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.

本文介绍了使用切比雪夫基函数求解具有边界条件的四阶微分方程的 Tau 方法的性能研究。研究了该方程解的存在性和唯一性，将其转化为Volterra-Fredholm积分方程。我们使用带有切比雪夫基函数的操作 Tau 矩阵表示来构造问题的代数等价表示。这种表示是一个特殊的半下三角系统，其解给出了向量解的分量。应用Gronwall's和广义Hardy's不等式，讨论了Tau方法的收敛性分析和误差估计。误差分析表明，当源函数足够平滑时，数值误差呈指数衰减。举例说明了所提方法的效率和准确性。此外，与现有结果进行了一些比较，使得在这种情况下，Tau 方法获得的结果比提出的方法更准确。