Boundary inverse problem for star-shaped graph with different densities strings-edges
Автор: Akhtyamov A.M., Mamedov Kh.R., Yilmazoglu E.N.
Рубрика: Математическое моделирование
Статья в выпуске: 3 т.11, 2018 года.
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The paper is devoted to the mathematical modelling of star-shaped geometric graphs with n rib-strings of different density and the solution of the boundary inverse spectral problem for Sturm-Liouville differential operators on these graphs. Earlier it was shown that if strings have the same length and densities, then the stiffness coefficients of springs at the ends of graph strings are not uniquely recovered from natural frequencies. They are found up to permutations of their places. We show, that if the strings have different densities, then the stiffness coefficients of springs on the ends of graph strings are uniquely recovered from all natural frequencies. Counterexamples are shown that for the unique recovery of the stiffness coefficients of springs on n dead ends of the graph, it is not enough to use n natural frequencies. Examples are also given showing that it is sufficient to use n+1 natural frequencies for the uniqueness of the recovery of the stiffness coefficients of springs at the n ends of the strings. Those, the uniqueness or non-uniqueness of the restoration of the stiffness coefficients of springs at the ends of the strings depends on whether the string densities are identical or different.
Natural frequencies, star-shaped graph, inverse problems, strings, densities, boundary conditions
Короткий адрес: https://sciup.org/147232899
IDR: 147232899 | DOI: 10.14529/mmp180301
Список литературы Boundary inverse problem for star-shaped graph with different densities strings-edges
- Levitan, B.M. Inverse Sturm -Liouville problems/B.M. Levitan. -Utrecht: VNU Science Press, 1987.
- Marchenko, V.A. Sturm -Liouville Operators and Applications/V.A. Marchenko. -Basel; Boston; Stuttgart: Birkhauser, 1986.
- Naimark, M.A. Linear Differential Operators. Part II. Linear Differential Operators in Hilbert Space/M.A. Naimark. -London, Toronto, Wellington, Sydney: Frederick Ungar Publishing, 1968.
- Levitan, B.M. Determination of a Differential Equation by Two of Its Spectra/B.M. Levitan, M.G. Gasymov//Russian Mathematical Surveys. -1964. -V. 19, № 2. -P. 1-63.
- Гасымов, М.Г. Обратная задача для оператора Штурма -Лиувилля с неразделенными самосопряженными граничными условиями/М.Г. Гасымов, И.М. Гусейнов, И.M. Набиев//Сибирский математический журнал. -1991. -Т. 31, № 6. -С. 46-54.
- Panakhov, E.S. Reconstruction Formula for the Potential Function of Sturm -Liouville Problem with Eigenparameter Boundary Condition/E.S. Panakhov, H. Koyunbakan, Ic. Unal//Inverse Problems in Science and Engineering. -2010. -V. 18, № 1. -P. 173-180.
- Mamedov, Kh.R. A Uniqueness Theorem for a Sturm -Liouville Equation with Spectral Parameter in Boundary Conditions/Kh.R. Mamedov, F.A. Cetinkaya//Applied Mathematics and Information Sciences. -2015. -V. 9, № 2. -P. 981-988.
- Sadovnichii, V.A. General Inverse Sturm -Liouville Problem with Symmetric Potential/V.A. Sadovnichii, Ya.T. Sultanaev, A.M. Akhtyamov//Azerbaijan Journal of Mathematics. -2015. -V. 5, № 2. -P. 96-108.
- Akhtyamov, A.M. Inverse Problem for the Diffusion Operator with Symmetric Functions and General Boundary Conditions/A.M. Akhtyamov, V.A. Sadovnichii, Ya.T. Sultanaev//Eurasian Mathematical Journal. -2017. -V. 8, № 1. -P. 10-22.
- Kadchenko, S.I. A Numerical Method for Solving Inverse Problems Generated by the Perturbed Self-Adjoint Operators/S.I. Kadchenko//Вестник ЮУрГУ. Серия: Математическое моделирование и программирование. -2013. -Т. 6, № 4. -С. 15-25.
- Kadchenko, S.I. A Numerical Method for Inverse Spectral Problems/S.I. Kadchenko, G.A. Zakirova//Вестник ЮУрГУ. Серия: Математическое моделирование и программирование. -2015. -Т. 8, № 3. -С. 116-126.
- Покорный, Ю.В. Дифференциальные уравнения на графах/Ю.В. Покорный, O.В. Пенкин, В.Л. Прядиев, A.В. Боровских, K.П. Лазарев, С.A. Шабров. -M.: Физмалит, 2005.
- Faddeev, M.D. Model of Free Electrons and the Scattering Problem/M.D. Faddeev, B.S. Pavlov//Theoretical and Mathematical Physics. -1983. -V. 55, № 2. -P. 485-492.
- Kottos, T. Quantum Chaos on Graphs/T. Kottos, U. Smilansky//Physical Review Letters. -1997. -V. 79. -P. 4794-4797.
- Langese, J.E. Modelling, Analysis and Control of Dynamic Elastic Multi-Link Structures/J.E. Langese, G. Leugering, J.P. Schmidt. -Boston: Birkhäuser, 1994.
- Pokornyi, Yu.V. Differential Equations on Networks (Geometric Graphs)/Yu.V. Pokornyi, A.V. Borovskikh//Journal of Mathematical Sciences. -2004. -V. 119, № 6. -P. 691-718.
- Pokornyi, Yu.V. The Qualitative Sturm -Liouville Theory on Spatial Networks/Yu.V. Pokornyi, V. Pryadiev//Journal of Mathematical Sciences. -2004. -V. 119, № 6. -P. 788-835.
- Sobolev, A. Schrödinger Operator on Homogeneous Metric Trees: Spectrum in Gaps/A. Sobolev, M. Solomyak//Reviews in Mathematical Physics. -2002. -V. 14, № 5. -P. 421-467.
- Belishev, M.I. Boundary Spectral Inverse Problem on a Class of Graphs (Trees) by the BC Method/M.I. Belishev//Inverse Problems. -2004. -V. 20. -P. 647-672.
- Brown, B.M. A Borg -Levinson Theorem for Trees/B.M. Brown, R. Weikard//Proceedings of the Royal Society. A Mathematical Physical and Engineering Sciences. -2005. -V. 464, № 2062. -P. 3231-3243.
- Свиридюк, Г.A. Устойчивость решений линейных уравнений Осколкова на геометрическом графе/Г.A. Свиридюк, A.С. Шипилов//Вестник Самарского государственного технического университета. Серия: Физико-математические науки. -2009. -Т. 19, № 2. -С. 9-16.
- Свиридюк, Г.A., Устойчивость уравнения Хоффа на графе/Г.A. Свиридюк, С.A. Загребина, П.O. Пивоварова//Вестник Самарского государственного технического университета. Серия: Физико-математические науки. -2010. -Т. 20, № 1. -С. 6-15.
- Ахтямов, A.M. Теория идентификации краевых условий и ее приложений. -M.: Физмалит, 2009.
- Sadovnichii, V.A. Multiparameter Inverse Spectral Problems and Their Applications/V.A. Sadovnichii, Ya.T. Sultanaev, N.F. Valeev//Doklady Mathematics. -2009. -V. 79, № 3. -P. 390-393.
- Мартынова, Ю.В. Модель обратной обратной спектральной задачи для оператора Штурма -Лиувилля на геометрическом графе/Мартынова Ю.В.//Вестник Башкирского университета. -2011. -Т. 16, № 1. -С. 4-10.
- Ахтямов, A.M., Идентификация параметров упругого крепления механической системы от струн/A.M. Ахтямов, З.Ф. Аксенова//Современные проблемы науки и образования. -2015. -№ 1. -URL: www.science-education.com/121-18706.
- Kadchenko, S.I. Spectral Problems on Compact Graphs/S.I. Kadchenko, S.N. Kakushkin, G.A. Zakirova//Вестник ЮУрГУ. Серия: Математическое моделирование и программирование. -2017. -Т. 10, № 3. -С. 156-162.
- Akhtyamov, A.M. Identification of Boundary Conditions at Both Ends of a String from the Natural Vibration Frequencies/A.M. Akhtyamov, I.M. Utyashev//Acoustical Physics. -2015. -V. 61, № 6. -P. 615-622.
