A Chelyshkov-based spectral homotopy analysis method for singular nonlinear Lane–Emden-type equations

Mouaad Bouakkaz; Nouria Arar; Mabrouk Meflah; Kamal Shah; Bahaaeldin Abdalla; Thabet Abdeljawad; Mouaad Bouakkaz; Nouria Arar; Mabrouk Meflah; Kamal Shah; Bahaaeldin Abdalla; Thabet Abdeljawad

doi:10.3934/nhm.2026039

Networks and Heterogeneous Media

2026, Volume 21, Issue 3: 943-967. doi: 10.3934/nhm.2026039

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A Chelyshkov-based spectral homotopy analysis method for singular nonlinear Lane–Emden-type equations

1.
Laboratory of Applied Mathematics, Kasdi Merbah University, Ouargla 30000, Algeria
2.
Department of Mathematics, Constantine 1 University, Brothers Mentouri, Constantine 25017, Algeria
3.
Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia
4.
Department of Fundamental Sciences, Faculty of Engineering and Architecture, Istanbul Gelisim University, Avcilar, Istanbul 34310, Turkey
5.
Department of Medical Research, China Medical University, Taichung 40402, Taiwan, China
6.
Department of Mathematics and Applied Mathematics, School of Science and Technology, Sefako Makgatho Health Sciences University, Ga-Rankuwa, South Africa

Received: 26 January 2026 Revised: 19 February 2026 Accepted: 24 February 2026 Published: 12 May 2026

An enhanced numerical technique for solving the Lane–Emden equation is developed using the spectral homotopy analysis method (SHAM) with Chelyshkov polynomials and Gauss–Lobatto collocation nodes. The numerical results, including absolute errors and comparisons with the exact solution, demonstrate the convergence behavior and accuracy of the proposed approach. The results show that the Chelyshkov-based SHAM provides an effective framework for the numerical treatment of Lane–Emden-type problems.
- nonlinear equations,
- Lane–Emden equation,
- Chelyshkov polynomials,
- spectral methods,
- spectral homotopy analysis method
Citation: Mouaad Bouakkaz, Nouria Arar, Mabrouk Meflah, Kamal Shah, Bahaaeldin Abdalla, Thabet Abdeljawad. A Chelyshkov-based spectral homotopy analysis method for singular nonlinear Lane–Emden-type equations[J]. Networks and Heterogeneous Media, 2026, 21(3): 943-967. doi: 10.3934/nhm.2026039

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Abstract

An enhanced numerical technique for solving the Lane–Emden equation is developed using the spectral homotopy analysis method (SHAM) with Chelyshkov polynomials and Gauss–Lobatto collocation nodes. The numerical results, including absolute errors and comparisons with the exact solution, demonstrate the convergence behavior and accuracy of the proposed approach. The results show that the Chelyshkov-based SHAM provides an effective framework for the numerical treatment of Lane–Emden-type problems.

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