On GLM type integral equation for singular Sturm-Liouville operator which has discontinuous coefficient
Yıl 2022,
Cilt: 71 Sayı: 2, 305 - 325, 30.06.2022
Nilüfer Topsakal
,
Rauf Amirov
Öz
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.
Kaynakça
- 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.
Yıl 2022,
Cilt: 71 Sayı: 2, 305 - 325, 30.06.2022
Nilüfer Topsakal
,
Rauf Amirov
Kaynakça
- 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.