Analysis of Spectral Tau Method for Approximate Solution of Fourth‐Order BVP in Hilbert Spaces

This research explores the effectiveness of the spectral Tau method for solving fourth‐order differential boundary value problem (FBVP). We transform this FBVP into a Volterra–Fredholm integral equation (VFIE). By applying Banach’s fixed‐point theorem, we investigate the existence and uniqueness of the solution for the VFIE form of the FBVP equation. Through examples, we illustrate that the VFIE problem demonstrates superior numerical stability compared with the original FBVP formulation. For this purpose, the spectral Tau method is applied in two cases, one for the VFIE form of the problem and the other for the FBVP form. The notable numerical results from the VFIE problem, in contrast to those from the FBVP problem, highlight the efficiency of the proposed method. Additionally, we demonstrate the convergence theorem for the numerical solution of the Tau method in the VFIE problem and extend it to the FBVP problem.