Algorithm for Numerical Solution of Inverse Spectral Problems Generated by Sturm - Liouville Operators of an Arbitrary Even Order

The article is devoted to the construction of algorithm for solving inverse spectral problems generated by Sturm–Liouville differential operators of an arbitrary even order. The goal of solving inverse spectral problems is to recover operators from their spectral characteristics and spectral characteristics of auxiliary problems. In the scientific literature, we can not find examples of the numerical solution of inverse spectral problems for the Sturm–Liouville operator of higher than the second order. However, their solution is caused by the need to construct mathematical models of many processes arising in science and technology. Therefore, the development of computationally efficient algorithm for the numerical solution of inverse spectral problems generated by the Sturm–Liouville operators of an arbitrary even order is of great scientific interest.
In this article, we use linear formulas obtained earlier in order to find the eigenvalues of discrete semi-bounded operators and develop algorithm for solving inverse spectral problems for Sturm–Liouville operators of an arbitrary even order.
The results of the performed computational experiments show that the use of the algorithm developed in the article makes it possible to recover the values of the potentials in the Sturm–Liouville operators of any necessary even order.

Keywords

eigenvalues and eigenfunctions; discrete, self-adjoint and semi-bounded operators; Galerkin method; ill-posed problems; Fredholm integral equations of the first kind; asymptotic formulas.

References

1. Sadovnichy V.A., Dubrovsky V.V. [Remarks on One New Method for Calculating Eigenvalues and Eigenfunctions of Discrete Operators]. Trudy seminara imeni I.G. Petrovskogo, 1994, issue 17, pp. 244-248. (in Russian)
2. Kadchenko S.I., Kakushkin S.N. Numerical Method for Finding the Eigenvalues and Eigenfunctions of Perturbed Self-Adjoint Operators. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2012, no. 27 (286), pp. 45-57. (in Russian)
3. Kadchenko S.I., Zakirova G.A. A Numerical for Inverse Spectral Problem. 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
4. Dubrovsky V.V., Kadchenko S.I., Kravchenko V.F., Sadovnichy V.A. [Calculation of the First Eigenvalues of the Orr-Sommerfeld Boundary Value Problem Using the Theory of Regularized Traces]. Electromagnetic Waves and Electronic Systems, 1997, vol. 2, no. 6, pp. 13-19. (in Russian)
5. Zakirova G.A., Kadchenko S.I., Kadchenko A.I.,Ryazanova L.S. [Discrete Semi-Bounded Operators and the Galerkin Method]. Vth All-Russia Scientific-Practical Conference ``Mathematical Modeling of Processes and Systems'', Sterlitamak, 2016, pp. 266-272. (in Russian)
6. Kadchenko S.I., Zakirova G.A., Ryazanova L.S., Torshina O.A. Calculation of Eigenvalues with Large Numbers of Spectral Problems by the Modified Galerkin Method. Actual Problems of Modern Science, Technology and Education, 2019, vol. 10, no. 1, pp. 148-152.
7. Kadchenko S.I. Numerical Method for Solving Inverse Problems Generated by Perturbed Self-Adjoint Operators. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2013, vol. 6, no. 4, pp. 15-25. (in Russian)
8. Behiri S.E., Kazaryan A.R., Khachatryan I.G. Asymptotic Formula for Eigenvalues of a Regular Two-Term Differential Operator of Arbitrary Even Order. Scientific Notes of the Yerevan State University. Natural Sciences, 1994, no. 1, pp. 3-18. (in Russian)
9. Naimark М.А. Lineinye differencialnye operatory [Linear Differential Operators]. Moscow, Nauka, 1960. (in Russian)
10. Mikhailov V.P. [On Riesz Bases in L^2(0,1)]. Doklady Mathematics, 1962, vol. 144, no. 5, pp. 981-984. (in Russian)
11. Keselman G.M. On the Unconditional Convergence of Expansions in Eigenfunctions of Some Differential Operators. Russian Mathematics (Izvestiya VUZ. Matematika), 1964, no. 2 (39), pp. 82-93. (in Russian)
12. Tikhonov A.N., Arsenin V.Y. Metody reshenija nekorrektnyh zadach [Methods for Solving Ill-Posed Problems]. Moscow, Nauka, 1979. (in Russian)
13. Tikhonov A.N. [Regularization of Ill-Posed Problems]. Doklady Mathematics, 1963, vol. 153, no. 1, pp. 49-52. (in Russian)
14. Tikhonov A.N. On Ill-Posed Problems in Linear Algebra and a Stable Method for Their Solution. Doklady Mathematics, 1965, vol. 163, no. 6, pp. 591-595. (in Russian)
15. Golikov A.I., Evtushenko Yu.G. Regularization and Normal Solutions of Systems of Linear Equations and Inequalities. Proceedings of the Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, 2014, vol. 20, no. 2, pp. 113-121. (in Russian)
16. Chechkin А.V. Special Regulator A.N. Tikhonov for Integral Equations of the First Kind. Computational Mathematics and Mathematical Physics, 1970, vol. 10, no. 2, pp. 453-461. (in Russian)