An efficient spectral algorithm for non-linear astrophysical Lane–Emden problem using pseudo-Chebyshev wavelets with error analysis

Mother wavelets play a crucial role in wavelet analysis, leading to the development of several well-known families such as Haar, Morlet, Legendre, and Chebyshev wavelets. In this work, we introduce the concept of Pseudo-Chebyshev Wavelets, which, unlike classical wavelet families, are specifically designed to enhance approximation accuracy through the exploitation of pseudo-Chebyshev function. This novel construction yields wavelets that preserve the spectral convergence properties of Chebyshev methods while providing greater flexibility in handling boundary conditions and singularities. Extending our earlier efforts, we define pseudo-Chebyshev wavelets on the generalized interval a,b, 0≤a≤b≤l, rather than restricting to the standard interval 0,1. This generalization is particularly significant since many engineering and astrophysical models, including Lane–Emden-type differential equations, are naturally posed on arbitrary finite domains. The proposed pseudo-Chebyshev wavelet operational matrix of integration is developed and incorporated into a new spectral algorithm, reducing Lane–Emden-type equations to tractable systems of algebraic equations. The novelty of our method lies in (i) its distinction from existing spectral and wavelet-based approaches by combining pseudo-polynomial structures with wavelet operational techniques, (ii) its capacity to achieve higher numerical stability and precision in nonlinear problems using pseudo-Chebyshev wavelets, (iii) for this specific wavelet with k=1 and μ=2, the operational matrix of integration for an arbitrary M is, to the best of our knowledge, developed and introduced here for the first time as a novel contribution to the existing framework. This matrix provides an effective tool for the accurate and efficient solution of Lane–Emden differential equations, as well as a wide class of related differential and integral equations, and it further offers significant potential for broader applications in applied mathematics and physics, (iv) its applicability to computational mathematics and astrophysics, where accurate and efficient solvers for Lane–Emden-type equations remain a central challenge. Numerical experiments and graphical comparisons with exact solutions confirm the effectiveness and reliability of the proposed approach. This study thus establishes pseudo-Chebyshev wavelets as a powerful new tool for spectral approximation and applied problem-solving.