in this paper, the ultraspherical operational matrices of derivatives are constructed. based on these operational matrices, two numerical algorithms are presented and analyzed for obtaining new approximate spectral solutions of a class of linear and nonlinear lane-emden type singular initial value problems. the basic idea behind the suggested algorithms is basically built on transforming the eq...

We investigate the effects of relativity on the gravitational instability of finite isothermal gaseous spheres. In the first part of the paper, we treat the gravitational field within the framework of Newtonian mechanics but we use a relativistic equation of state in the condition of hydrostatic equilibrium. In the second part of the paper, we study the full general relativistic problem for a g...

The isothermal gas sphere is well known as a powerful tool to model many problems in astrophysics, physics, chemistry, and engineering. This singular differential equation has not an exact solution solved only by numerical approximate methods. In the present paper within framework of Newtonian hydrostatic equilibrium, we have developed general analytical formulations for fractional sphere. To o...

We complete our previous investigations concerning the structure and the stability of “isothermal” spheres in general relativity. This concerns objects that are described by a linear equation of state, P = qǫ, so that the pressure is proportional to the energy density. In the Newtonian limit q → 0, this returns the classical isothermal equation of state. We specifically consider a self-gravitat...

We complete the existing literature on the structure and stability of polytropic gas spheres reported in the classical monograph of Chandrasekhar (1942). For isolated polytropes with index 1 < n < 5, we provide a new, alternative, proof that the onset of instability occurs for n = 3 and we express the perturbation profiles of density and velocity at the point of marginal stability in terms of t...

For non-isothermal operation in plate type gas absorber, the temperature changes and number of ideal trays can be determined with recourse to graphical method. In this paper there is an analysis illustrating that a Modified Kremser Equation can be used for plate type gas absorber in non-isothermal operation.

Self-similar solutions provide good descriptions for the gravitational collapse of spherical clouds or stars when the gas obeys a polytropic equation of state, p = Kργ (with γ ≤ 4/3, and γ = 1 corresponds to isothermal gas). We study the behaviors of nonradial (nonspherical) perturbations in the similarity solutions of Larson, Penston and Yahil, which describe the evolution of the collapsing cl...

We discuss the statistical mechanics of rotating self-gravitating systems by allowing properly for the conservation of angular momentum. We study analytically the case of slowly rotating isothermal spheres by expanding the solutions of the Boltzmann-Poisson equation in a series of Legendre polynomials, adapting the procedure introduced by Chandrasekhar (1933) for distorted polytropes. We show h...

The Emden differential equation is one of the most widely studied and challenging nonlinear dynamics equations in literature. It finds applications various areas study such as celestial mechanics, fluid Steller structure, isothermal gas spheres, thermionic currents so on. Because importance equation, method generalized Sundman transformation (GST) proposed by Nakpim Meleshko used for linearizin...