How to solve smooth nonlinear PDEs in algebras of generalized functions with dense singularities

2004/06/29 by E. E. Rosinger, Elem'er E. Rosinger, Rosinger, E. E. · 3 citations
Computer Science · Mathematics · Physics and Astronomy · #35A #46F30 #Analysis of PDEs (math.AP) #B #D #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Iterative Methods for Nonlinear Equations #Mathematical Physics (math-ph) #Mathematical and Theoretical Analysis #Optimization and Variational Analysis #R #gr-qc #math-ph #math.AP #math.MP #msc:35A #msc:46F30

paper · pdf · doi:10.48550/arxiv.math/0406594

arxiv created 2004/06/29 · openalex publication_date 2004/06/29 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

As a significant strengthening of properties of earlier algebras of generalized functions, here are presented classes of such algebras which can deal with dense singularities. In fact, the cardinal of the set of singular points can be larger than that of the nonsingular points. This large class of singularities allows the solution of large classes of smooth nonlinear PDEs. This in certain ways overcomes the celebrated 1957 H. Lewy impossibility result.

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• New Symmetry Groups for Generalized Solutions of ODEs
• Junction Conditions, Resolution of Singularities and Nonlinear Equations of Physics
• Scattering in highly singular potentials
• Inevitable Infinite Branching in the Multiplication of Singularities

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