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Homogeneous trees of second order Sturm-Liouville equations: a general theory and program

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journal contribution
posted on 15.05.2015 by W.P. Howson, Andrew Watson
Quantum graph problems occur in many disciplines of science and engineering and they can be solved by viewing the problem as a structural engineering one. The Sturm–Liouville operator acting on a tree is an example of a quantum graph and the structural engineering analogy is the axial vibration of an assembly of bars connected together with a tree topology. Using the dynamic stiffness matrix method the natural frequencies of the system can be determined which are analogous to the eigenvalues of the quantum graph. Theory is presented that yields exact solutions to the Sturm–Liouville problem on homogeneous trees. This is accompanied by an extremely efficient and compact computer program that implements the theory. An understanding of the former is enhanced by recourse to a structural mechanics analogy, while the latter program is fully annotated and explained for those who might wish to extend its capability. In addition, the use of the program as a ‘black box’ is fully described and a small parametric study is undertaken to confirm the accuracy of the approach and indicate its range of application including the computation of negative eigenvalues.

History

School

  • Aeronautical, Automotive, Chemical and Materials Engineering

Department

  • Aeronautical and Automotive Engineering

Published in

COMPUTERS & STRUCTURES

Volume

104

Pages

13 - 20 (8)

Citation

HOWSON, A.P. and WATSON, A., 2012. Homogeneous trees of second order Sturm-Liouville equations: a general theory and program. Computers and Structures, 104-105, pp. 13 - 20.

Publisher

© Civil-Comp and Elsevier Ltd.

Version

SMUR (Submitted Manuscript Under Review)

Publisher statement

This work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/

Publication date

2012

Notes

This is the submitted version of an article that has been subsequently published in the journal, Computers and Structures [© Civil-Comp and Elsevier Ltd.] The definitive version is available at: http://dx.doi.org/10.1016/j.compstruc.2012.03.001.

ISSN

0045-7949

Language

en

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