Oscillation properties for non-classical Sturm-Liouville problems with additional transmission conditions
Abstract
This work is aimed at studying some comparison and oscillation properties of boundary value problems (BVP’s) of a new type, which differ from classical problems in that they are defined on two disjoint intervals and include additional transfer conditions that describe the interaction between the left and right intervals. This type of problems we call boundary value-transmission problems (BVTP’s). The main difficulty arises when studying the distribution of zeros of eigenfunctions, since it is unclear how to apply the classical methods of Sturm’s theory to problems of this type. We established new criteria for comparison and oscillation properties and new approaches used to obtain these criteria. The obtained results extend and generalizes the Sturm’s classical theorems on comparison and oscillation.
Keyword : non-classical SLP’s, transmission problems, comparison theorems, oscillatory solutions
How to Cite
Mukhtarov, O. S., & Aydemir, K. (2021). Oscillation properties for non-classical Sturm-Liouville problems with additional transmission conditions . Mathematical Modelling and Analysis, 26(3), 432-443. https://doi.org/10.3846/mma.2021.13216
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
B.P. Allahverdiev, E. Bairamov and E. Ugurlu. Eigenparameter dependent Sturm–Liouville problems in boundary conditions with transmission conditions. Journal of Mathematical Analysis and Applications, 401(1):388–396, 2013. https://doi.org/10.1016/j.jmaa.2012.12.020
B.P. Allahverdiev and H. Tuna. Eigenfunction expansion for singular SturmLiouville problems with transmission conditions. Electronic Journal of Differential Equations, 2019(3):1–10, 2019.
R.S. Anderssen and J.R. Cleary. Asymptotic structure in torsional free oscillations of the earthI. Overtone structure. Geophysical Journal International, 39(2):241–268, 1974. https://doi.org/10.1111/j.1365-246X.1974.tb05453.x
K. Aydemir and O.Sh. Mukhtarov. Completeness of one two-interval boundary value problem with transmission conditions. Miskolc Mathematical Notes, 15(2):293–303, 2014. https://doi.org/10.18514/MMN.2014.1229
K. Aydemir and O.Sh. Mukhtarov. Class of Sturm–Liouville problems with eigenparameter dependent transmission conditions. Numerical Functional Analysis and Optimization, 38(10):1260–1275, 2017. https://doi.org/10.1080/01630563.2017.1316995
K. Aydemir, H. Olğar, O.Sh. Muhtarov and F. Muhtarov. Differential operator equations with interface conditions in modified direct sum spaces. Filomat, 32(3):921–931, 2018. https://doi.org/10.2298/FIL1803921A
E. Bairamov and E. Ugurlu. On the characteristic values of the real component of a dissipative boundary value transmission problem. Applied Mathematics and Computation, 218(19):9657–9663, 2012. https://doi.org/10.1016/j.amc.2012.02.079
W.F. Bauer. Modified Sturm-Liouville systems. Quarterly of Applied Mathematics, 11(3):273–283, 1953. https://doi.org/10.1090/qam/64970
M. Bayramoglu, A. Bayramov and E. Sen. A regularized trace formula for a discontinuous Sturm-Liouville operator with delayed argument. Electronic Journal of Differential Equations, 2017(104):1–12, 2017.
P.A. Binding, H. Langer and M. Möller. Oscillation results for Sturm–Liouville problems with an indefinite weight function. Journal of computational and applied mathematics, 171(1):93–101, 2004.
J.R. Cannon and G.H. Meyer. On diffusion in a fractured medium. SIAM Journal on Applied Mathematics, 20(3):434–448, 1971. https://doi.org/10.1137/0120047
J.M.C. Duhamel. Mémoire sur les vibrations d’une corde flexible, chargée d’un ou de plusieurs curseurs. J. de lÉtcole Polytechnique, 1843.
A. Ergün and R. Amirov. Half inverse problem for diffusion operators with jump conditions dependent on the spectral parameter. Numerical Methods for Partial Differential Equations, 2020. https://doi.org/10.1002/num.22666
C.T. Fulton. Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions. Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 77(3-4):293–308, 1977. https://doi.org/10.1017/S030821050002521X
R.E. Gaskell. A problem in heat conduction and an expansion theorem. American Journal of Mathematics, 64(1):447–455, 1942. https://doi.org/10.2307/2371696
S.R. Grace and H.A. El-Morshedy. Oscillation criteria of comparison type for second order difference equations. Journal of Applied Analysis, 6(1):87–102, 2000. https://doi.org/10.1515/JAA.2000.87
M. Kandemir and O.Sh. Mukhtarov. Nonlocal Sturm-Liouville problems with integral terms in the boundary conditions. Electronic Journal of Differential Equations, 2017(11):1–12, 2017.
A. Kneser. Belastete integralgleichungen. Rendiconti del Circolo Matematico di Palermo (1884-1940), 37(1):169–197, 1914. https://doi.org/10.1007/BF03014816
K. Kreith. Oscillation theory. Lecture Notes in Mathematics, 7(314):301, 1973. https://doi.org/10.1007/BFb0067537
R. E. Langer. A problem in diffusion or in the flow of heat for a solid in contact with a fluid. Tohoku Mathematical Journal, First Series, 35:260–275, 1932.
G.W. Morgan. Some remarks on a class of eigenvalue problems with special boundary conditions. Quarterly of Applied Mathematics, 11(2):157–165, 1953. https://doi.org/10.1090/qam/54132
O.Sh. Muhtarov and K. Aydemir. The eigenvalue problem with interaction conditions at one interior singular point. Filomat, 31(17):5411–5420, 2017. https://doi.org/10.2298/FIL1717411M
H. Olğar, O. Sh. Mukhtarov and K. Aydemir. Some properties of eigenvalues and generalized eigenvectors of one boundary value problem. Filomat, 32(3):911–920, 2018. https://doi.org/10.2298/FIL1803911O
R.L. Peek. Solution to a problem in diffusion employing a nonorthogonal sine series. Annals of Mathematics, 30(1-4):265–269, 1928. https://doi.org/10.2307/1968278
S.D. Poisson. Mémoire sur la Manière d’exprimer les Fonctions par des Séries de quantités périodiques, et sur l’Usage de cette Transformation dans la Resolution de differens Problèmes. J. Ecole Polytechnique, 1820.
L. Rayleigh. JWS, The Theory of Sound, vol. 1. Macmillan, London (reprinted Dover, New York, 1945), 1894.
E. S¸en. Computation of eigenvalues and eigenfunctions of a Schrödinger-type boundary-value-transmission problem with retarded argument. Mathematical Methods in the Applied Sciences, 41(16):6604–6610, 2018. https://doi.org/10.1002/mma.5178
E. S¸en and O.Sh. Mukhtarov. Spectral properties of discontinuous Sturm–Liouville problems with a finite number of transmission conditions. Mediterranean Journal of Mathematics, 13(1):153–170, 2016. https://doi.org/10.1007/s00009-014-0487-x
C. Sturm. Mémoire sur les équations différentielles linéaires du second ordre. Journal de Mathématiques Pures et Appliquées, 1:106–186, 1836.
C.A. Swanson. Comparison and Oscillation Theory of Linear Differential Equations. Elsevier, 2000.
N. Yoshida. Oscillation theory of partial differential equations. World Scientific, 2008. https://doi.org/10.1142/7046
Y. Zhang, X. Zhang and Z. Wang. Oscillation problem of left-definite Sturm– Liouville problems with coupled BCs. Applied Mathematics and Computation, 219(18):9709–9716, 2013. https://doi.org/10.1016/j.amc.2013.03.054
B.P. Allahverdiev and H. Tuna. Eigenfunction expansion for singular SturmLiouville problems with transmission conditions. Electronic Journal of Differential Equations, 2019(3):1–10, 2019.
R.S. Anderssen and J.R. Cleary. Asymptotic structure in torsional free oscillations of the earthI. Overtone structure. Geophysical Journal International, 39(2):241–268, 1974. https://doi.org/10.1111/j.1365-246X.1974.tb05453.x
K. Aydemir and O.Sh. Mukhtarov. Completeness of one two-interval boundary value problem with transmission conditions. Miskolc Mathematical Notes, 15(2):293–303, 2014. https://doi.org/10.18514/MMN.2014.1229
K. Aydemir and O.Sh. Mukhtarov. Class of Sturm–Liouville problems with eigenparameter dependent transmission conditions. Numerical Functional Analysis and Optimization, 38(10):1260–1275, 2017. https://doi.org/10.1080/01630563.2017.1316995
K. Aydemir, H. Olğar, O.Sh. Muhtarov and F. Muhtarov. Differential operator equations with interface conditions in modified direct sum spaces. Filomat, 32(3):921–931, 2018. https://doi.org/10.2298/FIL1803921A
E. Bairamov and E. Ugurlu. On the characteristic values of the real component of a dissipative boundary value transmission problem. Applied Mathematics and Computation, 218(19):9657–9663, 2012. https://doi.org/10.1016/j.amc.2012.02.079
W.F. Bauer. Modified Sturm-Liouville systems. Quarterly of Applied Mathematics, 11(3):273–283, 1953. https://doi.org/10.1090/qam/64970
M. Bayramoglu, A. Bayramov and E. Sen. A regularized trace formula for a discontinuous Sturm-Liouville operator with delayed argument. Electronic Journal of Differential Equations, 2017(104):1–12, 2017.
P.A. Binding, H. Langer and M. Möller. Oscillation results for Sturm–Liouville problems with an indefinite weight function. Journal of computational and applied mathematics, 171(1):93–101, 2004.
J.R. Cannon and G.H. Meyer. On diffusion in a fractured medium. SIAM Journal on Applied Mathematics, 20(3):434–448, 1971. https://doi.org/10.1137/0120047
J.M.C. Duhamel. Mémoire sur les vibrations d’une corde flexible, chargée d’un ou de plusieurs curseurs. J. de lÉtcole Polytechnique, 1843.
A. Ergün and R. Amirov. Half inverse problem for diffusion operators with jump conditions dependent on the spectral parameter. Numerical Methods for Partial Differential Equations, 2020. https://doi.org/10.1002/num.22666
C.T. Fulton. Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions. Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 77(3-4):293–308, 1977. https://doi.org/10.1017/S030821050002521X
R.E. Gaskell. A problem in heat conduction and an expansion theorem. American Journal of Mathematics, 64(1):447–455, 1942. https://doi.org/10.2307/2371696
S.R. Grace and H.A. El-Morshedy. Oscillation criteria of comparison type for second order difference equations. Journal of Applied Analysis, 6(1):87–102, 2000. https://doi.org/10.1515/JAA.2000.87
M. Kandemir and O.Sh. Mukhtarov. Nonlocal Sturm-Liouville problems with integral terms in the boundary conditions. Electronic Journal of Differential Equations, 2017(11):1–12, 2017.
A. Kneser. Belastete integralgleichungen. Rendiconti del Circolo Matematico di Palermo (1884-1940), 37(1):169–197, 1914. https://doi.org/10.1007/BF03014816
K. Kreith. Oscillation theory. Lecture Notes in Mathematics, 7(314):301, 1973. https://doi.org/10.1007/BFb0067537
R. E. Langer. A problem in diffusion or in the flow of heat for a solid in contact with a fluid. Tohoku Mathematical Journal, First Series, 35:260–275, 1932.
G.W. Morgan. Some remarks on a class of eigenvalue problems with special boundary conditions. Quarterly of Applied Mathematics, 11(2):157–165, 1953. https://doi.org/10.1090/qam/54132
O.Sh. Muhtarov and K. Aydemir. The eigenvalue problem with interaction conditions at one interior singular point. Filomat, 31(17):5411–5420, 2017. https://doi.org/10.2298/FIL1717411M
H. Olğar, O. Sh. Mukhtarov and K. Aydemir. Some properties of eigenvalues and generalized eigenvectors of one boundary value problem. Filomat, 32(3):911–920, 2018. https://doi.org/10.2298/FIL1803911O
R.L. Peek. Solution to a problem in diffusion employing a nonorthogonal sine series. Annals of Mathematics, 30(1-4):265–269, 1928. https://doi.org/10.2307/1968278
S.D. Poisson. Mémoire sur la Manière d’exprimer les Fonctions par des Séries de quantités périodiques, et sur l’Usage de cette Transformation dans la Resolution de differens Problèmes. J. Ecole Polytechnique, 1820.
L. Rayleigh. JWS, The Theory of Sound, vol. 1. Macmillan, London (reprinted Dover, New York, 1945), 1894.
E. S¸en. Computation of eigenvalues and eigenfunctions of a Schrödinger-type boundary-value-transmission problem with retarded argument. Mathematical Methods in the Applied Sciences, 41(16):6604–6610, 2018. https://doi.org/10.1002/mma.5178
E. S¸en and O.Sh. Mukhtarov. Spectral properties of discontinuous Sturm–Liouville problems with a finite number of transmission conditions. Mediterranean Journal of Mathematics, 13(1):153–170, 2016. https://doi.org/10.1007/s00009-014-0487-x
C. Sturm. Mémoire sur les équations différentielles linéaires du second ordre. Journal de Mathématiques Pures et Appliquées, 1:106–186, 1836.
C.A. Swanson. Comparison and Oscillation Theory of Linear Differential Equations. Elsevier, 2000.
N. Yoshida. Oscillation theory of partial differential equations. World Scientific, 2008. https://doi.org/10.1142/7046
Y. Zhang, X. Zhang and Z. Wang. Oscillation problem of left-definite Sturm– Liouville problems with coupled BCs. Applied Mathematics and Computation, 219(18):9709–9716, 2013. https://doi.org/10.1016/j.amc.2013.03.054