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Existence and Uniqueness of Solutions in Inverse Sturm-Liouville Problems

Year 2023, Volume: 2023 Issue: 19, 19 - 26, 03.01.2024

Münevver Tuz

Abstract

In this article, we examined a boundary value problem for the Sturm-Liouville equation defined in the interval [0, L]. The problem with [0, L] corresponds to the small vibrations of a fixed-end straight rope. In these problems, the necessary and sufficient conditions for the unique determination of the potential by only one spectrum at certain parameters of the boundary conditions are investigated. For the inverse problem, it has been examined that the spectrum is effective in describing the potential of the Sturm-Liouville problem alone, thus the intensity of the array. Also, the uniqueness results of this problem are proved using the Leray-Schauder fixed point theorem in a Banach space. Thus, with a different method, an existence and uniqueness result was created for the problem with these boundary conditions.

Keywords

Sturm-Liouville problem boundary conditions Leray–Schauder eigenvalue spectrum.

Supporting Institution

ICEANS 2022 konferans (Kasım)

Project Number

no

Thanks

I would like to express my gratitude to all the referees and editors for their valuable comments and suggestions that improve the quality of this article.

References

  • [1] O'Regan D (1987). Topological transversality: Application to third-order boundary value problem, SIAM Journal on Mathematical Analysis, 19, 630-641.
  • [2] Cabada A (1994). The method of lower and upper solutions for second, third, fourth and higher order boundary value problems, Journal of Mathematical Analysis and Applications, 185, 302-320.
  • [3] Ma R (1998). Multiplicity results for a third order boundary value problem at resonance, Nonlinear Analysis, 32, 493-500.
  • [4] Marchenko VA (1950). Concerning the theory of a differential operator of the second order, Doklady Akademii nauk SSSR, 72, 457-460.
  • [5] Carlson R (1994). An inverse spectral problem for Sturm–Liouville operators with discontinuous coefficients, Proceedings of the American Mathematical Society, 120(2), 475–484.
  • [6] Levitan BM (1955). On the determination of a differential equation from its spectral function, American Mathematical Society Translations Series 2, 1, 253-304.
  • [7] Gasymov GM, Levitan BM (1968). On Sturm-Liouville differential operators with discrete spectra, American Mathematical Society Translations Series 2, 68, 21-33.
  • [8] Zhornitskaya LA, Serov VS (1994). Inverse eigenvalue problems for a singular Sturm Liouville operator on (0, 1), Inverse Problems, 10(4), 975-987.
  • [9] Borg G (1946). Eine Umkehrung der Sturm-Liouvillesehen Eigenwertaufgabe, Acta Mathematica, 78, 1-96.
  • [10] Hochstadt H(1973). The inverse Sturm-Liouville problem, Communications on Pure and Applied Mathematics, 26, 715-729.
  • [11] Pivovarchik V (2000). Inverse Problem for the Sturm--Liouville Equation on a Simple Graph. SIAM Journal on Mathematical Analysis, 32(4), 801-819.
  • [12] Sadovnichii VA, Sultanaev YT and Akhtyamov AM (2015). General inverse Sturm-Liouville problem with symmetric potential, Azerbaijan Journal of Mathematics, 5(2), 96-108.
  • [13] Buterin SA (2010). On the reconstruction of a convolution perturbation of the Sturm-Liouville operator from the spectrum, Differential Equations, 46(1), 150-154.
  • [14] Volkmer H and Zettl A (2007). Inverse spectral theory for Sturm-Liouville problems with finite spectrum, Proceedings of the American Mathematical Society, 135(4), 1129-1132.
  • [15] Yokuş A, Durur H, Duran S and Islam T (2022). and Ample felicitous wave structures for fractional foam drainage equation modeling for fluid-flow mechanism, Computational and Applied Mathematics, 41(4), 1-13.
  • [16] Yokuş A, Duran S and Durur H (2022). Analysis of wave structures for the coupled Higgs equation modelling in the nuclear structure of an atom, The European Physical Journal Plus, 137(9), 1-17.
  • [17] Durur H,Yokuş A (2021). Discussions on diffraction and the dispersion for traveling wave solutions of the (2+ 1)-dimensional paraxial wave equation, Mathematical Sciences, 16(3), 269-279.
  • [18] Yokuş A, Durur H and Duran S (2021). Simulation and refraction event of complex hyperbolic type solitary wave in plasma and optical fiber for the perturbed Chen-Lee-Liu equation, Optical and Quantum Electronics, 53(7), 1-17.

Year 2023, Volume: 2023 Issue: 19, 19 - 26, 03.01.2024

Münevver Tuz

Abstract

Keywords

Sturm-Liouville problem boundary conditions Leray–Schauder eigenvalue spectrum.

Project Number

no

References

  • [1] O'Regan D (1987). Topological transversality: Application to third-order boundary value problem, SIAM Journal on Mathematical Analysis, 19, 630-641.
  • [2] Cabada A (1994). The method of lower and upper solutions for second, third, fourth and higher order boundary value problems, Journal of Mathematical Analysis and Applications, 185, 302-320.
  • [3] Ma R (1998). Multiplicity results for a third order boundary value problem at resonance, Nonlinear Analysis, 32, 493-500.
  • [4] Marchenko VA (1950). Concerning the theory of a differential operator of the second order, Doklady Akademii nauk SSSR, 72, 457-460.
  • [5] Carlson R (1994). An inverse spectral problem for Sturm–Liouville operators with discontinuous coefficients, Proceedings of the American Mathematical Society, 120(2), 475–484.
  • [6] Levitan BM (1955). On the determination of a differential equation from its spectral function, American Mathematical Society Translations Series 2, 1, 253-304.
  • [7] Gasymov GM, Levitan BM (1968). On Sturm-Liouville differential operators with discrete spectra, American Mathematical Society Translations Series 2, 68, 21-33.
  • [8] Zhornitskaya LA, Serov VS (1994). Inverse eigenvalue problems for a singular Sturm Liouville operator on (0, 1), Inverse Problems, 10(4), 975-987.
  • [9] Borg G (1946). Eine Umkehrung der Sturm-Liouvillesehen Eigenwertaufgabe, Acta Mathematica, 78, 1-96.
  • [10] Hochstadt H(1973). The inverse Sturm-Liouville problem, Communications on Pure and Applied Mathematics, 26, 715-729.
  • [11] Pivovarchik V (2000). Inverse Problem for the Sturm--Liouville Equation on a Simple Graph. SIAM Journal on Mathematical Analysis, 32(4), 801-819.
  • [12] Sadovnichii VA, Sultanaev YT and Akhtyamov AM (2015). General inverse Sturm-Liouville problem with symmetric potential, Azerbaijan Journal of Mathematics, 5(2), 96-108.
  • [13] Buterin SA (2010). On the reconstruction of a convolution perturbation of the Sturm-Liouville operator from the spectrum, Differential Equations, 46(1), 150-154.
  • [14] Volkmer H and Zettl A (2007). Inverse spectral theory for Sturm-Liouville problems with finite spectrum, Proceedings of the American Mathematical Society, 135(4), 1129-1132.
  • [15] Yokuş A, Durur H, Duran S and Islam T (2022). and Ample felicitous wave structures for fractional foam drainage equation modeling for fluid-flow mechanism, Computational and Applied Mathematics, 41(4), 1-13.
  • [16] Yokuş A, Duran S and Durur H (2022). Analysis of wave structures for the coupled Higgs equation modelling in the nuclear structure of an atom, The European Physical Journal Plus, 137(9), 1-17.
  • [17] Durur H,Yokuş A (2021). Discussions on diffraction and the dispersion for traveling wave solutions of the (2+ 1)-dimensional paraxial wave equation, Mathematical Sciences, 16(3), 269-279.
  • [18] Yokuş A, Durur H and Duran S (2021). Simulation and refraction event of complex hyperbolic type solitary wave in plasma and optical fiber for the perturbed Chen-Lee-Liu equation, Optical and Quantum Electronics, 53(7), 1-17.
There are 18 citations in total.

Details

Primary Language English
Subjects Software Testing, Verification and Validation
Journal Section Research Article
Authors

Münevver Tuz 0000-0002-9620-247X

Project Number no
Early Pub Date December 27, 2023
Publication Date January 3, 2024
Submission Date November 25, 2022
Acceptance Date September 25, 2023
Published in Issue Year 2023 Volume: 2023 Issue: 19

Cite

APA Tuz, M. (2024). Existence and Uniqueness of Solutions in Inverse Sturm-Liouville Problems. Journal of New Results in Engineering and Natural Sciences, 2023(19), 19-26.