Study of fourth-order boundary value problem based on Volterra–Fredholm equation: numerical treatment

This paper presents a study of the performance of the Tau method using Chebyshev basis functions for solving fourth-order differential equation with boundary conditions. Existence and uniqueness of the solution of this equation are investigated transforming it into the Volterra–Fredholm integral equation. We use the operational Tau matrix representation with Chebyshev basis functions for constructing the algebraic equivalent representation of the problem.This representation is an special semi lower triangular system whose solution gives the components of the vector solution. Applying Gronwall’s and the generalized Hardy’s inequality, convergence analysis and error estimation of the Tau method are discussed. The error analysis indicates that the numerical errors decay exponentially when the source function are sufficiently smooth. Illustrative examples are given to represent the efficiency and the accuracy of the proposed method. Also, some comparisons are made with existing results such that the results obtained by Tau method are more accurate than the proposed methods in this case.