Gumede Sfundo C, Govinder Keshlan S, Maharaj Sunil D
Astrophysics Research Centre, School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X54001, Durban 4000, South Africa.
Department of Mathematical Sciences, Mangosuthu University of Technology, P.O. Box 12363, Jacobs 4026, South Africa.
Entropy (Basel). 2021 Nov 19;23(11):1539. doi: 10.3390/e23111539.
A single master equation governs the behaviour of shear-free neutral perfect fluid distributions arising in gravity theories. In this paper, we study the integrability of yxx=f(x)y2, find new solutions, and generate a new first integral. The first integral is subject to an integrability condition which is an integral equation which restricts the function f(x). We find that the integrability condition can be written as a third order differential equation whose solution can be expressed in terms of elementary functions and elliptic integrals. The solution of the integrability condition is generally given parametrically. A particular form of f(x)∼1x51-1x-15/7 which corresponds to repeated roots of a cubic equation is given explicitly, which is a new result. Our investigation demonstrates that complexity of a self-gravitating shear-free fluid is related to the existence of a first integral, and this may be extendable to general matter distributions.
一个主方程支配着引力理论中出现的无剪切中性完美流体分布的行为。在本文中，我们研究了(y_{xx}=f(x)y^2)的可积性，找到了新的解，并生成了一个新的第一积分。该第一积分受制于一个可积性条件，该条件是一个限制函数(f(x))的积分方程。我们发现可积性条件可以写成一个三阶微分方程，其解可以用初等函数和椭圆积分表示。可积性条件的解通常以参数形式给出。明确给出了对应于三次方程重根的(f(x)\sim\frac{1}{x^{5/1 - 1/x^{-15/7}}})的一种特殊形式，这是一个新结果。我们的研究表明，自引力无剪切流体的复杂性与第一积分的存在有关，并且这可能可以推广到一般物质分布。
