Results on generalized models and singular products of distributions in the Colombeau algebra $\mathcal {G}(\mathbb {R})$

Blagovest Damyanov

Displaying similar documents to “Results on generalized models and singular products of distributions in the Colombeau algebra $𝒢 (ℝ)$ ”

On Generalized Models and Singular Products of Distributions in Colombeau Algebra G(R)

Damyanov, Blagovest P. (2013)

Mathematica Balkanica New Series

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MSC 2010: 46F30, 46F10 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 Colombeau that model such singularities. Moreover, we evaluate some products of singularity-modelling generalized functions whenever the result admits an associated distribution.

Singular Products of Distributions.

A.K. Tiwari (1979)

Mathematische Annalen

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Singular solutions to systems of conservation laws and their algebraic aspects

V. M. Shelkovich* (2010)

Banach Center Publications

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We discuss the definitions of singular solutions (in the form of integral identities) to systems of conservation laws such as shocks, δ-, δ’-, and $δ^{(n)}$ -shocks (n = 2,3,...). Using these definitions, the Rankine-Hugoniot conditions for δ- and δ’-shocks are derived. The weak asymptotics method for the solution of the Cauchy problems admitting δ- and δ’-shocks is briefly described. The algebraic aspects of such singular solutions are studied. Namely, explicit formulas for flux-functions of...

Results on Colombeau product of distributions

Blagovest Damyanov (1997)

Commentationes Mathematicae Universitatis Carolinae

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The differential $ℂ$ -algebra $𝒢 (ℝ^{m})$ of generalized functions of J.-F. Colombeau contains the space $𝒟^{'} (ℝ^{m})$ of Schwartz distributions as a $ℂ$ -vector subspace and has a notion of ‘association’ that is a faithful generalization of the weak equality in $𝒟^{'} (ℝ^{m})$ . This is particularly useful for evaluation of certain products of distributions, as they are embedded in $𝒢 (ℝ^{m})$ , in terms of distributions again. In this paper we propose some results of that kind for the products of the widely used distributions $x_{\pm}^{a}$ and $δ^{(p)} (x)$ ,...

Taylor formula for distributions

Bogdan Ziemian

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CONTENTS0. Introduction................................................................................................................................................................51. Preliminary remarks...................................................................................................................................................62. Hyperfunctions and their generalizations.................................................................................................................103....

Balanced Colombeau products of the distributions $x_{\pm}^{- p}$ and $x^{- p}$

Blagovest Damyanov (2005)

Czechoslovak Mathematical Journal

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Results on singular products of the distributions $x_{\pm}^{- p}$ and $x^{- p}$ for natural $p$ are derived, when the products are balanced so that their sum exists in the distribution space. These results follow the pattern of a known distributional product published by Jan Mikusiński in 1966. The results are obtained in the Colombeau algebra of generalized functions, which is the most relevant algebraic construction for tackling nonlinear problems of Schwartz distributions.

A contribution to the equivalence results for the product of distributions

Jiří Jelínek (1994)

Commentationes Mathematicae Universitatis Carolinae

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Products $[S] \cdot [T]$ and $[S] \cdot T$ , defined by model delta-nets, are equivalent.

A characterization of the space $D_{F}^{'}$

S. Pilipović (1982)

Matematički Vesnik

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Displaying similar documents to “Results on generalized models and singular products of distributions in the Colombeau algebra 𝒢 ( ℝ ) ”

Displaying similar documents to “Results on generalized models and singular products of distributions in the Colombeau algebra $𝒢 (ℝ)$ ”