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

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