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On GLM type integral equation for singular Sturm-Liouville operator which has discontinuous coefficient

Year 2022, Volume: 71 Issue: 2, 305 - 325, 30.06.2022

Nilüfer Topsakal Rauf Amirov

https://doi.org/10.31801/cfsuasmas.923029

Abstract

In this study, we derive Gelfand-Levitan-Marchenko type main
integral equation of the inverse problem for singular Sturm-Liouville equation
which has discontinuous coefficient. Then we prove the unique solvability of
the main integral equation.

Keywords

Main integral equation GLM equation inverse problem

References

  • Shepelsky, D. G., The inverse problem of reconstruction of the medium’s conductivity in a class of discontinuous and increasing functions, Adv. Soviet Math., 19 (1994), 209-231.
  • Anderssen, R. S., The effect of discontinuities in density and shear velocity on the asypmtotic overtone sturcture of toritonal eigenfrequencies of the Earth, Geophys, J. R. Astr. Soc., 50 (1997), 303-309.
  • Amirov, R. Kh., Topsakal, N., On Sturm-Liouville operators with Coulomb potential which have discontinuity conditions inside an interval, Integral Transforms Spec. Funct., 19(12) (2008), 923-937. http://dx.doi.org/10.1080/10652460802420386
  • Adiloglu, A., Nabiev, Amirov, R. Kh., On the boundary value problem for the Sturm-Liouville equation with the discontinuous coefficients, , Mathematical methods in the Applied Sciences, 36 (2013). http://dx.doi.org/1685-1700.10.1002/mma.2714
  • Akhmedova, E.N., Huseyin, H.M., On inverse problem for the Sturm-Liouville operator with the discontinuous coefficients, Proc. of Saratov University, New ser., Ser.Math., Mech., and Inf., 10(1) (2010), 3-9.
  • Litvinenko, O. N., Soshnikov, V. I., The Theory of Heterogeneous Lines and Their Applications in Radio Engineering, Radio, Moscow (in Russian) 1964.
  • Krueger, R. J., Inverse problems for nonabsorbing media with discontinuous material properties, J. Math. Phys., 23(3) (1982), 396-404.
  • Savchuk, A.M., Shkalikov, A.A., Sturm-Liouville operator with singular potentials, Mathematical Notes, 66(6) (1999), 741-753. https://doi.org/10.1007/BF02674332
  • Savchuk, A.M., Shkalikov, A.A., Trace formula for Sturm-Liouville operator with singular potentials, Mathematical Notes, 69(3) (2001), 427-442. https://doi.org/10.4213/mzm515
  • Savchuk, A.M., On the eigenvalues and eigenfunctions of the Sturm-Liouville operator with a singular potential, Mathematical Notes, 69(2) (2001), 277-285. https://doi.org/10.4213/mzm502
  • Hryniv, R., Mykityuk, Y., Inverse spectral problems for Sturm-Liouville operators with singular potentials, Inverse Problems, 19(3) (2003), 665-684. http://dx.doi.org/10.1088/0266-5611/19/3/312
  • Hryniv, R., Mykityuk, Y., Transformation operators for Sturm-Liouville operators with singular potentials, Math. Phys. Anal. and Geometry, 7(2) (2004), 119-149. http://dx.doi.org/10.1023/B:MPAG.0000024658.58535
  • Hryniv, R., Mykityuk, Y., Eigenvalue asymptotics for Sturm-Liouville operators with singular potentials,arXivpreprint math/0407252.
  • Hryniv, R., Mykityuk, Y., Inverse spectral problems for Sturm-Liouville operators with singular potentials, II. Reconstruction by two spectra, North-Holland Mathematics Studies, 197 (2004), 97-114.
  • Amirov, R. Kh., Topsakal, N., A representation for solutions of Sturm-Liouville equations with Coulomb Potential inside finite interval, Journal of Cumhuriyet University Natural Sciences, 28(2) (2007), 11-38.
  • Topsakal, N., Amirov, R. Kh., Inverse problem for Sturm-Liouville operators with Coulomb potential which have discontinuity conditions inside an interval. Math. Phys. Anal. Geom. 13(1) (2010), 29–46. http://dx.doi.org/10.1007/s11040-009-9066-y
  • Naimark, M. A., Linear Differential Operators, Moscow, Nauka, (in Russian) 1967.
  • Marchenko, V. A., Sturm-Liouville Operators and Their Applications, Naukova Dumka, Kiev, Birkhauser, Basel, 1986.
  • Levitan, B. M., Inverse Sturm-Louville Problems, Nauka, Moscow, 1984. English transl.:VNU Sci. Press, Utrecht, 1987.
  • Yurko, V. A., Inverse Spectral Problems of Differential Operators and Their Applications, Gordon and Breach, New York, 2000.

Year 2022, Volume: 71 Issue: 2, 305 - 325, 30.06.2022

Nilüfer Topsakal Rauf Amirov

https://doi.org/10.31801/cfsuasmas.923029

Abstract

References

  • Shepelsky, D. G., The inverse problem of reconstruction of the medium’s conductivity in a class of discontinuous and increasing functions, Adv. Soviet Math., 19 (1994), 209-231.
  • Anderssen, R. S., The effect of discontinuities in density and shear velocity on the asypmtotic overtone sturcture of toritonal eigenfrequencies of the Earth, Geophys, J. R. Astr. Soc., 50 (1997), 303-309.
  • Amirov, R. Kh., Topsakal, N., On Sturm-Liouville operators with Coulomb potential which have discontinuity conditions inside an interval, Integral Transforms Spec. Funct., 19(12) (2008), 923-937. http://dx.doi.org/10.1080/10652460802420386
  • Adiloglu, A., Nabiev, Amirov, R. Kh., On the boundary value problem for the Sturm-Liouville equation with the discontinuous coefficients, , Mathematical methods in the Applied Sciences, 36 (2013). http://dx.doi.org/1685-1700.10.1002/mma.2714
  • Akhmedova, E.N., Huseyin, H.M., On inverse problem for the Sturm-Liouville operator with the discontinuous coefficients, Proc. of Saratov University, New ser., Ser.Math., Mech., and Inf., 10(1) (2010), 3-9.
  • Litvinenko, O. N., Soshnikov, V. I., The Theory of Heterogeneous Lines and Their Applications in Radio Engineering, Radio, Moscow (in Russian) 1964.
  • Krueger, R. J., Inverse problems for nonabsorbing media with discontinuous material properties, J. Math. Phys., 23(3) (1982), 396-404.
  • Savchuk, A.M., Shkalikov, A.A., Sturm-Liouville operator with singular potentials, Mathematical Notes, 66(6) (1999), 741-753. https://doi.org/10.1007/BF02674332
  • Savchuk, A.M., Shkalikov, A.A., Trace formula for Sturm-Liouville operator with singular potentials, Mathematical Notes, 69(3) (2001), 427-442. https://doi.org/10.4213/mzm515
  • Savchuk, A.M., On the eigenvalues and eigenfunctions of the Sturm-Liouville operator with a singular potential, Mathematical Notes, 69(2) (2001), 277-285. https://doi.org/10.4213/mzm502
  • Hryniv, R., Mykityuk, Y., Inverse spectral problems for Sturm-Liouville operators with singular potentials, Inverse Problems, 19(3) (2003), 665-684. http://dx.doi.org/10.1088/0266-5611/19/3/312
  • Hryniv, R., Mykityuk, Y., Transformation operators for Sturm-Liouville operators with singular potentials, Math. Phys. Anal. and Geometry, 7(2) (2004), 119-149. http://dx.doi.org/10.1023/B:MPAG.0000024658.58535
  • Hryniv, R., Mykityuk, Y., Eigenvalue asymptotics for Sturm-Liouville operators with singular potentials,arXivpreprint math/0407252.
  • Hryniv, R., Mykityuk, Y., Inverse spectral problems for Sturm-Liouville operators with singular potentials, II. Reconstruction by two spectra, North-Holland Mathematics Studies, 197 (2004), 97-114.
  • Amirov, R. Kh., Topsakal, N., A representation for solutions of Sturm-Liouville equations with Coulomb Potential inside finite interval, Journal of Cumhuriyet University Natural Sciences, 28(2) (2007), 11-38.
  • Topsakal, N., Amirov, R. Kh., Inverse problem for Sturm-Liouville operators with Coulomb potential which have discontinuity conditions inside an interval. Math. Phys. Anal. Geom. 13(1) (2010), 29–46. http://dx.doi.org/10.1007/s11040-009-9066-y
  • Naimark, M. A., Linear Differential Operators, Moscow, Nauka, (in Russian) 1967.
  • Marchenko, V. A., Sturm-Liouville Operators and Their Applications, Naukova Dumka, Kiev, Birkhauser, Basel, 1986.
  • Levitan, B. M., Inverse Sturm-Louville Problems, Nauka, Moscow, 1984. English transl.:VNU Sci. Press, Utrecht, 1987.
  • Yurko, V. A., Inverse Spectral Problems of Differential Operators and Their Applications, Gordon and Breach, New York, 2000.
There are 20 citations in total.

Details

Primary Language English
Subjects Applied Mathematics
Journal Section Research Articles
Authors

Nilüfer Topsakal 0000-0001-7216-044X

Rauf Amirov 0000-0001-6754-2283

Publication Date June 30, 2022
Submission Date April 20, 2021
Acceptance Date August 17, 2021
Published in Issue Year 2022 Volume: 71 Issue: 2

Cite

APA Topsakal, N., & Amirov, R. (2022). On GLM type integral equation for singular Sturm-Liouville operator which has discontinuous coefficient. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 71(2), 305-325. https://doi.org/10.31801/cfsuasmas.923029
AMA Topsakal N, Amirov R. On GLM type integral equation for singular Sturm-Liouville operator which has discontinuous coefficient. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. June 2022;71(2):305-325. doi:10.31801/cfsuasmas.923029
Chicago Topsakal, Nilüfer, and Rauf Amirov. “On GLM Type Integral Equation for Singular Sturm-Liouville Operator Which Has Discontinuous Coefficient”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71, no. 2 (June 2022): 305-25. https://doi.org/10.31801/cfsuasmas.923029.
EndNote Topsakal N, Amirov R (June 1, 2022) On GLM type integral equation for singular Sturm-Liouville operator which has discontinuous coefficient. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71 2 305–325.
IEEE N. Topsakal and R. Amirov, “On GLM type integral equation for singular Sturm-Liouville operator which has discontinuous coefficient”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 71, no. 2, pp. 305–325, 2022, doi: 10.31801/cfsuasmas.923029.
ISNAD Topsakal, Nilüfer - Amirov, Rauf. “On GLM Type Integral Equation for Singular Sturm-Liouville Operator Which Has Discontinuous Coefficient”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71/2 (June 2022), 305-325. https://doi.org/10.31801/cfsuasmas.923029.
JAMA Topsakal N, Amirov R. On GLM type integral equation for singular Sturm-Liouville operator which has discontinuous coefficient. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2022;71:305–325.
MLA Topsakal, Nilüfer and Rauf Amirov. “On GLM Type Integral Equation for Singular Sturm-Liouville Operator Which Has Discontinuous Coefficient”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 71, no. 2, 2022, pp. 305-2, doi:10.31801/cfsuasmas.923029.
Vancouver Topsakal N, Amirov R. On GLM type integral equation for singular Sturm-Liouville operator which has discontinuous coefficient. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2022;71(2):305-2.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

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