- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Recently Accepted Articles ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 361989, 10 pages
Inverse Problems for the Quadratic Pencil of the Sturm-Liouville Equations with Impulse
1Department of Mathematics, Faculty of Sciences, Cumhuriyet University, 58140 Sivas, Turkey
2Department of Secondary Science and Mathematics Education, Faculty of Education, Cumhuriyet University, 58140 Sivas, Turkey
Received 22 November 2012; Revised 2 March 2013; Accepted 4 March 2013
Academic Editor: Juan J. Nieto
Copyright © 2013 Rauf Kh. Amırov and A. Adiloglu Nabıev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- V. A. Marchenko, Sturm-Liouville Operators and Applications, vol. 22 of Operator Theory: Advances and Applications, Birkhäuser, Basel, Switzerlands, 1986.
- B. M. Levitan, Inverse Sturm-Liouville Problems, Nauka, Moscow, Russia, 1987.
- R. Beals, P. Deift, and C. Tomei, Direct and Inverse Scattering on the Line, vol. 28 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, USA, 1988.
- V. A. Yurko, Introduction to the Theory of Inverse Spectral Problems, Fizmatlit, Moscow, Russia, 2007, English translation, Inverse Spectral Problems for Differential Operators and Their Applications, Gordon and Breach, Amsterdam, The Netherlands, 2000.
- B. M. Levitan and M. G. Gasymov, “Determination of a differential equation by two spectra,” Uspekhi Matematicheskikh Nauk, vol. 19, no. 2, pp. 3–63, 1964.
- M. Jaulent and C. Jean, “The inverse problem for the one-dimensional Schrödinger equation with an energy-dependent potential. I, II,” Annales de l'Institut Henri Poincaré A, vol. 25, no. 2, pp. 105–137, 1976.
- M. G. Gasymow, “On the spectral theory of dierential operators polnomially depending on a parameter,” Uspekhi Matematicheskikh Nauk, vol. 37, no. 4, p. 99, 1982 (Russian).
- V. A. Yurko, “An inverse problem for differential operator pencils,” Matematicheskiĭ Sbornik, vol. 191, no. 10, pp. 137–160, 2000, English translation, Sbornik: Mathematics vol. 191, no. 10, pp. 1561–1586, 2000.
- M. G. Gasymov and G. Š. Guseĭnov, “Determination of a diffusion operator from spectral data,” Akademiya Nauk Azerbaĭdzhanskoĭ SSR. Doklady, vol. 37, no. 2, pp. 19–23, 1981 (Russian).
- G. Sh. Guseĭnov, “On the spectral analysis of a quadratic pencil of Sturm-Liouville operators,” Doklady Akademii Nauk SSSR, vol. 285, no. 6, pp. 1292–1296, 1985, English translation, Soviet Mathematics Doklady, vol. 32, no.3, pp. 859–862, 1985.
- G. Sh. Guseĭnov, “Inverse spectral problems for a quadratic pencil of Sturm-Liouville operators on a finite interval,” in Spectral Theory of Operators and Its Applications, pp. 51–101, Èlm, Baku, Azerbaijan, 1986.
- I. M. Nabiev, “Inverse periodic problem for a diffusion operator,” Transactions of Academy of Sciences of Azerbaijan, vol. 23, no. 4, pp. 125–130, 2003.
- I. M. Nabiev, “The inverse spectral problem for the diffusion operator on an interval,” Matematicheskaya Fizika, Analiz, Geometriya, vol. 11, no. 3, pp. 302–313, 2004 (Russian).
- V. N. Pivovarchik, “Reconstruction of the potential of the Sturm-Liouville equation from three spectra of boundary value problems,” Funktsional'nyi Analiz i ego Prilozheniya, vol. 33, no. 3, pp. 87–90, 1999, English translation, Functional Analysis and Its Applications, vol. 33, no. 3, pp. 233–235, 1999.
- V. P. Meshonav and A. L. Feldstein, Automatic Design of Directional Couplers, Sviaz, Moscow, Russia, 1980.
- O. N. Litvinenko and V. I. Soshnikov, The Theory of Heterogeneous Lines and Their Applications in Radio Engineering, Radio, Moscow, Russia, 1964.
- R. J. Krueger, “Inverse problems for nonabsorbing media with discontinuous material properties,” Journal of Mathematical Physics, vol. 23, no. 3, pp. 396–404, 1982.
- D. G. Shepelsky, “The inverse problem of reconstruction of the medium's conductivity in a class of discontinuous and increasing functions,” Advances in Soviet Mathematics, vol. 19, pp. 303–309, 1997.
- F. R. Lapwood and T. Usami, Free Oscillation of the Earth, Cambridge University Press, Cambridge, UK, 1981.
- G. Borg, “Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe,” Acta Mathematica, vol. 78, pp. 1–96, 1946.
- B. M. Levitan and I. S. Sargsyan, Introduction To Spectral Theory, vol. 39 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, USA, 1975.
- J. Pöschel and E. Trubowitz, Inverse Spectral Theory, vol. 130 of Pure and Applied Mathematics, Academic Press, Boston, Mass, USA, 1987.
- V. A. Yurko, Inverse spectral problems for differential operators and their applications, vol. 2 of Analytical Methods and Special Functions, Gordon and Breach, Amsterdam, The Netherlands, 2000.
- G. Freiling and V. Yurko, Inverse Sturm-Liouville problems and their applications, Nova Science, Huntington, NY, USA, 2001.
- J. R. McLaughlin, “Analytical methods for recovering coefficients in differential equations from spectral data,” SIAM Review, vol. 28, no. 1, pp. 53–72, 1986.
- O. H. Hald, “Discontinuous inverse eigenvalue problems,” Communications on Pure and Applied Mathematics, vol. 37, no. 5, pp. 539–577, 1984.
- A. McNabb, R. S. Anderssen, and E. R. Lapwood, “Asymptotic behavior of the eigenvalues of a Sturm-Liouville system with discontinuous coefficients,” Journal of Mathematical Analysis and Applications, vol. 54, no. 3, pp. 741–751, 1976.
- W. W. Symes, “Impedance profile inversion via the first transport equation,” Journal of Mathematical Analysis and Applications, vol. 94, no. 2, pp. 435–453, 1983.
- T. Aktosun, M. Klaus, and C. van der Mee, “Inverse wave scattering with discontinuous wave speed,” Journal of Mathematical Physics, vol. 36, no. 6, pp. 2880–2928, 1995.
- W. Eberhard, G. Freiling, and A. Schneider, “On the distribution of the eigenvalues of a class of indefinite eigenvalue problems,” Differential and Integral Equations, vol. 3, no. 6, pp. 1167–1179, 1990.
- R. Carlson, “An inverse spectral problem for Sturm-Liouville operators with discontinuous coefficients,” Proceedings of the American Mathematical Society, vol. 120, no. 2, pp. 475–484, 1994.
- R. Kh. Amirov, “On Sturm-Liouville operators with discontinuity conditions inside an interval,” Journal of Mathematical Analysis and Applications, vol. 317, no. 1, pp. 163–176, 2006.
- V. A. Yurko, “On higher-order differential operators with a singular point,” Inverse Problems, vol. 9, no. 4, pp. 495–502, 1993.
- V. A. Yurko, “On higher-order differential operators with a regular singularity,” Matematicheskiĭ Sbornik, vol. 186, no. 6, pp. 133–160, 1995, English translation, Sbornik: Mathematics, vol. 186, no. 6, pp. 901–928, 1995.
- V. Yurko, “Integral transforms connected with differential operators having singularities inside the interval,” Integral Transforms and Special Functions, vol. 5, no. 3-4, pp. 309–322, 1997.
- R. Kh. Amirov and V. A. Yurko, “On differential operators with a singularity and discontinuity conditions inside an interval,” Ukrainian Mathematical Journal, vol. 53, no. 11, pp. 1443–1457, 2001.
- G. Freiling and V. Yurko, “Inverse spectral problems for singular non-selfadjoint differential operators with discontinuities in an interior point,” Inverse Problems, vol. 18, no. 3, pp. 757–773, 2002.
- R. Bellman and K. L. Cooke, Differential-Difference Equations, Academic Press, New York, NY, USA, 1963.
- B. Ya. Levin, Entire Functions, MGV, Moscow, Russia, 1971.
- B. F. Jdanovich, “Formulae for the zeros of Drichlet polynomials and quasi-polynomials,” Doklady Akademii Nauk SSSR, vol. 135, no. 8, pp. 1046–1049, 1960.
- M. Kreĭn and B. Ya. Levin, “On entire almost periodic functions of exponential type,” Doklady Akademii Nauk SSSR, vol. 64, pp. 285–287, 1949.