Numerical solution and error analysis of the Thomas–Fermi type equations with integral boundary conditions by the modified collocation techniques

Abstract

This paper introduces modified collocation techniques that utilize Chebyshev and Legendre polynomials for solving Thomas–Fermi type equations with integral-type boundary conditions. Firstly, the problems under consideration have been transformed into corresponding integral equations to avoid singularity at the origin. Then, the modified collocation methods have been applied to achieve practical numerical solutions. The existence of unique solutions for each integral equation is established through corresponding theorems. Notably, proposed techniques do not require approximation of higher derivatives such as x′ and x′′. This aspect not only simplifies the computational process but also enhances the precision of the method by avoiding potential errors associated with derivative approximations. To ensure the reliability of the current method, error analysis is conducted for each approach under general conditions. Furthermore, the effectiveness of the technique is substantiated by including practical examples in the paper. These examples serve as empirical evidence, showcasing the efficiency of proposed approaches in realistic problem-solving scenarios.