Asian Research Journal of Current Science

This study presents a Variational Iteration Algorithm (VIA) for obtaining accurate numerical solutions of linear thirteenth-order boundary value problems using Chebyshev polynomials of the fourth kind as basis functions. High-order boundary value problems arise in various areas of mathematical modelling and scientific computing, where efficient and reliable numerical techniques are required to obtain accurate solutions. The proposed approach combines the correction functional of the Variational Iteration Algorithm with the approximation properties of fourth-kind Chebyshev polynomials to construct accurate approximations. The approximate solution is expressed as a finite expansion of appropriately shifted fourth-kind Chebyshev polynomials, while the unknown coefficients are determined by imposing the prescribed boundary conditions. To assess the accuracy and effectiveness of the proposed method, two numerical examples are considered, and the obtained results are compared with solutions reported in the existing literature. The numerical results demonstrate that the proposed approach produces highly accurate approximations and performs competitively with other established methods. The study further illustrates the suitability of fourth-kind Chebyshev polynomials as basis functions for the numerical treatment of high-order boundary value problems. Therefore, the proposed Variational Iteration Algorithm provides an efficient, accurate, and computationally attractive framework for solving linear thirteenth-order boundary value problems and has the potential for extension to other classes of high-order differential equations. All computations were performed using Maple 2018.