Related papers: Colombeau generalized functions and solvability of…
This paper is devoted to strictly hyperbolic systems and equations with non-smooth coefficients. Below a certain level of smoothness, distributional solutions may fail to exist. We construct generalised solutions in the Colombeau algebra of…
In this article we introduce the notion of fundamental solution in the Colombeau context as an element of the dual $\LL(\Gc(\R^n),\wt{\C})$. After having proved the existence of a fundamental solution for a large class of partial…
We present an extension of J. F. Colombeau's theory of nonlinear generalized functions to spaces of generalized sections of vector bundles. Our construction builds on classical functional analytic notions, which is the key to having a…
This is a gentle introduction to Colombeau nonlinear generalized functions, a generalization of the concept of distributions such that distributions can freely be multiplied. It is intended to physicists and applied mathematicians who…
We present some remarks about the embedding of spaces of Schwartz distributions into spaces of Colombeau generalized functions. We show that the various constructions of such embeddings existing in the literature lead in fact to the same…
The paper addresses linear hyperbolic systems in one space dimension with random field coefficients. In many applications, a low degree of regularity of the paths of the coefficients is required, which is not covered by classical stochastic…
Modelling of singularities given by discontinuous functions or distributions by means of generalized functions has proved useful in many problems posed by physical phenomena. We introduce in a systematic way generalized functions of…
The objective of this introduction to Colombeau algebras of generalized-functions (in which distributions can be freely multiplied) is to explain in elementary terms the essential concepts necessary for their application to basic non-linear…
Modelling of singularities given by discontinuous functions or distributions by means of generalized functions has proved useful in many problems posed by physical phenomena. We introduce in a systematic way generalized functions of…
The definition of a non-trivial space of generalized functions of a complex variable allowing to consider derivatives of continuous functions is a non-obvious task, e.g. because of Morera theorem, because distributional Cauchy-Riemann…
We characterize microlocal regularity of Colombeau generalized functions by an appropriate extension of the classical notion of micro-ellipticity to pseudodifferential operators with slow scale generalized symbols. Thus we obtain an…
Colombeau's generalized functions are used to adapt the distributional approach to singular hypersurfaces in general relativity with signature change. Equations governing the dynamics of singular hypersurface is obtained and it is shown…
We present a solution of the problem of multiplication of Schwartz distributions by embedding the space of distributions into a differential algebra of generalized functions, called in the paper ``asymptotic function'', similar to but…
We tackle the problem of finding a suitable categorical framework for generalized functions used in mathematical physics for linear and non-linear PDEs. We are looking for a Cartesian closed category which contains both Schwartz…
We describe partial differential operators for which we can construct generalised integral means satisfying Pizzetti-type formulas. Using these formulas we give a new characterisation of summability of formal power series solutions to some…
A new method is presented for assigning distributional curvature, in an invariant manner, to a space-time of low differentiability, using the techniques of Colombeau's `new generalised functions'. The method is applied to show that…
We construct an algebra of generalized functions $^*\mathcal{E}(\mathbb{R}^d)$. We also construct an embedding of the space of Schwartz distributions $\mathcal{D}^\prime(\mathbb{R}^d)$ into $^*\mathcal{E}(\mathbb{R}^d)$ and thus present a…
We briefly review results on Colombeau type generalized solutions to the Cauchy problem for linear Schr\"odinger-type equations with non-smooth principal part and their compatibility with classical and distributional solutions. In the main…
In this paper we review the extent to which one can use classical distribution theory in describing solutions of Einstein's equations. We show that there are a number of physically interesting cases which cannot be treated using…
The purpose of this paper is to investigate the non-constant entire as well as meromorphic solutions of the Fermat-type partial differential-difference equation: \[\left(\sum_{j=1}^m\frac{\partial f(z_1, z_2, \ldots, z_m)}{\partial…