New integrability case for the Riccati equation

Mak, M. K.; Harkó, Tiberiu

doi:10.1016/j.amc.2012.04.058

Cited by 20 publications

(24 citation statements)

References 14 publications

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“…On this note, the integrability conditions for Riccati type equations in Refs. [51][52][53] were also considered. However, these did not lead useful results in the considered context.…”

Section: Discussionmentioning

confidence: 99%

Static compact objects in Einstein-Cartan theory

Luz

Carloni

2019

Phys. Rev. D

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We generalize the Tolman-Oppenheimer-Volkoff equations for space-times endowed with a Weyssenhoff like torsion field in the Einstein-Cartan theory. The new set of structure equations clearly show how the presence of torsion affects the geometry of the space-time. We obtain new exact solutions for compact objects with non-null intrinsic spin surrounded by vacuum, explore their properties and discuss how these solutions should be smoothly matched to an exterior space-time.We study how the intrinsic spin of matter changes the Buchdahl limit for the maximum compactness of stars. Moreover, under rather generic conditions, we prove that in the context of a Weyssenhoff like torsion, no static, spherically symmetric compact objects supported only by the intrinsic spin can exist. We also provide some algorithms to generate new solutions. * paulo.luz@ist.utl.pt † sante.carloni@gmail.com 1 We should remark that here, and in the following, the word "spin" will be used exclusively to represent the quantum spin of the particles that source the gravitational field equations. In no case the word spin will be associated to any form of rotation of the compact objects we will analyze.

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“…On this note, the integrability conditions for Riccati type equations in Refs. [51][52][53] were also considered. However, these did not lead useful results in the considered context.…”

Section: Discussionmentioning

confidence: 99%

Static compact objects in Einstein-Cartan theory

Luz

Carloni

2019

Phys. Rev. D

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“…Integrability condition for the Riccati equation has been studied by Mak and Harko and a new method to generate analytical solutions of the Riccati equation was presented [5]. Mortici gives a new method of the variation of constants which leads directly to an equation with separable variables.…”

Section: Introductionmentioning

confidence: 99%

“…(1) into a first order linear equation. For the case of ( ) ≠ 0 , several methods are available in the literature [1][2][3][4][5][6][7][8][9][10][11]. The equation in this case is also known as Bernoulli equation.…”

Section: Introductionmentioning

confidence: 99%

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A New Analytical Method for Solving General Riccati Equation

Pala¹,

Ertas²

2017

ujam

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In this paper, the general Riccati equation is analytically solved by a new transformation. By the method developed, looking at the transformed equation, whether or not an explicit solution can be obtained is readily determined. Since the present method doesn't require a proper solution for the general solution, it is especially suitable for equations whose proper solutions cannot be seen at a first glance. Since the transformed second order linear equation obtained by the present transformation has the simplest form it can have, it is immediately seen whether or not the original equation can be solved analytically. The present method is exemplified by several examples.

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“…Very recently, the isotropic pressure equation was reformulated as a Riccati equation. By using the general integrability condition for the Riccati equation proposed in Mak & Harko (2012, 2013a, an exact non-singular solution of the interior field equations for a fluid star expressed in the form of infinite series was obtained in Mak & Harko (2013b). The astrophysical analysis indicates that this power series solution can be used as a realistic model for static general relativistic high density objects, for example neutron stars.…”

Section: Introductionmentioning

confidence: 99%

Exact power series solutions of the structure equations of the general relativistic isotropic fluid stars with linear barotropic and polytropic equations of state

Harkó

Mak

2016

Astrophys Space Sci

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Exact power series solutions of the structure equations of the general relativistic isotropic fluid stars with linear barotropic and polytropic equations of state T. Harko 1,2 • M. K. Mak 3Abstract Obtaining exact solutions of the spherically symmetric general relativistic gravitational field equations describing the interior structure of an isotropic fluid sphere is a long standing problem in theoretical and mathematical physics. The usual approach to this problem consists mainly in the numerical investigation of the Tolman-Oppenheimer-Volkoff and of the mass continuity equations, which describes the hydrostatic stability of the dense stars. In the present paper we introduce an alternative approach for the study of the relativistic fluid sphere, based on the relativistic mass equation, obtained by eliminating the energy density in the Tolman-Oppenheimer-Volkoff equation. Despite its apparent complexity, the relativistic mass equation can be solved exactly by using a power series representation for the mass, and the Cauchy convolution for infinite power series. We obtain exact series solutions for general relativistic dense astrophysical objects described by the linear barotropic and the polytropic equations of state, respectively. For the polytropic case we obtain the exact power series solution corresponding to arbitrary values of the polytropic index n. The explicit form of the solution is presented for the polytropic index n = 1, and for the indexes n = 1/2 and n = 1/5, respectively. The case of n = 3 is also considered. In each case the exact power series solution is compared with the exact numerical solutions, which are reproduced by the power series solutions truncated to seven

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New integrability case for the Riccati equation

Cited by 20 publications

References 14 publications

Static compact objects in Einstein-Cartan theory

Static compact objects in Einstein-Cartan theory

A New Analytical Method for Solving General Riccati Equation

Exact power series solutions of the structure equations of the general relativistic isotropic fluid stars with linear barotropic and polytropic equations of state

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