关于具有无穷多个导数的线性微分方程
On linear differential equations with infinitely many derivatives
摘要：
具有无穷多个导数的微分方程，有时也被称为"非定域“微分方程，经常出现在现代物理学的分支中，如弦论，引力和宇宙学。本文的目的是展示如何正确地解释和解决这些方程，特别侧重于基于Borel变换的求解方法。这种方法是对以前方法的一种意义深远的推广（N。巴纳比和N. Kamran，Dynamics with Infinite Many Derivatives：The Initial Value Problem。{ \em J. High Energy Physics} 2008 no. 02，Paper 008，40 pp.; P. Górka，H. Prado和E.G. Reyes，通过拉普拉斯变换和具有无穷多个导数的方程的函数演算。{ \em Journal of数学物理} 51（2010），103512; P. Górka，H. Prado和E.G. Reyes，具有无穷多个导数的常方程的初值问题。{ \em Classical and量子引力} 29（2012），065017）。特别是，我们重新考虑广义初值问题，并反驳了现代文献中发现的各种aptures。我们说明了各种现象，可以发生与具体的例子，我们也对待有效的实现理论。
英文摘要：
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. Górka, 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. Górka, 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.
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