Postprint version. Published in Integral Transforms and Special Functions, Volume 22, Issue 4-5, September 1, 2011.
Copyright © 2011 Taylor & Francis. This is an electronic version of an article published in Integral Transforms and Special Functions. The definitive version is available at http://dx.doi.org/10.1080/10652469.2010.541057.
We offer an axiomatic definition of a differential algebra of generalized functions over an algebraically closed non-Archimedean field. This algebra is of Colombeau type in the sense that it contains a copy of the space of Schwartz distributions. We study the uniqueness of the objects we define and the consistency of our axioms. Next, we identify an inconsistency in the conventional Laplace transform theory. As an application we offer a free of contradictions alternative in the framework of our algebra of generalized functions. The article is aimed at mathematicians, physicists and engineers who are interested in the non-linear theory of generalized functions, but who are not necessarily familiar with the original Colombeau theory. We assume, however, some basic familiarity with the Schwartz theory of distributions.