As written in L. Schwartz' book, Heaviside's theory of cables is an important source of the theory of generalized functions. The partial differential equations he discussed were the usual heat equation and the simplest hyperbolic equations of one space dimension, but he had to solve them as evolution equations in the unusual direction of the distance along which the electric signals propagate. Although he obtained explicit expressions of solutions, which were of great economical values, it has not yet been clarified completely how he derived and proved them. The author gives easy proofs of Heaviside's results based only on the classical theory of Laplace transforms and their reciprocal, the Bromwich integrals. At the end it is indicated that an abstract version of the Fatou theorem on bounded harmonic functions on a half space implies the uniqueness of solutions for the Thomson cables.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-bc88-0-13,
author = {Hikosaburo Komatsu},
title = {Heaviside's theory of signal transmission on submarine cables},
journal = {Banach Center Publications},
volume = {89},
year = {2010},
pages = {159-173},
zbl = {1207.46036},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc88-0-13}
}
Hikosaburo Komatsu. Heaviside's theory of signal transmission on submarine cables. Banach Center Publications, Tome 89 (2010) pp. 159-173. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc88-0-13/