İç Tekil Noktası Bulunan Bir Periyodik Sturm-Liouville Probleminin Bazı Nitel Özellikleri

Year 2023, Volume: 12 Issue: 3, 236 - 243, 31.12.2023

Ümmügülsüm Esen , Kadriye Aydemir , Oktay Mukhtarov

Abstract

Bu makalede ortak uç noktası olan iki ayrık aralıkta tanımlı olan kendine eşlenik ikinci mertebeden diferansiyel denklemden (Sturm-Liouville denklemi olarak adlandırılan diferansiyel denklemden), periyodik sınır şartlarından ve verilmiş aralıkların ortak uç noktasında verilmiş iki tane ek geçiş şartlarından oluşmuş yeni tip bir sınır değer problemini inceledik. İncelediğimiz sınır değer probleminin bazı spektral özelliklerini ispat ettik. Ayrıca modifiye edilmiş (biçimi değiştirilmiş) Rayleigh oranından yararlanarak esas özdeğer için bir tahmin elde ettik. \gamma=\delta=1 olduğu özel durumda, elde edilen sonuçlar uygun gelen klasik sonuçlara indirgeniyor. Bu nedenle elde edilen sonuçlar klasik sonuçları genelleştiriyor.

Keywords

Sturm-Liouville problemi, periyodik sınır şartı, geçiş şartı, Rayleigh oranı.

Some Qualitative Properties of a Periodic Sturm-Liouville Problem With an Inner Singular Point

Abstract

In this paper we study a new type of boundary value problem consisting of a self-adjoint second-order differential equation (the so-called Sturm-Liouville equation) defined on two non-intersecting intervals with a common end, periodic boundary conditions and two additional transmission conditions specified at the common endpoint of the considered intervals. We proved some spectral properties of the boundary value problem consideration. In particular, we obtained an estimate of the principal eigenvalue using a modified Rayleigh quotient. In the special case where γ=δ=1, the obtained results are reduced to the corresponding classical results, so our results, generalize the classical results

Keywords

Sturm-Liouville problem, periodic boundary condition, transmission condition, Rayleigh quotient

References

• Allahverdiev, B. P., Tuna, H., 2019. Eigenfunction Expansion for Singular Sturm-Liouville Problems with Transmission Conditions, Electronic Journal of Differential Equations, 2019(03), 1-10.
• Ao, J., Sun, J., 2014. Matrix representations of Sturm-Liouville problems with coupled eigenparameter- dependent boundary conditions, Applied Mathematics and Computation 244 (2014) 142-148
• Aydemir, K., Mukhtarov, O. Sh., 2016. Qualitative analysis of eigenvalues and eigenfunctions of one boundary value-transmission problem, Boundary Value Problems, 1-16.
• Aydemir, K., Olğar, H., Mukhtarov, O. Sh., Muhtarov, F., 2018. Differential Operator Equations with Interface Conditions in Modified Direct Sum Spaces, Filomat 32(3), 921-931.
• Cannon, J. R., Meyer, G.H., 1971. On a Diffusion in a Fractured Medium, SIAM J. Appl. Math., 3 (1971), pp. 434-448.
• Edmonds, A. R. 1973. Studies of the quadratic Zeeman effect. I. Application of the sturmian functions. Journal of Physics B: Atomic and Molecular Physics, 6(8), 1603.
• Gesztesy, F., Macdeo, C., Streit, L. 1985. An exactly solvable periodic Schrodinger operator. Journal of Physics A: Mathematical and General, 18(9), L503.
• Huy, H. P., Sánchez-Palencia, E. 1974. Phénomènes de transmission à travers des couches minces de conductivitéélevée. Journal of Mathematical Analysis and Applications, 47(2), 284-309.
• Kong, Q., Wu, H., Zettl, A. Geometric aspects of Sturm-Liouville problems. Preprint.
• Mukhtarov, O. Sh., Aydemir, K., 2015. Eigenfunction expansion for Sturm-Liouville problems with transmission conditions at one interior point. Acta Mathematica Scientia, 35(3), 639-649.3.
• Mukhtarov, O. Sh., Aydemir, K., 2021. Two-linked periodic Sturm-Liouville problems with transmission conditions, Mathematical Methods In The Applied Sciences 44 (18),14664-14676.
• Mukhtarov, O. S., Yücel, M., 2020. A study of the eigenfunctions of the singular Sturm–Liouville problem using the analytical method and the decomposition technique. Mathematics, 8(3), 415.
• Mukhtarov, O. S., Yücel, M., Aydemir, K., 2020. Treatment a new approximation method and its justification for Sturm–Liouville problems. Complexity, 2020, 1-8.
• Sherstyuk, A. I. 1988. Problems of Theoretical Physics. Leningrad. Gos. Univ., Leningrad.
• Şen, E., 2021. Spectrum, Trace and Nodal Points of a Sturm-Liouville Type Delayed Differential Operator with Interface Conditions Rocky Mountain Journal of Mathematics (2021) 51 (1), 283-294.
• Titeux, I., Yakubov, Y. 1997. Completeness of root functions for thermal conduction in a strip with piecewise continuous coefficients. Mathematical Models and Methods in Applied Sciences, 7(07), 1035-1050.
• Ugurlu, E., 2020. On the characteristic values of the real component of a dissipative boundary value transmission problem,Quaestiones Mathematicae 43.4 (2020): 507-521.
• Yücel, M., Muhtarov, F., 2023. Parameterized Differential Transform Method and Its Application to Boundary Value Transmission Problems. Yüzüncü Yıl Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 28(2), 431- 442.
• Yücel, M., Mukhtarov, O. S., Aydemir, K., 2023. Computation of eigenfunctions of nonlinear boundary-value- transmission problems by developing some approximate techniques. Boletim da Sociedade Paranaense de Matemática, 41, 1-12.
• Wang, A., Sun, J., Hao, X., Yao, S. 2009. Completeness of eigenfunctions of Sturm-Liouville problems with transmission conditions.