Journal of Applied Physical Science International

This study presents a numerical scheme for solving linear fourteenth-order boundary value problems by combining the variational iteration method with shifted Chebyshev polynomials of the fourth kind. The proposed approach constructs a correction functional for the governing differential equation, with the associated Lagrange multiplier determined through variational principles. Shifted Chebyshev polynomials of the fourth kind are then employed as basis functions to approximate the solution and determine the unknown coefficients under the prescribed boundary conditions. The method was applied to three benchmark linear fourteenth-order boundary value problems with known exact solutions. The accuracy of the approximate solutions was evaluated through absolute-error comparisons and graphical agreement between the numerical and exact solutions. At the tabulated points, the proposed scheme produced absolute errors ranging from 0 to 1.0 × 10⁻⁹ in the first example, from 1.41 × 10⁻⁷ to 2.27 × 10⁻⁷ in the second example, and from 0 to 2.0 × 10⁻¹⁰ in the third example. The results indicate close agreement between the computed and exact solutions for the selected test problems. Comparisons with existing numerical results suggest that the proposed method provides competitive accuracy for the considered class of linear high-order boundary value problems. These findings support the use of variational iteration combined with shifted Chebyshev polynomial approximation for selected high-order linear differential problems.