Integrable Abel equations and Vein's Abel equation

We present an analysis of the Rayleigh-Plesset equation for a three dimensional vacuous bubble in water. In the simplest case when the effects of surface tension are neglected, the known parametric solutions for the radius and time evolution of the bubble in terms of a hypergeometric function are briefly reviewed. By including the surface tension, we show the connection between the Rayleigh-Plesset equation and Abel's equation, and obtain the parametric rational Weierstrass periodic solutions following the Abel route. In the same Abel approach, we also provide a discussion of the nonintegrable case of nonzero viscosity for which we perform a numerical integration.

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“…where we define K 1 = P∞−P ρw and K 2 = 2σ ρw . Proceeding as in [12], we first show that solutions to a general second order ODE of type…”

Section: Surface Tension Included Via Abel's Equationmentioning

confidence: 95%

Evolution of spherical cavitation bubbles: Parametric and closed-form solutions

Mancas

Rosu

2016

Physics of Fluids

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“…Actually, the cause of this discrepancy in relic density obtained between numerical and analytical solution occurs when we simplify the Eq.2.20 to Eq.2.25 to only retain terms of the order ∼ ∆ 3 . If we consider second order term in ∆(x), the equation looks like that of Abel equation of first kind [42], solution of that will mimic the numerical solution even more closely.…”

Section: Approximate Analytical Solution To Boltzmann Equationmentioning

confidence: 99%

SIMPler realisation of scalar dark matter

Bhattacharya

Ghosh

Verma

2020

J. Cosmol. Astropart. Phys.

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With growing agony of not finding a dark matter (DM) particle in direct search experiments so far (for example in XENON1T), frameworks where the freeze-out of DM is driven by number changing processes within the dark sector itself and do not contribute to direct search, like Strongly Interacting Massive Particle (SIMP) are gaining more attention.In this analysis, we ideate a simple scalar DM framework stabilised by Z 3 symmetry to serve with a SIMP-like DM (χ) with additional light scalar mediation (φ) to enhance DM self interaction. We identify that a large parameter space for such DM is available from correct relic density and self interaction constraints coming from Bullet or Abell cluster data. We derive an approximate analytic solution for freeze-out of the SIMP like DM in Boltzmann equation describing 3 DM → 2 DM number changing process within the dark sector. We also provide a comparative analysis of the SIMP like solution with the Weakly Interacting Massive Particle (WIMP) realisation of the same model framework here.

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“…From an historical standpoint, we believe that Abel's equation of the first kind should be called Riccati–Liouville equation, and Abel's equation of the second kind must be renamed Abel–Appell equation. Abel's equations arise in the integration of several models of dissipative systems . Our main task here is to lift Liouville–Appell theory of integration of Abel's equation of the first kind

ż = c_{0} (t) + 3 c_{1} z (t) + 3 c_{2} (t) z^{2} + c_{3} (t) z^{3},

where the overdot is differentiation with respect to t an the functions z and c i 's are valued in a commutative hypercomplex, to systems that we shall call Abel‐type systems.…”

Section: Integration Of Systems Of Ordinary Differential Equations Bymentioning

confidence: 99%

Hypercomplex analysis and integration of systems of ordinary differential equations

Soh

Mahomed

2016

Math Methods in App Sciences

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We review the theory of hypercomplex numbers and hypercomplex analysis with the ultimate goal of applying them to issues related to the integration of systems of ordinary differential equations (ODEs). We introduce the notion of hypercomplexification, which allows the lifting of some results known for scalar ODEs to systems of ODEs. In particular, we provide another approach to the construction of superposition laws for some Riccati‐type systems, we obtain invariants of Abel‐type systems, we derive integrable Ermakov systems through hypercomplexification, we address the problem of linearization by hypercomplexification, and we provide a solution to the inverse problem of the calculus of variations for some systems of ODEs. Copyright © 2016 John Wiley & Sons, Ltd.

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Integrable Abel equations and Vein's Abel equation

Cited by 16 publications

References 17 publications

Evolution of spherical cavitation bubbles: Parametric and closed-form solutions

Evolution of spherical cavitation bubbles: Parametric and closed-form solutions

SIMPler realisation of scalar dark matter

Hypercomplex analysis and integration of systems of ordinary differential equations

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