Differential chain of algebras of generalized functions

2021 ◽  
Vol Accepted ◽  
Author(s):  
Dennis Ferdinand Agbebaku ◽  
Jan Harm van der Walt
Keyword(s):  
Generalized Functions ◽  
Algebras Of Generalized Functions

scholarly journals Algebras of generalized functions with smooth parameter dependence

2012 ◽  
Vol 55 (1) ◽  
pp. 105-124 ◽  
Author(s):  
Annegret Burtscher ◽  
Michael Kunzinger
Keyword(s):  
Systematic Study ◽  
Generalized Functions ◽  
Parameter Dependence ◽  
Order Structure ◽  
Colombeau Generalized Functions ◽  
Algebras Of Generalized Functions ◽  
Algebraic Properties

AbstractWe show that spaces of Colombeau generalized functions with smooth parameter dependence are isomorphic to those with continuous parametrization. Based on this result we initiate a systematic study of algebraic properties of the ring $\tilde{\mathbb{K}}_\mathrm{sm}$ of generalized numbers in this unified setting. In particular, we investigate the ring and order structure of $\tilde{\mathbb{K}}_\mathrm{sm}$ and establish some properties of its ideals.

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 A New Approach to Diffeomorphism Invariant Algebras of Generalized Functions

2015 ◽  
Vol 58 (3) ◽  
pp. 717-738
Author(s):  
E. A. Nigsch
Keyword(s):  
Generalized Functions ◽  
New Approach ◽  
Local Setting ◽  
Algebras Of Generalized Functions ◽  
Nonlinear Generalized Functions

AbstractWe develop the diffeomorphism invariant Colombeau-type algebra of nonlinear generalized functions in a modern and compact way. Using a unifying formalism for the local setting and on manifolds, the construction becomes simpler and more accessible than before.

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