Bounds for the Solution of the Sturm-Liouville Problem with Application to Finite Difference Methods

Previous article Next article Bounds for the Solution of the Sturm-Liouville Problem with Application to Finite Difference MethodsA. K. Aziz and B. E. HubbardA. K. Aziz and B. E. Hubbardhttps://doi.org/10.1137/0112015PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] J. H. Bramble and , B. E. Hubbard, Some higher order integral identities with application to bounding techniques, J. Res. Nat. Bur. Standards Sect. B, 65B (1961), 261–268 MR0132909 0104.34901 CrossrefISIGoogle Scholar[2] J. H. Bramble and , L. E. Payne, Bounds for solutions of second-order elliptic partial differential equations, Contributions to Differential Equations, 1 (1963), 95–127 MR0163049 0141.09801 Google Scholar[3] J. H. Bramble and , L. E. Payne, Some integral inequalities for uniformly elliptic operators, Contributions to Differential Equations, 1 (1963), 129–135 MR0163050 0141.09802 Google Scholar[4] J. H. Bramble and , L. E. Payne, Bounds for derivatives in the Dirichlet problem for Poisson's equation, J. Soc. Indust. Appl. Math., 10 (1962), 370–380 10.1137/0110027 MR0155994 0113.08903 LinkISIGoogle Scholar[5] Lothar Collatz, The numerical treatment of differential equations. 3d ed, Translated from a supplemented version of the 2d German edition by P. G. Williams. Die Grundlehren der mathematischen Wissenschaften, Bd. 60, Springer-Verlag, Berlin, 1960xv+568 pp. (1 plate) MR0109436 0086.32601 CrossrefGoogle Scholar[6] L. Collatz, Konvergenzbeweis und Fehlerabschätzung für das Differenzverfahren bei Eigenwertproblemen gewöhnlicher Differentialgleichungen zweiter und vierter Ordnung, Deutsche Math., 2 (1937), 189–215 Google Scholar[7] H. Lewy, , R. Courant and , K. Friedrichs, Über die partiellen Differenzengleichungen der mathematischen Physik, Math. Ann., 100 (1928), 32–74 10.1007/BF01448839 MR1512478 CrossrefGoogle Scholar[8] Hiroshi Fujita, On the error of the finite difference method in 1-dimensional boundary value problems, J. Phys. Soc. Japan, 11 (1956), 160–169 MR0088806 CrossrefGoogle Scholar[9] B. E. Hubbard, Bounds for eigenvalues of the Sturm-Liouville problem by finite difference methods, Arch. Rational Mech. Anal., 10 (1962), 171–179 10.1007/BF00281184 MR0145670 0196.49802 CrossrefISIGoogle Scholar[10] V. P. Il'in, Estimate of error in the Ritz method for ordinary differential equations, Trudy Mat. Inst. Steklov., 53 (1959), 43–63 MR0111148 Google Scholar[11] L. V. Kantorovich and , V. I. Krylov, Approximate methods of higher analysis, Translated from the 3rd Russian edition by C. D. Benster, Interscience Publishers, Inc., New York, 1958xv+681 MR0106537 0083.35301 Google Scholar[12] L. E. Payne and , H. F. Weinberger, New bounds for solutions of second order elliptic partial differential equations, Pacific J. Math., 8 (1958), 551–573 MR0104047 0093.10901 CrossrefGoogle Scholar[13A] Johann Schröder, Über das Differenzenverfahren bei nichtlinearen Randwertaufgaben. II, Z. Angew. Math. Mech., 36 (1956), 443–455 MR0085607 0072.33602 CrossrefGoogle Scholar[13B] Johann Schröder, Über das Differenzenverfahren bei nichtlinearen Randwertaufgaben. I, Z. Angew. Math. Mech., 36 (1956), 319–331 MR0092233 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails On the Eigenvalues of Second Order Elliptic Difference OperatorsSeymour V. Parter31 July 2006 | SIAM Journal on Numerical Analysis, Vol. 19, No. 3AbstractPDF (942 KB)Properties of Some Tridiagonal Matrices and Their Application to Boundary Value ProblemsCharlotte F. Fischer and Riaz A. Usmani14 July 2006 | SIAM Journal on Numerical Analysis, Vol. 6, No. 1AbstractPDF (1103 KB)Numerical Methods for Generalized Axially Symmetric PotentialsSeymour V. Parter14 July 2006 | Journal of the Society for Industrial and Applied Mathematics Series B Numerical Analysis, Vol. 2, No. 3AbstractPDF (1353 KB) Volume 12, Issue 1| 1964Journal of the Society for Industrial and Applied Mathematics History Submitted:01 November 1962Accepted:01 August 1963Published online:13 July 2006 InformationCopyright © 1964 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0112015Article page range:pp. 163-178ISSN (print):0368-4245ISSN (online):2168-3484Publisher:Society for Industrial and Applied Mathematics