An improved Tau method for a class of Sturm-Liouville problems

We consider the numerical solution of Sturm-Liouville eigenvalue problems by Legendre-Gauss Tau method. The latter approximates the solution of differential equations as a finite sum of Legendre polynomials {Lk(x);k∈N}. We propose an improved approach which seeks approximants in terms of a finite sum of exponentially weighted Legendre polynomials eωkxLk(x);k∈N for some real or complex frequencies {ωk}. With the introduction of such exponentials, Legendre-Gauss Tau method can detect the sharp variations exhibited by the highly indexed Sturm-Liouville eigenfunctions. The efficiency of our results is illustrated through numerical examples.