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Lane-Emden differential equations describe different physical and astrophysical phenomena that include forms of stellar structure, isothermal gas spheres, gas spherical cloud thermal history, and thermionic currents. This paper presents a computational approach to solve the problems related to fractional Lane-Emden differential equations based on neural networks. Such a solution will help solve the fractional polytropic gas spheres problems which have different applications in physics, astrophysics, engineering, and several real-life issues. We used Artificial Neural Network (ANN) framework in its feedforward back propagation learning scheme. The efficiency and accuracy of the presented algorithm are checked by testing it on four fractional Lane-Emden equations and compared with the exact solutions for the polytopic indices n=0,1,5 and those of the series expansions for the polytropic index n=3. The results we obtained prove that using the ANN method is feasible, accurate, and may outperform other methods.

Lane -Emden differential equation of the polytropic gas sphere could be used to construct simple models of stellar structures, star clusters and many configurations in astrophysics.This differential equation suffers from the singularity at the center and has an exact solution only for the polytropic index n=0, 1 and 5 . In the present paper, we present an analytical solution to the fractional polytropic gas sphere via accelerated series expansion. The solution is performed in the frame of conformable fractional derivatives. The calculated models recover the well-known series of solutions when 1   . Physical parameters such as mass-radius relation, density ratio, pressure ratio and temperature ratio for different fractional models have been calculated and investigated.We found that the present models of the conformable fractional stars have smaller volume and mass than that of both the integer star and fractional models performed in the frame of modified Rienmann Liouville derivatives.

Abstract.The spherical isothermal Lane-Emden equation is a second order non-linear differential equation

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Lane-Emden equation could be used to model stellar interiors, star clusters and many configurations in astrophysics. Unfortunately, there is an exact solution only for the polytropic index n=0, 1 and 5 . In the present paper, a series solution for the fractional Lane-Emden equation is presented. The solution is performed in the frame of modified Rienmann Liouville derivatives. The obtained results recover the well-known series solutions when 1   . Fractional model of n=3 has been calculated and mass-radius relation, density ratio, pressure ratio and temperature ratio have been investigated. We found that the fractional star has a smaller volume and mass than that of the integer star.

We carried out a spectroscopic survey of several B, Be, and shell stars in optical and near-infrared regions. Line profiles of the Hα line and of selected Fe ii and O i lines are presented.

We investigated the luminosity function (LF) and initial mass function (IMF) of some open clusters having different ages. To calculate the LF, we followed the classical definition by van Rhijn (1936). Statistical investigation of the dispersion around a range of magnitudes concerning what is called Wielen dip revealed that the dip is unreal. To confirm the unreality of the dip, we computed the IMF for these open clusters, the statistical investigation of the IMF confirmed the results obtained using the LF, that is, there is no dip for these open clusters under study.