An efficient collocation scheme for new type of variable-order fractional Lane–Emden equation

The fractional Lane–Emden model illustrates different phenomena in astrophysics and mathematical physics. This paper involves the Vieta–Lucas (Vt-L) bases to solve types of variable-order (V-O) fractional Lane–Emden equation (linear and nonlinear). The operational matrix of the V-O fractional derivative is obtained for the Vt-L polynomials. In the established approach, these polynomials are applied to transform the main problem into an algebraic equations system. To indicate the performance and capability of the scheme, a number of examples are presented for various types of V-O fractional Lane–Emden equations. Also, for one example, a comparison is done between the calculated results by our technique and those obtained via the Bernoulli polynomials. Overall, this paper introduces a new methodology for solving V-O fractional Lane–Emden equations using Vt-L bases. The derived operational matrix and the transformation to an algebraic equation system offer practical advantages in solving these equations efficiently. The presented examples and comparative analysis highlight the effectiveness and validity of the proposed technique, contributing to the understanding and advancement of fractional Lane–Emden models in astrophysics and mathematical physics.