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

摘要

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.