Inverse Spectral Problems and Mathematical Models of Continuum Mechanics

The article contains results in the field of spectral problems for mathematical models with discrete semi-bounded operator. The theory is based on linear formulas for calculating the eigenvalues of a discrete operator. The main idea is to reduce spectral problem to the Fredholm integral equation of the first kind. A computationally efficient numerical method for solving inverse spectral problems is developed. The method is based on the Galerkin method for discrete semi-bounded operators. This method allows to reconstruct the coefficient functions of boundary value problems with a high accuracy. The results obtained in the article are applicable to the study of problems for differential operators of any order. The results of a numerical solution of the inverse spectral problem for a fourth-order perturbed differential operator are presented. We study some mathematical models of continuum mechanics based on spectral problems for a discrete semi-bounded operator.

Keywords

inverse spectral problem; discrete operator; fourth order operator; self-adjoint operator; eigenvalues; eigenfunctions; ill-posed problems.

References

1. Akhtyamov A.M., Galeeva D.R. Identification of Length, Density and Elastic Modulus of Corrosion Part of the Rod by Natural Frequencies of Longitudinal Vibrations. Bulletin of Bashkir University, 2015, vol. 20, no. 2, pp. 398-402. (in Russian)
2. Anderssen R.S. The Effect of Discontinuities in Destiny and Shear Velocity on the Asymptotic Overtone Structure of Tortional Eigenfrequencies of the Earth. Astronomical Sciences, 1997, vol. 50, pp. 303-309. DOI: 10.1111/j.1365-246X.1977.tb04175.x
3. Anosova Ye.A., Potetunko E.N., Scherbak Ye.N. Parameters of Physically Non-Homogenous Media Reconstructed from the Eigenfrequencies of Their Free Oscillations. Journal of Engineering Mathematics, 2006, vol. 55, pp. 339-356. DOI: 10.1007/s10665-006-9032-7
4. Anosova E., Herskowitz I., Potetynko E.N., Srubshchik L.S. Assesment of the Efliqciency of the Structure Foundation by the Resonanse Frequencies of its Anti Planar Vibrations. International Congress 'Strures Congress and the Forensic Engineering Symposium', N.Y., 2005, pp. 1-12. DOI: 10.1061/40753(171)119
5. Vibracii v tekhnike: Kolebaniya lineynyh sistem [Vibrations in Technology: Fluctuations of Linear Systems]. Moscow, Mashinostroenie, 1978. (in Russian)
6. Bykov A.A., Dremin I.M., Leonidov A.V. [Potential Models of Quarkonia]. Uspekhi fizicheskih nauk [Physics-Uspekhi], 1984, vol. 143, no. 1, pp. 3-32. DOI: 10.3367/UFNr.0143.198405a.0003 (in Russian)
7. Cherkesov L.V., Potetunko E.N., Shubin D.S. Reconstruction of Ocean Density Distribution from Its Wave Spectrum. International Journal of Fluid Mechanics Research, 2003, vol. 30, no. 1, pp. 11-23. DOI: 10.1615/InterJFluidMechRes.v30.i1.20
8. Dubrovskii V.V., Nagorny A.N. On an Inverse Problem for the Laplace Operator with Continuous Potential. Differential Equations, 1990, vol. 26, no. 9. pp. 1156-1159.
9. Dubrovskii V.V., Nagorny A.N. The Inverse Problem for Degree of Laplace Operator with a Potential from L^2. Differential Equations, 1992, vol. 28, no. 9, pp. 1552-1561.
10. Dubrovskii V.V. On Stability of Inverse Problems of Spectral Analysis for Equations of Mathematical Physics. Russian Mathematical Surveys, 1994, vol. 49, no. 3, pp. 183-184.
11. Dubrovskii V.V. Operator Recovery by Eigenvalues of Various Problems. Russian Mathematical Surveys, 1996, vol. 51, no. 4, pp. 732-733. DOI: 10.1070/RM1996v051n04ABEH002978
12. Dubrovskii V.V. The Existence Theorem in the Inverse Problem of Spectral Analysis. Differential Equations, 1997, vol. 33, no. 12, pp. 1707-1709.
13. Dubrovskii V.V. Obratnye zadachi spektral'nogo analiza dlya nekotoryh differencial'nyh operatorov v chastnyh proizvodnyh [Inverse Spectral Analysis Problems for Some Partial Differential Operators: PhD Thesis]. Magnitogorsk, Magnitogorsk State University, 2006.
14. Egorov A.I. Osnovy teorii upravleniya [Fundamentals of Control Theory]. Moscow, Fizmatlit, 2004. (in Russian)
15. Graham M., Glehdvell L. Obratnye zadachi teorii kolebaniy [Inverse Problems of the Theory of Oscillations]. Izhevsk, RHD, 2008. (in Russian)
16. Favini A., Sviridyuk G.A., Manakova N.A. Linear Sobolev Type Equations with Relatively p-Sectorial Operators in Space of 'Noises'. Abstract and Applied Analysis, 2015, vol. 2015, article ID: 697410, 8 p. DOI: 10.1155/2015/697410
17. Kadchenko S.I., Kakushkin S.N. Chislennye metody regulyarizovannyh sledov spektral'nogo analiza [Numerical Methods of Regularized Traces of Spectral Analysis]. Chelyabinsk, Publishing SUSU, 2015. (in Russian)
18. Kadchenko S.I. Numerical Method for Solving Inverse Spectral Problems Generated by Perturbed Self-Adjoint Operators. Vestnik SamGU. Estestvennonauchnaya seriya, 2013, no. 9, pp. 5-11. (in Russian)
19. Kadchenko S.I., Zakirova G.A. A Numerical Method for Inverse Spectral Problems. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2015, vol. 8, no. 3, pp. 116-126. DOI: 10.14529/mmp150307
20. Kadchenko S.I. Numerical Method for Solving of Inverse Spectral Problems Generated by Perturbed Self-Adjoint Operators. Vestnik Samarskogo Universiteta. Estestvenno-Nauchnaya Seriya, 2013, no. 9, pp. 5-11.
21. Kachalov A., Kurylev Ya., Lassas M. Inverse Boundary Spectral Problems. Florida, CRC Press, 2001. DOI: 10.1201/9781420036220
22. Litvinenko O.N., Soshnikov V.I. Teoriya neodnorodnyh linij i ih primenenie v radiotekhnike [Theory of Inhomogeneous Lines and Their Application in Radio Engineering]. Moscow, Sovetskoe radio, 1964. (in Russian)
23. Meshchanov V.P., Feldshtein A.L. Avtomatizirovannoe proektirovanie napravlennyh otvetvitelej SVCH [Automated Design of Directed Branches of SVCH]. Moscow, Svyaz, 1980. (in Russian)
24. Miropolskii Yu.Z. Dinamika vnutrennih gravitacionnyh voln v okeane [Dynamics of Internal Gravity Waves in the Ocean]. Leningrad, Gidrometeoizdat, 1981. (in Russian)
25. Maximov V.I. Zadacha dinamicheskogo vosstanovleniya vhodov beskonechnomernyh sistem [The Problem of Dynamic Input Restoration of Infinite-Dimensional Systems]. Ekaterinburg, URO RAN, 2000. (in Russian)
26. Polskii N.I. [Some Generalizations of the Galerkin Method]. Doklady Akademii Nauk SSSR, 1952, vol. 46, no. 1, pp. 469-472. (in Russian)
27. Polskii N.I. [On the Convergence of Some Approximate Methods of Analysis]. Ukrainian Mathematical Journal, 1955, vol. 7, no. 1, pp. 56-70. (in Russian)
28. Polskii N.I. [About One General Scheme of Using Approximate Methods]. Doklady Akademii Nauk SSSR, 1956, vol. 111, no. 6, pp. 1181-1183. (in Russian)
29. Potetyunko E.N. [Determining the Density of the Ocean by a Single Frequency and the Corresponding Wave Number in the Problem of Free Oscillations of the Ocean]. Mezhdunarodnyj zhurnal prikladnyh i fundamental'nyh issledovanii, 2011, no. 10, pp. 55-58.
30. Potetyunko E.N. [Inverse Spectral Problems]. Uspekhi sovremennogo estestvoznaniya, 2011, no. 2, pp. 99-104. (in Russian)
31. Potetunko E.N., Scherbak Ye.N. The Inverse Spectral Problem in the Detection of the Defect End Heteroge. Mexico City, Eneititles os the Civil Engeneering, 2005. DOI: 10.1061/40794(179)106
32. Potetyunko E.N., Cherkesov L.V., Shubin D.S., Scherbak E.N. Svobodnye kolebaniya i obratnye spektral'nye zadachi. Volnovye dvizheniya neodnorodnoj zhidkosti [Free Vibrations and Inverse Spectral Problems. Wave Motion Heterogeneous Fluid]. Moscow, Vuzovskaya kniga, 2001. (in Russian)
33. Polyakov A.V. Determination of Atmospheric Gas Composition and Aerosol Characteristics by the Eclipsing Method: D.Sc. Thesis. Saint-Petersburg, 2006.
34. Sadovnchii V.A., Dubrovskii V.V. [A Note on One New Method for Computing Eigenvalues and Eigenfunctions of Discrete Operators]. Trudy Seminara imeni I.G. Petrovskogo, 1994, no. 17, pp. 244-248. (in Russian)
35. Sadovnchii V.A., Dubrovskii V.V., Smirnova L.V. [On the Uniqueness of the Solution of Inverse Problems of Spectral Analysis]. Doklady Akademii Nauk SSSR, 2000, vol. 370, no. 3, pp. 19-321. (in Russian)
36. Sadovnchii V.A., Dubrovskii V.V., Puzankova E.A. The Inverse Problem of Spectral Analysis for a Power of the Laplace Operator in the Rectangle. Differential Equations, 2000, vol. 6, no. 12, pp. 1859-1862.
37. Sharp T.E. Potential-Energy Curves for Molecular Hydrogen and Its Ions. Atomic Data and Nuclear Data Tables, 1971, vol. 2, pp. 119-169. DOI: 10.1016/S0092-640X(70)80007-9
38. Sedov A.I. On the Approximate Solution of the Inverse Problem of Spectral Analysis for the Degree of the Laplace Operator on a Rectangle. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2010, no. 16 (192), issue 5, pp. 73-78. (in Russian)
39. Sedov A.I., Dubrovskii V.V. [The Inverse Problem of Spectral Analysis for a Single Partial Differential Operator with a Non-Nuclear Resolvent]. Elektromagnitnye volny i ehlektronnye sistemy, 2005, vol. 10, no. 1, pp. 4-9.
40. Vo CHong Thak. Numerical Investigation of Models of Wave and Quantum Physics in the Formulation of Inverse Parametric Spectral Problem: PhD. Thesis. Moscow, Moscow State Technological University 'STANKIN', 2013. (in Russian)
41. Valeev N.F. [On One Model for Controlling the Own Oscillations of Dynamic Systems]. Vestnik Ufimskogo gosudarstvennogo aviacionnogo tekhnicheskogo universiteta, 2008, no. 2, pp. 45-46. (in Russian)
42. Valeev N.F., Yumagulov M.G. [Inverse Spectral Problems of the Theory of Identification of Linear Dynamic Systems]. Avtomatika i Telemekhanika, 2009, no. 11, pp. 1776-1782.
43. Yurko V.A. Ob odnoy zadache teorii uprugosti [Introduction to the Theory of Inverse Spectral Problems]. Moscow, Fizmatlit, 2007. (in Russian)
44. Yurko V.A. A Problem in Elasticity Theory. Journal of Applied Mathematics and Mechanics, 1990, vol. 54, no. 6, pp. 820-824. DOI: 10.1016/0021-8928(90)90017-5
45. Zahariev B.N., Kostov N.A., Plekhanov E.B. Exactly Solvable One- and Multi-Channel Models (Lessons on Quantum Intuition). Physics of Elementary Particles and Atomic Nuclei, 1990, vol. 21, issue 4, pp. 914-962. (in Russian)
46. Znamenskaya L.N. Upravlenie uprugimi kolebaniyami [Control of Elastic Vibrations]. Moscow, Fizmatlit, 2004. (in Russian)
47. Zakirova G.A., Sedov A.I. [The Inverse Problem of Spectral Analysis for the Perturbed Degree of the Laplace Operator in the Case of the Neumann Problem on a Parallelepiped]. Vestnik ChelGU. Matematika. Mekhanika. Informatika, 2008, vol. 10, no. 6, pp. 63-68. (in Russian)
48. Zakirova G.A. Approximate Solution of the Inverse Spectral Problem for the Laplace Operator. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2008, vol. 2, no. 27 (127), pp. 19-27. (in Russian)
49. Zakirova G.A., Manakova N., Sviridyuk G. The Asymptotics of Eigenvalues of a Differential Operator in the Stochastic Models with 'White Noise'. Applied Mathematical Sciences, 2014, no. 8, pp. 8747-8754. DOI: 10.12988/ams.2014.49756