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 { L k ( x ) ; k ∈ N } . We propose an improved approach which seeks approximants in terms of...

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Bibliographic Details
Published in:Applied mathematics and computation 2010-06, Vol.216 (7), p.1923-1937
Main Authors: El-Daou, Mohamed K., Al-Matar, Nadia R.
Format: Article
Language:English
Subjects:
Approximants
Approximation
Computation
Differential equations
Eigenfunctions
Eigenvalues
Exact sciences and technology
Fourier analysis
Gauss points
Legendre polynomials
Mathematical analysis
Mathematical models
Mathematics
Nonlinear algebraic and transcendental equations
Numerical analysis
Numerical analysis. Scientific computation
Numerical linear algebra
Ordinary differential equations
Sciences and techniques of general use
Sturm–Liouville problem
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Summary: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 { L k ( x ) ; k ∈ N } . We propose an improved approach which seeks approximants in terms of a finite sum of exponentially weighted Legendre polynomials e ω k x L k ( x ) ; k ∈ N for some real or complex frequencies { ω k } . 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.
ISSN:0096-3003
1873-5649
DOI:10.1016/j.amc.2010.03.022