Solving second-order discrete Sturm-Liouville BVP using matrix pencils

Michael K. Wilson, Aihua Li

Research output: Chapter in Book/Report/Conference proceedingConference contribution

1 Citation (Scopus)

Abstract

This paper deals with discrete second order Sturm-Liouville Boundary Value Problems (DSLBVP) where the parameter λ, as part of the difference equation, appears nonlinearly in the boundary conditions. We focus on the case where the boundary condition is given by a cubic equation in λ. We first describe the problem by a matrix equation with nonlinear variables such that solving the DSLBVP is equivalent to solving the matrix equation. We develop methods to finding roots of the characteristic polynomial (in the variable λ) of the involved matrix. We further reduce the problem to finding eigenvalues of a matrix pencil in the form of A-λB. Under certain conditions, such a matrix pencil eigenvalue problem can be reduced to a stabdard eigenvalue problem, so that existing computational tools can be used to solve the problem. The main results of the paper provide the reduction procedure and rules to identify the cubic DSLBVPs which can be reduced to standard eigenvalue problems. We also investigate the structure of the matrix form of a DSLBVP and its effect on the reality of the eigenvalues of the problem. We give a class of DSLBVPs which have only real eigenvalues.

Original languageEnglish
Title of host publicationAdvances in Applied Mathematics and Approximation Theory
Subtitle of host publicationContributions from AMAT 2012
PublisherSpringer New York LLC
Pages201-214
Number of pages14
ISBN (Print)9781461463924
DOIs
StatePublished - 1 Jan 2013

Publication series

NameSpringer Proceedings in Mathematics and Statistics
Volume41
ISSN (Print)2194-1009
ISSN (Electronic)2194-1017

Fingerprint

Matrix Pencil
Sturm-Liouville
Eigenvalue Problem
Boundary Value Problem
Matrix Equation
Eigenvalue
Boundary conditions
Root-finding
Cubic equation
Characteristic polynomial
Difference equation
Form

Cite this

Wilson, M. K., & Li, A. (2013). Solving second-order discrete Sturm-Liouville BVP using matrix pencils. In Advances in Applied Mathematics and Approximation Theory: Contributions from AMAT 2012 (pp. 201-214). (Springer Proceedings in Mathematics and Statistics; Vol. 41). Springer New York LLC. https://doi.org/10.1007/978-1-4614-6393-1_12
Wilson, Michael K. ; Li, Aihua. / Solving second-order discrete Sturm-Liouville BVP using matrix pencils. Advances in Applied Mathematics and Approximation Theory: Contributions from AMAT 2012. Springer New York LLC, 2013. pp. 201-214 (Springer Proceedings in Mathematics and Statistics).
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Wilson, MK & Li, A 2013, Solving second-order discrete Sturm-Liouville BVP using matrix pencils. in Advances in Applied Mathematics and Approximation Theory: Contributions from AMAT 2012. Springer Proceedings in Mathematics and Statistics, vol. 41, Springer New York LLC, pp. 201-214. https://doi.org/10.1007/978-1-4614-6393-1_12

Solving second-order discrete Sturm-Liouville BVP using matrix pencils. / Wilson, Michael K.; Li, Aihua.

Advances in Applied Mathematics and Approximation Theory: Contributions from AMAT 2012. Springer New York LLC, 2013. p. 201-214 (Springer Proceedings in Mathematics and Statistics; Vol. 41).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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Wilson MK, Li A. Solving second-order discrete Sturm-Liouville BVP using matrix pencils. In Advances in Applied Mathematics and Approximation Theory: Contributions from AMAT 2012. Springer New York LLC. 2013. p. 201-214. (Springer Proceedings in Mathematics and Statistics). https://doi.org/10.1007/978-1-4614-6393-1_12