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Differential Equations with Infinitely Many Derivatives and the Borel Transform

Carlsson, Marcus LU ; Prado, Humberto and Reyes, Enrique G. (2016) In Annales Henri Poincare 17(8). p.2049-2074
Abstract

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. We properly interpret and solve linear equations in this class with a special focus on a solution method based on the Borel transform. This method is a far-reaching generalization of previous studies of nonlocal equations via Laplace and Fourier transforms, see for instance (Barnaby and Kamran, J High Energy Phys 02:40, 2008; Górka et al., Class Quantum Gravity 29:065017, 2012; Górka et al., Ann Henri Poincaré 14:947–966, 2013). We reconsider “generalized” initial value problems within the present approach and we... (More)

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. We properly interpret and solve linear equations in this class with a special focus on a solution method based on the Borel transform. This method is a far-reaching generalization of previous studies of nonlocal equations via Laplace and Fourier transforms, see for instance (Barnaby and Kamran, J High Energy Phys 02:40, 2008; Górka et al., Class Quantum Gravity 29:065017, 2012; Górka et al., Ann Henri Poincaré 14:947–966, 2013). We reconsider “generalized” initial value problems within the present approach and we disprove various conjectures found in modern physics literature. We illustrate various analytic phenomena that can occur with concrete examples, and we also treat efficient implementations of the theory.

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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
in
Annales Henri Poincare
volume
17
issue
8
pages
26 pages
publisher
Birkhauser Verlag Basel
external identifiers
  • scopus:84946867146
  • wos:000379847700004
ISSN
1424-0637
DOI
10.1007/s00023-015-0447-4
language
English
LU publication?
yes
id
dbb43371-f57b-42b2-9951-5cb466de1a83
date added to LUP
2016-12-30 10:35:57
date last changed
2017-09-18 11:34:42
@article{dbb43371-f57b-42b2-9951-5cb466de1a83,
  abstract     = {<p>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. We properly interpret and solve linear equations in this class with a special focus on a solution method based on the Borel transform. This method is a far-reaching generalization of previous studies of nonlocal equations via Laplace and Fourier transforms, see for instance (Barnaby and Kamran, J High Energy Phys 02:40, 2008; Górka et al., Class Quantum Gravity 29:065017, 2012; Górka et al., Ann Henri Poincaré 14:947–966, 2013). We reconsider “generalized” initial value problems within the present approach and we disprove various conjectures found in modern physics literature. We illustrate various analytic phenomena that can occur with concrete examples, and we also treat efficient implementations of the theory.</p>},
  author       = {Carlsson, Marcus and Prado, Humberto and Reyes, Enrique G.},
  issn         = {1424-0637},
  language     = {eng},
  month        = {08},
  number       = {8},
  pages        = {2049--2074},
  publisher    = {Birkhauser Verlag Basel},
  series       = {Annales Henri Poincare},
  title        = {Differential Equations with Infinitely Many Derivatives and the Borel Transform},
  url          = {http://dx.doi.org/10.1007/s00023-015-0447-4},
  volume       = {17},
  year         = {2016},
}