zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Computation of eigenvalues and solutions of regular Sturm-Liouville problems using Haar wavelets. (English) Zbl 1152.65086

Summary: The paper presents a novel method for the computation of eigenvalues and solutions of Sturm-Liouville eigenvalue problems using truncated Haar wavelet series. This is an extension of the technique proposed by C.-H. Hsiao [Math. Comput. Simul. 64, No. 5, 569–585 (2004; Zbl 1039.65051)] to solve discretized version of variational problems via Haar wavelets. The proposed method aims to cover a wider class of problems, by applying it to historically important and a very useful class of boundary value problems, thereby enhancing its applicability.

To demonstrate the effectiveness and efficiency of the method various celebrated Sturm-Liouville problems are analyzed for their eigenvalues and solutions. Also, eigensystems are investigated for their asymptotic and oscillatory behavior. The proposed scheme, unlike the conventional numerical schemes, such as Rayleigh quotient and Rayleigh-Ritz approximation, gives eigenpairs simultaneously and provides upper and lower estimates of the smallest eigenvalue, and it is found to have quadratic convergence with increase in resolution.

MSC:
65L15Eigenvalue problems for ODE (numerical methods)
34L16Numerical approximation of eigenvalues and of other parts of the spectrum
65L20Stability and convergence of numerical methods for ODE
References:
[1]Bender, C. M.; Orszag, S. A.: Advanced mathematical methods for scientists and engineers, (1987)
[2]Binding, P.; Volmer, H.: Eigencurves for two-parameter Sturm – Liouville equations, SIAM rev. 38, 27-48 (1996) · Zbl 0869.34020 · doi:10.1137/1038002
[3]Van Brunt, B.: The calculus of variations, (2003)
[4]Chen, C. F.; Hsiao, C. H.: A Walsh series direct method for solving variational problems, J. franklin inst. 300, 265-280 (1975) · Zbl 0339.49017 · doi:10.1016/0016-0032(75)90199-4
[5]Elsgolts, L.: Differential equations and calculus of variations, (1970) · Zbl 0212.39902
[6]Everitt, W. N.: Oscillation of eigenfunctions of weighed regular Sturm – Liouville problems, J. London math. Soc. 27, No. 2, 106-120 (1983) · Zbl 0529.34039 · doi:10.1112/jlms/s2-27.1.106
[7]Wan, Y. M. Frederick: Introduction to the calculus of variations and its applications, (1995) · Zbl 0843.49001
[8]Hsiao, C. H.: Haar wavelet direct method for solving variational problems, Math. comput. Simulation 64, 560-585 (2004) · Zbl 1039.65051 · doi:10.1016/j.matcom.2003.11.012
[9]Razzaghi, M.; Seperhrian, B.: Single term Walsh series direct method for the solution of non-linear problems in the calculus of variations, J. vibration and control 10, No. 7, 1071-1081 (2004) · Zbl 1092.49022 · doi:10.1177/1077546304042071
[10]Simmons, G. F.: Differential equations with applications and historical notes, (1992)
[11]Strauss, W. A.: Partial differential equations: an introduction, (1992) · Zbl 0817.35001
[12]Strchartz, R. S.: Construction of orthonormal wavelets, Wavelets mathematics and application (1993)