Skip to content
DraconiticResearch
Search Sources API
190,860 studies31 sourcesUpdated 22 min ago
PreprintCC BY-NC-ND 4.0arXiv

Exact Integration and Invertibility Completeness of the Nonlinear Relaxation Equation

  • Lukáš Mrazík
Open at sourcePDFDOICite

Abstract

We examine the nonlinear initial value problem ẏ = C(1 − yα) modeling polytropic gas permeation. For rational α, we derive the exact primitive via cyclotomic partial fraction decomposition, establishing its real trigonometric representation. By synthesizing Ritt's theorem with Rosenlicht's differential algebraic residue criterion, we prove the exact step response y(t) is an elementary function if and only if α∈ {1, 2}. Admitting the Lambert W function extends closed-form solvability strictly to α∈ {−1, 1/2}. Finally, the asymptotic limit α→ 0 converges to a singular dynamics solved by the logarithmic integral. These results constitute a complete classification of all explicit closed-form solutions admitted by this system.

Related

  1. Extension Theorems in Measure and Integration Theory as Completeness Axioms
    Rafael Reno S. Cantuba
    arXivPreprintCC BY 4.0Full textMathematical physicsarXiv
  2. Nonlinear Lebesgue spaces: Dense subspaces, completeness and separability
    Guillaume Sérieys, Alain Trouvé
    HALPreprintCC BY-NC-SA 4.0Full textAnalysis & PDEsHAL
  3. Extended Generalized Contractions, Metric Completeness, and Applications to Solving Nonlinear Diophantine Equations
    Ravindra K. Bisht, Manuel De La Sen
    AxiomsPeer-reviewedCC BY 4.0General mathematicsCrossref
  4. An Initial-Value Problem for an Ordinary Differential Equation Involving Multi-Term Time-Fractional Derivatives with Relaxation Factors
    Fikret Gölgeleyen, Seher Seda Sarıköse Bayram, Masahiro Yamamoto
    Fractal and FractionalPeer-reviewedCC BY 4.0Analysis & PDEsCrossref
  5. Many-point tropical relaxation and the Monge--Ampère equation
    Nikita Kalinin, Ernesto Lupercio, Higinio Serrano +1
    arXivPreprintarXiv licenceFull textAlgebra & number theoryAnalysis & PDEsarXiv

Cite

BibTeXRIS
BibTeX
@misc{mrazik2026exact,
  title = {{Exact Integration and Invertibility Completeness of the Nonlinear Relaxation Equation}},
  author = {Lukáš Mrazík},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.31748},
  eprint = {2609.31748},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.31748},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Exact Integration and Invertibility Completeness of the Nonlinear Relaxation Equation
AU  - Lukáš Mrazík
PY  - 2026
DA  - 2026/09/23
PB  - arXiv
DO  - 10.48550/arxiv.2609.31748
UR  - https://arxiv.org/abs/2609.31748
AB  - We examine the nonlinear initial value problem ẏ = C(1 − y^α) modeling polytropic gas permeation. For rational α, we derive the exact primitive via cyclotomic partial fraction decomposition, establishing its real trigonometric representation. By synthesizing Ritt's theorem with Rosenlicht's differential algebraic residue criterion, we prove the exact step response y(t) is an elementary function if and only if α∈ {1, 2}. Admitting the Lambert W function extends closed-form solvability strictly to α∈ {−1, 1/2}. Finally, the asymptotic limit α→ 0 converges to a singular dynamics solved by the logarithmic integral. These results constitute a complete classification of all explicit closed-form solutions admitted by this system.
ER  -