scholarly journals On Products of Distributions in Colombeau Algebra

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Marija Miteva ◽  
Biljana Jolevska-Tuneska ◽  
Tatjana Atanasova-Pacemska
Keyword(s):  
Differential Algebra ◽  
Generalized Functions ◽  
Colombeau Algebra ◽  
Schwartz Distributions ◽  
Products Of Distributions ◽  
Renormalization Theory

Results on products of distributionsx+-kandδ(p)(x)are derived. They are obtained in Colombeau differential algebra𝒢(R)of generalized functions that contains the space𝒟'(R)of Schwartz distributions as a subspace. Products of this form are useful in quantum renormalization theory in Physics.

scholarly journals An embedding of Schwartz distributions in the algebra of asymptotic functions

1998 ◽  
Vol 21 (3) ◽  
pp. 417-428 ◽  
Author(s):  
Michael Oberguggenberger ◽  
Todor Todorov
Keyword(s):  
Differential Algebra ◽  
Generalized Functions ◽  
Schwartz Distributions ◽  
Asymptotic Functions ◽  
Space Of Distributions ◽  
Asymptotic Function

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 different from J. F Colombeau's algebras of new generalized functions.

scholarly journals Further Results on Colombeau Product of Distributions

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Biljana Jolevska-Tuneska ◽  
Tatjana Atanasova-Pacemska
Keyword(s):  
Differential Algebra ◽  
Generalized Functions ◽  
Schwartz Distributions

Results on Colombeau product of distributions and are derived. They are obtained in Colombeau differential algebra of generalized functions that contains the space of Schwartz distributions as a subspace and has a notion of “association” that is a faithful generalization of the weak equality in .

Singular fractional evolution differential equations

Open Physics ◽  
2013 ◽  
Vol 11 (10) ◽  
Author(s):  
Mirjana Stojanovic
Keyword(s):  
Differential Equations ◽  
Evolution Equations ◽  
Nonlinear Models ◽  
Uniqueness Result ◽  
Generalized Functions ◽  
Colombeau Algebra ◽  
Fractional Evolution Equations ◽  
Linear And Nonlinear ◽  
Duhamel Principle

AbstractWe give an existence-uniqueness result for linear and nonlinear time fractional evolution equations with singularities in corresponding norm in extended Colombeau algebra of generalized functions using fractional analog for Duhamel principle. Paper deals with some nonlinear models with singularities appearing in viscoelasticity and in anomalous processes which have met great interest among researchers who consider them as a challenge in recent years.

scholarly journals The classical theory of calculus of variations for generalized functions

2017 ◽  
Vol 8 (1) ◽  
pp. 779-808 ◽  
Author(s):  
Alexander Lecke ◽  
Lorenzo Luperi Baglini ◽  
Paolo Giordano
Keyword(s):  
Calculus Of Variations ◽  
Classical Theory ◽  
Sufficient Conditions ◽  
Generalized Functions ◽  
Conjugate Points ◽  
Lagrange Equations ◽  
Smooth Functions ◽  
Nonlinear Properties ◽  
Schwartz Distributions ◽  
Necessary And Sufficient

Abstract We present an extension of the classical theory of calculus of variations to generalized functions. The framework is the category of generalized smooth functions, which includes Schwartz distributions, while sharing many nonlinear properties with ordinary smooth functions. We prove full connections between extremals and Euler–Lagrange equations, classical necessary and sufficient conditions to have a minimizer, the necessary Legendre condition, Jacobi’s theorem on conjugate points and Noether’s theorem. We close with an application to low regularity Riemannian geometry.

FOUNDATION OF THE FRACTIONAL CALCULUS IN GENERALIZED FUNCTION ALGEBRAS

2012 ◽  
Vol 10 (04) ◽  
pp. 439-467 ◽  
Author(s):  
MIRJANA STOJANOVIĆ
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivatives ◽  
Generalized Functions ◽  
Generalized Function ◽  
Colombeau Algebra ◽  
Nonlinear Odes ◽  
Function Algebras ◽  
Algebras Of Generalized Functions ◽  
Special Algebra

We introduce an approach to fractional derivatives involving singularities based on the theory of algebras of generalized functions in the sense of Colombeau. We are interested in solving fractional nonlinear ODEs and PDEs with singularities with a lack of solutions in the space of classical functions or distributions. For these purposes, we embed different forms of fractional derivatives into space of Colombeau special algebra of generalized functions using appropriate techniques such as the regularization with delta sequences and multiplication with different cut-off functions. Finally, we present an example for application of the ideas presented in paper to confirm the reason of introducing fractional derivatives into Colombeau algebra of generalized functions.

scholarly journals Maximal closed ideals of the Colombeau Algebra of Generalized functions

2020 ◽  
Author(s):  
A. Khelif ◽  
D. Scarpalezos
Keyword(s):  
Hausdorff Space ◽  
Continuous Functions ◽  
Generalized Functions ◽  
Colombeau Algebra ◽  
Rings Of Continuous Functions ◽  
Maximal Ideals

Abstract In this paper we investigate the structure of the set of maximal ideals of $${{\mathcal {G}}}(\Omega )$$ G ( Ω ) . The method of investigation passes through the use of the $$m-$$ m - reduction and the ideas are analoguous to those in Gillman and Jerison (Rings of Continuous Functions, N.J. Van Nostrand, Princeton, 1960) for the investigation of maximal ideals of continuous functions on a Hausdorff space K.

scholarly journals On the Stability of Trigonometric Functional Equations in Distributions and Hyperfunctions

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Jaeyoung Chung ◽  
Jeongwook Chang
Keyword(s):  
Functional Equations ◽  
Generalized Functions ◽  
Ulam Stability ◽  
Schwartz Distributions ◽  
The Stability

We consider the Hyers-Ulam stability for a class of trigonometric functional equations in the spaces of generalized functions such as Schwartz distributions and Gelfand hyperfunctions.

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