On linear differential equations with infinitely many derivatives

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.