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Eigenvalues of regular fourth-order Sturm-Liouville problems with transmission conditions. (English) Zbl 1371.34128

Summary: In this paper, a class of fourth-order Sturm-Liouville problems with transmission conditions is considered. The eigenvalues depend not only continuously but also smoothly on the problem. An expression for the derivative of the eigenvalues with respect to a given parameter: an endpoint, a boundary condition, a transmission condition, a coefficient, or the weight function, is found.

MSC:

34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
34B24 Sturm-Liouville theory

Software:

SLEUTH; SLEDGE
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Full Text: DOI

References:

[1] Likov, The Theory of Heat and Mass Transfer (1963)
[2] Buschmann, One-dimensional schrödinger operators with local point interactions, Journal Für Die Reine Und Angewandte Mathematik 467 pp 169– (1995) · Zbl 0833.34082
[3] Aydemir, Variational principles for spectral analysis of one Sturm-Liouville problem with transmission conditions, Advances in Difference Equations 2016 pp 1– (2016) · Zbl 1419.34105 · doi:10.1186/s13662-016-0800-z
[4] Tunc, Fundamental solutions and eigenvalues of one boundary-value problem with transmission conditions, Applied Mathematics and Computation 157 (2) pp 347– (2004) · Zbl 1060.34007 · doi:10.1016/j.amc.2003.08.039
[5] Zhang, Dependence of eigenvalues of Sturm-Liouville problems with interface conditions, Applied Mathematics and Computation 265 pp 31– (2015) · Zbl 1410.34260 · doi:10.1016/j.amc.2015.05.002
[6] Mukhtarov, Problems for ordinary differential equations with transmission conditions, Applicable Analysis 81 pp 1033– (2002) · Zbl 1062.34094 · doi:10.1080/0003681021000029853
[7] Zettl, Ajoint and self-adjoint boundary value problems with interface conditions, SIAM Journal on Applied Mathematics 16 pp 851– (1968) · Zbl 0162.11201 · doi:10.1137/0116069
[8] Allahverdiev, Eigenparameter dependent Sturm-Liouville problems in boundary conditions with transmission conditions, Journal of Mathematical Analysis and Applications 401 pp 388– (2013) · Zbl 1271.34032 · doi:10.1016/j.jmaa.2012.12.020
[9] Greenberg, The code SLEUTH for solving fourth order Sturm-Liouville problems, ACM Transactions on Mathematical Software 23 pp 453– (1997) · Zbl 0912.65073 · doi:10.1145/279232.279231
[10] Suo, Eigenvalues of a class of regular fourth-order Sturm-Liouville problems, Applied Mathematics and Computation 218 pp 9716– (2012) · Zbl 1268.34178 · doi:10.1016/j.amc.2012.03.015
[11] Zhang X The self-adjointness, dissipation and spectrum analysis of some classes high order differential operators with discontinuity Ph.D. thesis 2013
[12] Bailey, The SLEIGN2 Sturm-Liouville code, ACM Transactions on Mathematical Software 21 pp 143– (2001) · Zbl 1070.65576 · doi:10.1145/383738.383739
[13] Kong, Dependence of eigenvalues of Sturm-Liouville problems on the boundary, Journal of Differential Equations 126 pp 389– (1996) · Zbl 0856.34027 · doi:10.1006/jdeq.1996.0056
[14] Kong, Eigenvalues of regular Sturm-Liouville problems, Journal of Differential Equations 131 pp 1– (1996) · Zbl 0862.34020 · doi:10.1006/jdeq.1996.0154
[15] Kong, Dependence of eigenvalues on the problems, Mathematische Nachrichten 188 pp 173– (1997) · Zbl 0888.34017 · doi:10.1002/mana.19971880111
[16] Kong, Dependence of the nth Sturm-Lioouville eigenvalue on the problem, Journal of Differential Equations 156 pp 328– (1999) · Zbl 0932.34081 · doi:10.1006/jdeq.1998.3613
[17] Wang, The classification of self-adjoint boundary conditions: separated, coupled, and mixed, Journal of Functional Analysis 255 pp 1554– (2008) · Zbl 1170.34017 · doi:10.1016/j.jfa.2008.05.003
[18] Wang, Two-interval Sturm-Liouville operators in modified Hilbert spaces, Journal of Mathematical Analysis and Applications 328 pp 390– (2007) · Zbl 1116.47040 · doi:10.1016/j.jmaa.2006.05.058
[19] Hao, Canonical forms of self-adjoint boundary conditions for differential operators of order four, Journal of Mathematical Analysis and Applications 387 pp 1176– (2012) · Zbl 1245.34018 · doi:10.1016/j.jmaa.2011.10.025
[20] Dauge, Eigenvalues variation. I. Neumann problem for Sturm-Liouville operators, Journal of Differential Equations 104 pp 243– (1993) · Zbl 0784.34021 · doi:10.1006/jdeq.1993.1071
[21] Dauge, Eigenvalues variation. II. Multidimentional problems, Journal of Differential Equations 104 pp 263– (1993) · Zbl 0807.34033 · doi:10.1006/jdeq.1993.1072
[22] Zettl A Sturm-Liouville Theory
[23] Kong, WSSIAA, in: Inequalities and Applications pp 381– (1994) · doi:10.1142/9789812798879_0031
[24] Dieudonné, Foundations of Modern Analysis (1969)
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