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We extend the application of Legendre-Galerkin algorithms for sixth-orderelliptic problems with constant coefficients to sixth-order elliptic equations with variable polynomialcoefficients. The complexities of the algorithm are O(N) operations for a one-dimensional domain with(N−5) unknowns. An efficient and accurate direct solution for algorithms based on the Legendre-Galerkinapproximations developed for the two-dimensional sixth-order elliptic equations with variable coefficientsrelies upon a tensor product process. The proposed Legendre-Galerkin method for solvingvariable coefficients problem is more efficient than pseudospectral method. Numerical examples areconsidered aiming to demonstrate the validity and applicability of the proposed techniques

An Extension of the Legendre-Galerkin Method for Solving Sixth-Order Differential Equations with Variable Polynomial Coefficients