Traces and inverse nodal problems for a class of delay
Sturm–Liouville operators

Şen, Erdoğan

doi:10.3906/mat-2005-55

Traces and inverse nodal problems for a class of delay Sturm–Liouville operators

Erdoğan ŞEN (Tekirdağ Namık Kemal Üniversitesi, Tekirdağ, Türkiye)

Turkish Journal of Mathematics

1 0

Yıl: 2021 Cilt: 45 Sayı: 1 Sayfa Aralığı: 305 - 318 Metin Dili: İngilizce DOI: 10.3906/mat-2005-55 İndeks Tarihi: 06-07-2022

Traces and inverse nodal problems for a class of delay Sturm–Liouville operators

Öz:

In this study, we investigate the regularized sums of eigenvalues, oscillation of eigenfunctions and solutions of inverse nodal problems of discontinuous Sturm–Liouville operators with a delayed argument and with a finite number of transmission conditions. With this aim, we obtain asymptotic formulas for eigenvalues, eigenfunctions and nodal points of the problem. Moreover, some numerical examples are given to illustrate the results. The problem differs from the other discontinuous Sturm–Liouville problems with retarded argument in that it contains a spectral parameter in boundary conditions. If we take the delayed argument ∆ ≡ 0, the coefficients α + i = β + i = 0 ( i = 1, 2) in boundary conditions and the transmission coefficients δi = 1 ( i = 1, m − 1) the results obtained below coincide with corresponding results in the classical Sturm–-Liouville operator.

Anahtar Kelime:

Belge Türü: Makale Makale Türü: Araştırma Makalesi Erişim Türü: Erişime Açık

[1] Akgün FA, Bayramov A, Bayramoglu M. Discontinuous boundary value problems with retarded argument and eigenparameter dependent boundary conditions. Mediterranean Journal of Mathematics 2013; 10 (1): 277-288.
[2] Aydemir K, Mukhtarov OS. Asymptotic distribution of eigenvalues and eigenfunctions for a multi-point discontinuous Sturm–Liouville problem. Electronic Journal of Differential Equations 2016; 131: 1-14.
[3] Aydemir K, Mukhtarov OS. Class of Sturm–Liouville problems with eigenparameter dependent transmission conditions. Numerical Functional Analysis and Optimization 2017; 38 (10): 1260-1275.
[4] Bayramoglu M, Bayramov A, Şen E. A regularized trace formula for a discontinuous Sturm–Liouville operator with delayed argument. Electronic Journal of Differential Equations 2017; 104: 1-12.
[5] Bayramoğlu M, Köklü KÖ, Baykal O. On the spectral properties of the regular Sturm–Liouville problem with the lag argument for which its boundary conditions depends on the spectral parameter. Turkish Journal of Mathematics 2002; 26: 421-431.
[6] Bayramov A, C̣alıṣkan S, Uslu S. Computation of eigenvalues and eigenfunctions of a discontinuous boundary value problem with retarded argument. Applied Mathematics and Computation 2007; 191: 592-600.
[7] Cheng YH, Law CK. On the quasi-nodal map for the Sturm–Liouville problem. Proceedings of the Royal Society of Edinburgh Section A: Mathematics 2006; 136: 71-86.
[8] Dikii LA. Trace formulas for Sturm–Liouville differential equations. Uspekhi Matematicheskikh Nauk 1958; 12 (3): 111-143.
[9] Gelfand IM, Levitan BM. On a formula for eigenvalues of a differential operator of second order. Doklady Akademii Nauk SSSR 1953; 88: 593-596 (in Russian).
[10] Hald OH, McLaughlin JR. Solutions of inverse nodal problems. Inverse Problems 1989; 5: 307-347.
[11] Hira F. A trace formula for the Sturm–Liouville type equation with retarded argument. Communications Faculty of Sciences University of Ankara Series A1-Mathematics and Statistics 2017; 66 (1): 124-132.
[12] Kokiopoulou E, Chen J, Saad Y. Trace optimization and eigenproblems in dimension reduction methods. Numerical Linear Algebra and Applications 2010; 18 (3): 565-602.
[13] Mukhtarov OS, Kadakal M, Muhtarov FS. On discontinuous Sturm–Liouville problems with transmission conditions. Kyoto Journal of Mathematics 2004; 44 (4): 779-798.
[14] Ngo TT, Bellalij M, Saad Y. The Trace Ratio Optimization Problem. SIAM Review 2012; 54 (3): 545-569.
[15] Norkin SB. Differential equations of the second order with retarded argument. Providence, RI, USA: American Mathematical Society, 1972.
[16] Pikula M. Regularized traces of differential operator of Sturm–Liouville type with retarded argument. Differentsialnye Uravneniya 1990; 26 (1): 103-109 (in Russian); English translation: Differential Equations 1990; 26 (1): 92-96.
[17] Sadovnichii VA, Podolskii VE. Traces of operators. Uspekhi Matematicheskikh Nauk 2006; 61: 89-156 (in Russian); English translation: Russian Mathematical Surveys 2006; 61: 885-953.
[18] Shieh CT, Yurko VA. Inverse nodal and inverse spectral problems for discontinuous boundary value problem. Journal of Mathematical Analysis and Applications 2008; 347: 266-272.
[19] Şen E. A regularized trace formula and oscillation of eigenfunctions of a Sturm–Liouville operator with retarded argument at 2 points of discontinuity. Mathematical Methods in the Applied Sciences 2017; 40 (18): 7051-7061.
[20] Şen E. Spectrum, trace and nodal points of a Sturm–Liouville type delayed differential operator with interface conditions. To appear in: Rocky Mountain Journal of Mathematics 2020.
[21] Şen E, Bayramov A. Calculation of eigenvalues and eigenfunctions of a discontinuous boundary value problem with retarded argument which contains a spectral parameter in the boundary condition. Mathematical and Computer Modelling 2011; 54 (11-12): 3090-3097.
[22] Wang YP, Huang ZY, Yang CF. Reconstruction for the spherically symmetric speed of sound from nodal data. Inverse Problems in Science and Engineering 2013; 21: 1032-1046.
[23] Wang YP, Yurko VA. On the inverse nodal problems for discontinuous Sturm–Liouville operators. Journal of Differential Equations 2016; 260: 4086-4109.
[24] Yang CF. Trace and inverse problem of a discontinuous Sturm–Liouville operator with retarded argument. Journal of Mathematical Analysis and Applications 2012; 395: 30-41.
[25] Yang CF. Inverse nodal problems of discontinuous Sturm–Liouville operator. Journal of Differential Equations 2013; 254: 1992-2014.
[26] Yurko VA. Inverse Spectral Problems for Differential Operators and Their Applications. Amsterdam, Netherlands: Gordon and Breach, 2000.
[27] Yurko VA. Inverse nodal problems for Sturm–Liouville operators on star-type graphs. Journal of Inverse and IllPosed Problems 2008; 16: 715-722.

APA	Şen E (2021). Traces and inverse nodal problems for a class of delay Sturm–Liouville operators. , 305 - 318. 10.3906/mat-2005-55
Chicago	Şen Erdoğan Traces and inverse nodal problems for a class of delay Sturm–Liouville operators. (2021): 305 - 318. 10.3906/mat-2005-55
MLA	Şen Erdoğan Traces and inverse nodal problems for a class of delay Sturm–Liouville operators. , 2021, ss.305 - 318. 10.3906/mat-2005-55
AMA	Şen E Traces and inverse nodal problems for a class of delay Sturm–Liouville operators. . 2021; 305 - 318. 10.3906/mat-2005-55
Vancouver	Şen E Traces and inverse nodal problems for a class of delay Sturm–Liouville operators. . 2021; 305 - 318. 10.3906/mat-2005-55
IEEE	Şen E "Traces and inverse nodal problems for a class of delay Sturm–Liouville operators." , ss.305 - 318, 2021. 10.3906/mat-2005-55
ISNAD	Şen, Erdoğan. "Traces and inverse nodal problems for a class of delay Sturm–Liouville operators". (2021), 305-318. https://doi.org/10.3906/mat-2005-55

APA	Şen E (2021). Traces and inverse nodal problems for a class of delay Sturm–Liouville operators. Turkish Journal of Mathematics, 45(1), 305 - 318. 10.3906/mat-2005-55
Chicago	Şen Erdoğan Traces and inverse nodal problems for a class of delay Sturm–Liouville operators. Turkish Journal of Mathematics 45, no.1 (2021): 305 - 318. 10.3906/mat-2005-55
MLA	Şen Erdoğan Traces and inverse nodal problems for a class of delay Sturm–Liouville operators. Turkish Journal of Mathematics, vol.45, no.1, 2021, ss.305 - 318. 10.3906/mat-2005-55
AMA	Şen E Traces and inverse nodal problems for a class of delay Sturm–Liouville operators. Turkish Journal of Mathematics. 2021; 45(1): 305 - 318. 10.3906/mat-2005-55
Vancouver	Şen E Traces and inverse nodal problems for a class of delay Sturm–Liouville operators. Turkish Journal of Mathematics. 2021; 45(1): 305 - 318. 10.3906/mat-2005-55
IEEE	Şen E "Traces and inverse nodal problems for a class of delay Sturm–Liouville operators." Turkish Journal of Mathematics, 45, ss.305 - 318, 2021. 10.3906/mat-2005-55
ISNAD	Şen, Erdoğan. "Traces and inverse nodal problems for a class of delay Sturm–Liouville operators". Turkish Journal of Mathematics 45/1 (2021), 305-318. https://doi.org/10.3906/mat-2005-55