### An Efficient Method Based on Lucas Polynomials for Solving High-Order Linear Boundary Value Problems

#### Abstract

In this paper, a new collocation method based on Lucas polynomials for solving high-order linear differential equations with variable coefficients under the boundary conditions is presented by transforming the problem into a system of linear algebraic equations with Lucas coefficients. The proposed approach is applied to fourth, fifth, sixth and eighth-order two-point boundary values problems occurring in science and engineering, and compared by existing methods. The technique gives better approximations than other methods, and has a lower computational cost. In addition, the error analysis based on residual function is developed for our method and the improved approximate solution is obtained. Moreover, numerical examples are included to illustrate the practical usefulness and efficiency of the method.

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