Two Integrable Classes of Emden–Fowler Equations with Applications in Astrophysics and Cosmology
| Autor: | H. C. Rosu, Stefan C. Mancas |
|---|---|
| Rok vydání: | 2018 |
| Předmět: |
Physics
Class (set theory) Reduction (recursion theory) Integrable system General relativity 010102 general mathematics FOS: Physical sciences General Physics and Astronomy Mathematical Physics (math-ph) Astrophysics Function (mathematics) Type (model theory) Lambda 01 natural sciences 010305 fluids & plasmas 0103 physical sciences 0101 mathematics Physical and Theoretical Chemistry Parametric equation Mathematical Physics |
| Zdroj: | Zeitschrift für Naturforschung A. 73:805-814 |
| ISSN: | 1865-7109 0932-0784 |
| DOI: | 10.1515/zna-2018-0062 |
| Popis: | We show that some Emden-Fowler (EF) equations encountered in astrophysics and cosmology belong to two EF integrable classes of the type d^2z/d{\chi}^2=A \chi^{-\lambda-2}z^n for \lambda=(n-1)/2 (class one), and \lambda=n+1 (class two). We find their corresponding invariants which reduce them to first order nonlinear ordinary differential equations. Using particular solutions of such EF equations, the two classes are set in the autonomous nonlinear oscillator form d^2\nu/dt^2 +ad \nu/dt +b(\nu-\nu^n)=0, where the coefficients a,b depend only on \lambda, n. For both classes, we write closed-form solutions in parametric form. The illustrative examples from astrophysics and general relativity correspond to two n=2 cases from class one and two, and one n=5 case from class one, all of them yielding Weierstrass elliptic solutions. It is also noticed that when n=2, EF equations can be studied using the Painlev\'e reduction method, since they are a particular case of equations of the type d^2z/d{\chi}^2=F(\chi)z^2, where F(\chi) is the Kustaanheimo-Qvist function Comment: 10 pages, no figures, final version |
| Databáze: | OpenAIRE |
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