Computational Method for a Fractional Model of the Helium Burning Network

TL;DR

This paper introduces a computational fractional differential equation model for helium burning in stellar cores, providing analytic solutions and examining the effects of fractional parameters on element synthesis.

Contribution

It presents a novel fractional differential equation approach to model helium burning, with solutions that account for non-integer dynamics in stellar processes.

Findings

Good agreement with integer model at =1, max error 0.003

Analytic expressions for element abundances over time

Fractional parameters significantly affect element synthesis

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- {\alpha} 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 \alpha=1 , which represents the integer solution, with the numerical solution revealed a good agreement with maximum error \epsilon =0.003. The product abundances are calculated at\alpha=0.25, 0.5, 0.75 to declare the effects of changing the fractional parameters on the calculations.