012), 065017). In particular we reconsider generalized initial value problems and disprove various conjectures found in the modern literature. We illustrate various phenomena that can occur with concrete examples, and we also treat efficient implementations of the theory.