An efficient numerical algorithm for Lane–Emden–Fowler functional BVPs arising in isothermal gas spheres and heat diffusion models

Abstract

Over recent decades, the Lane–Emden–Fowler equation (LEFE) has become an important model with wide-ranging applications, including astrophysical processes and heat transfer in biological systems. Since the evolution of such systems depends on past and future states, it can be effectively described using functional-type boundary value problems. This study proposes an extended LEFE model incorporating functional arguments along with informative nonlocal three-point boundary conditions. The existence and uniqueness of solutions to the resulting functional boundary value problem are established. Due to the inherent complexity of LEFE-type functional problems, obtaining accurate and efficient numerical solutions remains challenging. To address this, a polynomial-based collocation approach using Bernstein polynomials in \(L^{2}[0, 1]\) space is developed. The wide applicability of the proposed method is further demonstrated through the well-known isothermal gas spheres and heat diffusion models, considered as a particular case of the proposed problem. The results confirm that the approach achieves faster convergence, higher accuracy, and improved computational efficiency compared to existing numerical schemes. A convergence analysis further supports the effectiveness of the method for solving complex functional-type differential equations.