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Advanced Nonlinear Studies

Editor-in-Chief: Ahmad, Shair


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Volume 13, Issue 2

Issues

Ultrafunctions and Generalized Solutions

Vieri Benci
  • Dipartimento di Matematica Applicata Università degli Studi di Pisa Via F. Buonarroti 1/c, Pisa, ITALY and Department of Mathematics, College of Science King Saud University Riyadh, 11451, SAUDI ARABIA
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Published Online: 2016-03-10 | DOI: https://doi.org/10.1515/ans-2013-0212

Abstract

The theory of distributions provides generalized solutions for problems which do not have a classical solution. However, there are problems which do not have solutions, not even in the space of distributions. As model problem you may think of

with Dirichlet boundary conditions in a bounded open star-shaped set. Having this problem in mind, we construct a new class of functions called ultrafunctions in which the above problem has a (generalized) solution. In this construction, we apply the general ideas of Non Archimedean Mathematics (NAM) and some techniques of Non Standard Analysis. Also, some possible applications of ultrafunctions are discussed.

Keywords: Non Archimedean Mathematics; Non Standard Analysis; ultrafunctions; distributions; generalized solutions; Sobolev critical exponent; formalism of Quantum Mechanics

About the article

Published Online: 2016-03-10

Published in Print: 2013-05-01


Citation Information: Advanced Nonlinear Studies, Volume 13, Issue 2, Pages 461–486, ISSN (Online) 2169-0375, ISSN (Print) 1536-1365, DOI: https://doi.org/10.1515/ans-2013-0212.

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Discrete and Continuous Dynamical Systems - Series S, 2014, Volume 7, Number 4, Page 593
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Arabian Journal of Mathematics, 2015, Volume 4, Number 4, Page 231
[5]
Vieri Benci and Isaia Nisoli
Arab Journal of Mathematical Sciences, 2015, Volume 21, Number 2, Page 144

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