# 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 language English Advances in Applied Mathematics and Approximation Theory Contributions from AMAT 2012 Springer New York LLC 201-214 14 9781461463924 https://doi.org/10.1007/978-1-4614-6393-1_12 Published - 1 Jan 2013

### Publication series

Name Springer Proceedings in Mathematics and Statistics 41 2194-1009 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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N2 - 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.

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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