Two robust operational matrix algorithms for solving non-linear Lane–Emden-type equations in astrophysics using Hermite and Fermat polynomials

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Abstract

Operational matrix-based algorithms have been identified as efficient tools for solving non-linear and fractional differential equations in engineering. Several wavelet-based algorithms have been developed for linear, non-linear and fractional differential equations. Wavelet-based spectral methods have also been identified as efficient tools for non-linear problems in astrophysics. In this paper, two reliable and efficient computational algorithms using Hermite wavelet and Fermat’s polynomial collocation methods are introduced to solve a class of non-linear Lane–Emden-type equations in astrophysics. Lane–Emden models have been characterised to predict the dynamics in various astrophysical contexts, such as stellar structure, white dwarfs and polytropic models. The main idea of the proposed wavelet and spectral algorithms is that the non-linear singular differential equations are converted into a system of algebraic equations using the operational matrix of derivatives. To the best of our knowledge, so far no rigorous Hermite wavelet and Fermat’s operational matrix of derivatives has been reported for the proposed models. The accuracy and efficiency of the proposed methods are confirmed by means of the comparison with other approximation algorithms. The proposed method can also be easily utilised to solve other types of non-linear differential equations in astrophysics.

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NRIAG Journal of Astronomy and Geophysics
Volume 14, Issue 1
December 2025
Pages 1-15

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