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We offer an axiomatic definition of a differential algebra of generalizedfunctions over an algebraically closed non-Archimedean field. This algebra isof Colombeau type in the sense that it contains a copy of the space of Schwartzdistributions. We study the uniqueness of the objects we define and theconsistency of our axioms. Next, we identify an inconsistency in theconventional Laplace transform theory. As an application we offer a free ofcontradictions alternative in the framework of our algebra of generalizedfunctions. The article is aimed at mathematicians, physicists and engineers whoare interested in the non-linear theory of generalized functions, but who arenot necessarily familiar with the original Colombeau theory. We assume,however, some basic familiarity with the Schwartz theory of distributions.

An axiomatic approach to the non-linear theory of generalized functions and consistency of Laplace transforms