Computational method for a fractional model of the helium burning network

Abstract

Stellar cores may be considered as a nuclear reactor that play important role in injecting new synthesized elements in the interstellar medium Helium burning is an important stage that contribute to the synthesis of key elements such as carbon, through the triple-α process, and oxygen. In the present paper, we introduce a computational method for the fractional model of the nuclear helium burning in stellar cores. The system of fractional differential equations is solved simultaneously using series expansion method. The calculations are performed in the sense of modified Riemann-Liouville fractional derivative. Analytic expressions are obtained for the abundance of each element as a function of time. Comparing the abundances calculated at the fractional parameter 1   , which represents the integer solution, with the numerical solution revealed a good agreement with maximum error 0.003   . The product abundances are calculated at 0.25, 0.5, 0.75   to declare the effects of changing the fractional parameters on the calculations.

A novel fractal-fractional analysis of the stellar helium burning network using extended operational matrix method

“…The comparison between the computed fractal-fractional models ensures the results obtained by the previous investigations, Nouh [10] and [35]; the fractional HB models may have an important impact during stellar model calculations. The findings of this study demonstrate that modelling nuclear burning networks with the OM Fractal-Fractional Derivative yields extremely good results and supports it as an accurate, resilient, and trustworthy approach for solving and modelling other stellar nuclear networks.…”

“…In addition to the comparison with the integer model, which sets the three parameters to 1 (i.e., α = β = η = 1), we compared the fractal-fractional derivative model (model #5; α = 0.9, β = η = 1) with two different fractional HB models; the conformable fractional HBN (C-HBN, [10]) and AB operator using a generalized Mittag-Leffler function (AB-HBN, [35]). We plotted in figure 11 the product abundances of the elements 4 He, 12 C, 16 O, and 20 Ne as a function of time for the three HB fractional models.…”

“…An Emden-Fowler family of differential equations was developed by El-Nabulsi [9] , which featured the introduction of a generalised derivatives operator. Nouh [10] solved the stellar helium burning phase in a fractional differential form and has discussed possibilities in stellar model calculations to apply the fractional model. Yousif et al [11] solved the conformable isothermal gas sphere and compared the conformable models with Riemann-Liouville fractional models.…”

“…Figure 11. Comparison three fractional HBN models; the C-HBN model (α = 0.9,[10]), the AB-HBN model (α = 0.9,[35]), and model #5 (α = 0.9, β = η = 1).…”