Guaranteed Error Bounds for Eigenvalues of Singular Sturm-Liouville Problems

Brown, B. M.; McCormack, D. K. R.; Marletta, Marco

doi:10.1002/(sici)1522-2616(200005)213:1<17::aid-mana17>3.0.co;2-r

Cited by 4 publications

(3 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A parametric study is finally undertaken that confirms those results that are available in the literature and extends them by considering the effects of changing the parameters L, p, q and w. 6 [9]. Furthermore, in both the regular and singular cases, precision can be given to these results by enclosing them in intervals whose bounds can be proven to be correct [10].…”

Section: Introductionsupporting

confidence: 61%

Homogeneous trees of second order Sturm–Liouville equations: A general theory and program

Howson¹,

Watson²

2012

Computers & Structures

View full text Add to dashboard Cite

show abstract

Section: Introductionsupporting

confidence: 61%

Homogeneous trees of second order Sturm–Liouville equations: A general theory and program

Howson¹,

Watson²

2012

Computers & Structures

View full text Add to dashboard Cite

show abstract

“…This work has been motivated by the need to have accurate estimates of the eigenvalues of equation (1.1), despite the analytic difficulties, which are such that few examples have so far been analysed. Furthermore, in both the regular and singular cases, precision can be given to these results by enclosing them in intervals whose bounds can be proven to be correct (see the recent paper by Brown et al . (2000)).…”

Section: Introductionmentioning

confidence: 99%

Application of the Wittrick—Williams algorithm to the Sturm—Liouville problem on homogeneous trees: a structural mechanics analogy

Williams

Howson

Watson

2004

Proc. R. Soc. Lond. A

View full text Add to dashboard Cite

The Wittrick-Williams (WW) algorithm was developed over 30 years ago and has been applied with increasing sophistication to problems in structural mechanics ever since. Much wider applications, to any field requiring eigenvalues of self-adjoint systems of differential equations, are possible based on a theorem due to Balakrishnan that underpins the algorithm. These can be calculated to machine accuracy and none are missed. Here the value of the algorithm in mathematics is illustrated by studying in depth Sturm-Liouville equations on large homogeneous trees. These typically involve 10 13 equations and eigenvalues, which often coincide to form high multiplicity ones. Computation is quick (e.g. 1 s) and numerically stable because the multi-level subsysteming corollary of the theorem underpinning the WW algorithm is used.Our numerical results confirm the recent theoretical bounds of Sobolev & Solomyak on the bands into which the spectrum is divided by gaps split by one very high multiplicity eigenvalue. Additionally, an analogy based on structural mechanics and confirmed by numerical results gives exact equations for the high multiplicities of the gap eigenvalue and of those in the band. These cover any b and n, the branching number and number of levels of the tree. When these equations are divided by the number of eigenvalues in one band-gap interval, dimensionless results are obtained which become exact for n → ∞. Finally, the fragmentation of multiple eigenvalues caused by introducing a potential is studied numerically and interpreted using the structural mechanics analogy.

show abstract

“…This work was motivated by the need for accurate eigenvalue estimates for (1.1), despite the analytic difficulties, which are such that few examples have so far been analysed. Furthermore, in both the regular and singular cases, precision can be given to these results by enclosing them in intervals whose bounds can be proven to be correct (see the recent paper by Brown et al (2000)). There is also much current interest in the problem of homogeneous trees (Evans & Harris 1993;Evans et al 2001) with the recent work of Sobolev & Solomyak (2002) being of particular relevance because they show that for an infinite tree the eigenvalues of the free Laplacian in one dimension form bands of absolutely continuous spectra with eigenvalues of infinite multiplicity in the gaps.…”

Section: Introductionmentioning

confidence: 99%

Exact solutions for Sturm–Liouville problems on trees via novel substitute systems and the Wittrick–Williams algorithm

et al. 2007

View full text Add to dashboard Cite

Since 1970, the Wittrick–Williams algorithm has been applied with increasing sophistication in structural mechanics to guarantee that eigenvalues cannot be missed and are calculated accurately. The underlying theorem enables its application to any discipline requiring eigenvalues of self-adjoint systems of differential equations. Its value in mathematics was recently illustrated by studying Sturm–Liouville equations on large homogeneous trees with Dirichlet boundary conditions and n (≤43) levels. Recursive subsysteming was applied n −1 times to assemble the tree progressively from sub-trees. Hence, numerical results confirmed the recent theoretical bounds of Sobolev & Solomyak for n →∞. In addition, a structural mechanics analogy yielded a proof that many eigenvalues had high multiplicities determined by n and the branching number b . Inspired by the structural mechanics analogy, we now prove that all eigenvalues of the tree are obtainable from n substitute chains r (=1, 2, …, n ) which involve only r consecutively linked differential equations and which have only singlefold eigenvalues. Equations are also derived for the multiplicities these eigenvalues have for the tree. Hence, double precision calculations on a PC readily gave eigenvalues for n =10 6 and b =10, i.e. ≃10 999 999 linked Sturm–Liouville equations. Moreover, a simple equation is derived which gives all the eigenvalues of uniform trees with Dirichlet conditions at both ends, and band-gap spectra are numerically demonstrated and theoretically justified for trees with the Dirichlet conditions at either end replaced by Neumann ones. Additionally, even if each multiple eigenvalue would be counted as if it were singlefold, the proportion of eigenvalues that are multiple is proved to approach unity as n →∞.

show abstract

Guaranteed Error Bounds for Eigenvalues of Singular Sturm-Liouville Problems

Cited by 4 publications

References 14 publications

Homogeneous trees of second order Sturm–Liouville equations: A general theory and program

Homogeneous trees of second order Sturm–Liouville equations: A general theory and program

Application of the Wittrick—Williams algorithm to the Sturm—Liouville problem on homogeneous trees: a structural mechanics analogy

Exact solutions for Sturm–Liouville problems on trees via novel substitute systems and the Wittrick–Williams algorithm

Contact Info

Product

Resources

About