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Yıl 2021, Cilt: 70 Sayı: 1, 341 - 356, 30.06.2021

Emrah Yılmaz Hikmet Kemaloğlu

https://doi.org/10.31801/cfsuasmas.733215

Öz

Kaynakça

  • Law, C. K., Tsay, J. On the well-posedness of the inverse nodal problem, Inverse Problems, 17(5) (2001), 1493-1512.
  • Cheng, Y. H., Law, C. K., The inverse nodal problem for Hill's equation, Inverse Problems, 22(3) (2006), 891-901.
  • Ambarzumyan, V. A., Über eine Frage der Eigenwerttheorie, Zeitschrift für Physik, 53 (1929), 690-695.
  • Levitan, B. M., Sargsyan, I. S., Introduction to spectral theory: self adjoint ordinary differential operators, American Mathematical Society, Providence, Rhode Island, 1975.
  • McLaughlin, J. R., Analytical methods for recovering coefficients in differential equations from spectral data, SIAM, 28(1) (1986), 53-72.
  • Pöschel, J., Trubowitz, E., Inverse spectral theory, Academic Press, Orlando, 1987.
  • Pivovarchik, V., Direct and inverse three-point Sturm-Liouville problem with parameter-dependent boundary conditions, Asymptotic Analysis, 26(3) (2001), 219-238.
  • Shieh, C. T., Buterin, S. A., Ignatiev, M., On Hochstadt-Lieberman theorem for Sturm-Liouville operators, Far East Journal of Applied Mathematics, 52(2) (2011), 131-146.
  • Rundell, W., Sacks, P. E., The reconstruction of Sturm-Liouville operators, Inverse Problems, 8(3) (1992), 457-482.
  • Hryniv, R., Pronska, N., Inverse spectral problems for energy-dependent Sturm-Liouville equations, Inverse Problems, 28(8) (2012), 085008.
  • McLaughlin, J. R., Inverse spectral theory using nodal points as data a uniqueness result, Journal of Differential Equations, 73(2) (1988), 354-362.
  • Hald, O. H., McLaughlin, J. R., Solution of inverse nodal problems, Inverse Problems, 5(3) (1989), 307-347.
  • Shen, C. L., On the nodal sets of the eigenfunctions of the string equation, SIAM Journal on Mathematical Analysis, 19 (1988), 1419-1424.
  • Browne, P. J., Sleeman, B. D., Inverse nodal problems for Sturm-Liouville equations with eigenparameter dependent boundary conditions, Inverse Problems, 12(4) (1996), 377-381.
  • Kuryshova, Y. V., Shieh, C. T., An inverse nodal problem for integro-differential operators, Journal of Inverse and III-posed Problems, 18(4) (2010), 357-369.
  • Yilmaz, E., Koyunbakan, H., Reconstruction of potential function and its derivatives for Sturm-Liouville problem with eigenvalues in boundary condition, Inverse Problems in Science and Engineering, 18(7) (2010), 935-944.
  • Guo, Y., Wei, G., Inverse nodal problem for Dirac equations with boundary conditions polynomially dependent on the spectral parameter, Results in Mathematics, 67(1-2) (2015), 95-110.
  • Law, C. K., Yang, C. F., Reconstructing the potential function and its derivatives using nodal data, Inverse Problems, 14(2) (1998), 299-312.
  • Panakhov, E. S., Yilmaz, E., Koyunbakan, H., Inverse nodal problem for Dirac operator, World Applied Sciences Journal, 11(8) (2010), 906-911.
  • Moses, H. E., Calculation of the scattering potential from refection coefficients, Physical Review, 102(2) (1956), 559-567.
  • Prats, F., Toll, J. S., Construction of the Dirac equation central potential from phase shifts and bound states, Physical Review, 113(1) (1959), 363-370.
  • Verde, M., The inversion problem in wave mechanics and dispersion relations, Nuclear Physics, 9 (1958), 255-266.
  • Gasymov, M. G., Levitan, B. M., The inverse problem for the Dirac System, Doklady Akademi NAauk SSSR, 167 (1966), 967-970.
  • Panakhov, E. S., The defining of Dirac system in two incompletely set collection of eigenvalues, Doklady Akademi AzSSR, 5 (1985), 8-12.
  • Gasymov, M. G. Dzhabiev, T. T., Determination of a system of Dirac differential equations using two spectra, Transactions of the summer school on spectral Theory of Operators, (1975), 46-71.
  • Joa, I., Minkin, A., Eigenfunction estimate for a Dirac operator, Acta Mathematica Hungarica, 76(4) (1997), 337-349.
  • Watson, B. A., Inverse spectral problems for weighted Dirac systems, Inverse Problems, 15(3) (1999), 793-805.
  • Kerimov, N. B., A boundary value problem for the Dirac system with a spectral parameter in the boundary conditions, Differential Equations, 38(2) (2002), 164-174.
  • Annaby, M. H., Tharwat, M. M., On sampling and Dirac systems with eigenparameter in the boundary conditions, Journal of Applied Mathematics and Computing, 36(1-2) (2011), 291-317.
  • Amirov, R. Kh., Keskin, B., Özkan, A. S., Direct and inverse problems for the Dirac operator with a spectral parameter linearly contained in a boundary condition, Ukrainian Mathematical Journal, 61(9) (2009), 1365-1379.
  • Albeverio, S., Hryniv, R. O., Mykytyuk, Ya. V., Inverse spectral problems for Dirac operators with summable potentials, Russian Journal of Mathematical Physics, 12(4) (2005), 406-423.
  • Hryniv, R. O., Mykytyuk, Ya. V., On zeros of some entire functions, Transactions of the American Mathematical Society, 361(4) (2009), 2207-2223.
  • Mykytyuk, Ya. V., Puyda, D. V., Inverse spectral problems for Dirac operators on a finite interval, Journal of Mathematical Analysis and Applications, 386(1) (2012), 177-194.
  • Thaller, B., The Dirac Equation, Springer, Berlin, 1992.
  • Puyda, D. V., Inverse spectral problems for Dirac operators with summable matrix-valued potentials, Integral Equations and Operator Theory, 74(3) (2012), 417-450.
  • Yang, C. F., Huang, Z. Y., Reconstruction of the Dirac operator from nodal data, Integral Equations and Operator Theory, 66(4) (2010), 539-551.
  • Gulsen, T., Yilmaz, E., Koyunbakan, H., Inverse nodal problem for p-laplacian Dirac system, Mathematical Methods in the Applied Sciences, 40(2017), 2329-2335.
  • Yang, C. F., Pivovarchik, V. N., Inverse nodal problem for Dirac system with spectral parameter in boundary conditions, Complex Analysis and Operator Theory, 7(4) (2013), 1211-1230.
  • Marchenko, V. A., Maslov, K. V., Stability of the problem of recovering the Sturm-Liouville operator from the spectral function, Mathematics of the USSR Sbornik, 81(4) (1970), 525-551.
  • McLaughlin, J. R., Stability theorems for two inverse spectral problems, Inverse Problems, 4(2) (1988), 529-540.

On the Lipschitz stability of inverse nodal problem for Dirac system

Yıl 2021, Cilt: 70 Sayı: 1, 341 - 356, 30.06.2021

Emrah Yılmaz Hikmet Kemaloğlu

https://doi.org/10.31801/cfsuasmas.733215

Öz

Inverse nodal problem on Dirac operator is determination problem of the parameters in the boundary conditions, number m and potential function V by using a set of nodal points of a component of two component vector eigenfunctions as the given spectral data. In this study, we solve a stability problem using nodal set of vector eigenfunctions and show that the space of all V functions is homeomorphic to the partition set of all space of asymptotically equivalent nodal sequences induced by an equivalence relation. Moreover, we give a reconstruction formula for the potential function as a limit of a sequence of functions and associated nodal data of one component of vector eigenfunction. Our technique depends on the explicit asymptotic expressions of the nodal parameters and, it is basically similar to [1, 2] which is given for Sturm-Liouville and Hill's operators, respectively.

Anahtar Kelimeler

Dirac System inverse nodal problem Lipschitz stability

Kaynakça

  • Law, C. K., Tsay, J. On the well-posedness of the inverse nodal problem, Inverse Problems, 17(5) (2001), 1493-1512.
  • Cheng, Y. H., Law, C. K., The inverse nodal problem for Hill's equation, Inverse Problems, 22(3) (2006), 891-901.
  • Ambarzumyan, V. A., Über eine Frage der Eigenwerttheorie, Zeitschrift für Physik, 53 (1929), 690-695.
  • Levitan, B. M., Sargsyan, I. S., Introduction to spectral theory: self adjoint ordinary differential operators, American Mathematical Society, Providence, Rhode Island, 1975.
  • McLaughlin, J. R., Analytical methods for recovering coefficients in differential equations from spectral data, SIAM, 28(1) (1986), 53-72.
  • Pöschel, J., Trubowitz, E., Inverse spectral theory, Academic Press, Orlando, 1987.
  • Pivovarchik, V., Direct and inverse three-point Sturm-Liouville problem with parameter-dependent boundary conditions, Asymptotic Analysis, 26(3) (2001), 219-238.
  • Shieh, C. T., Buterin, S. A., Ignatiev, M., On Hochstadt-Lieberman theorem for Sturm-Liouville operators, Far East Journal of Applied Mathematics, 52(2) (2011), 131-146.
  • Rundell, W., Sacks, P. E., The reconstruction of Sturm-Liouville operators, Inverse Problems, 8(3) (1992), 457-482.
  • Hryniv, R., Pronska, N., Inverse spectral problems for energy-dependent Sturm-Liouville equations, Inverse Problems, 28(8) (2012), 085008.
  • McLaughlin, J. R., Inverse spectral theory using nodal points as data a uniqueness result, Journal of Differential Equations, 73(2) (1988), 354-362.
  • Hald, O. H., McLaughlin, J. R., Solution of inverse nodal problems, Inverse Problems, 5(3) (1989), 307-347.
  • Shen, C. L., On the nodal sets of the eigenfunctions of the string equation, SIAM Journal on Mathematical Analysis, 19 (1988), 1419-1424.
  • Browne, P. J., Sleeman, B. D., Inverse nodal problems for Sturm-Liouville equations with eigenparameter dependent boundary conditions, Inverse Problems, 12(4) (1996), 377-381.
  • Kuryshova, Y. V., Shieh, C. T., An inverse nodal problem for integro-differential operators, Journal of Inverse and III-posed Problems, 18(4) (2010), 357-369.
  • Yilmaz, E., Koyunbakan, H., Reconstruction of potential function and its derivatives for Sturm-Liouville problem with eigenvalues in boundary condition, Inverse Problems in Science and Engineering, 18(7) (2010), 935-944.
  • Guo, Y., Wei, G., Inverse nodal problem for Dirac equations with boundary conditions polynomially dependent on the spectral parameter, Results in Mathematics, 67(1-2) (2015), 95-110.
  • Law, C. K., Yang, C. F., Reconstructing the potential function and its derivatives using nodal data, Inverse Problems, 14(2) (1998), 299-312.
  • Panakhov, E. S., Yilmaz, E., Koyunbakan, H., Inverse nodal problem for Dirac operator, World Applied Sciences Journal, 11(8) (2010), 906-911.
  • Moses, H. E., Calculation of the scattering potential from refection coefficients, Physical Review, 102(2) (1956), 559-567.
  • Prats, F., Toll, J. S., Construction of the Dirac equation central potential from phase shifts and bound states, Physical Review, 113(1) (1959), 363-370.
  • Verde, M., The inversion problem in wave mechanics and dispersion relations, Nuclear Physics, 9 (1958), 255-266.
  • Gasymov, M. G., Levitan, B. M., The inverse problem for the Dirac System, Doklady Akademi NAauk SSSR, 167 (1966), 967-970.
  • Panakhov, E. S., The defining of Dirac system in two incompletely set collection of eigenvalues, Doklady Akademi AzSSR, 5 (1985), 8-12.
  • Gasymov, M. G. Dzhabiev, T. T., Determination of a system of Dirac differential equations using two spectra, Transactions of the summer school on spectral Theory of Operators, (1975), 46-71.
  • Joa, I., Minkin, A., Eigenfunction estimate for a Dirac operator, Acta Mathematica Hungarica, 76(4) (1997), 337-349.
  • Watson, B. A., Inverse spectral problems for weighted Dirac systems, Inverse Problems, 15(3) (1999), 793-805.
  • Kerimov, N. B., A boundary value problem for the Dirac system with a spectral parameter in the boundary conditions, Differential Equations, 38(2) (2002), 164-174.
  • Annaby, M. H., Tharwat, M. M., On sampling and Dirac systems with eigenparameter in the boundary conditions, Journal of Applied Mathematics and Computing, 36(1-2) (2011), 291-317.
  • Amirov, R. Kh., Keskin, B., Özkan, A. S., Direct and inverse problems for the Dirac operator with a spectral parameter linearly contained in a boundary condition, Ukrainian Mathematical Journal, 61(9) (2009), 1365-1379.
  • Albeverio, S., Hryniv, R. O., Mykytyuk, Ya. V., Inverse spectral problems for Dirac operators with summable potentials, Russian Journal of Mathematical Physics, 12(4) (2005), 406-423.
  • Hryniv, R. O., Mykytyuk, Ya. V., On zeros of some entire functions, Transactions of the American Mathematical Society, 361(4) (2009), 2207-2223.
  • Mykytyuk, Ya. V., Puyda, D. V., Inverse spectral problems for Dirac operators on a finite interval, Journal of Mathematical Analysis and Applications, 386(1) (2012), 177-194.
  • Thaller, B., The Dirac Equation, Springer, Berlin, 1992.
  • Puyda, D. V., Inverse spectral problems for Dirac operators with summable matrix-valued potentials, Integral Equations and Operator Theory, 74(3) (2012), 417-450.
  • Yang, C. F., Huang, Z. Y., Reconstruction of the Dirac operator from nodal data, Integral Equations and Operator Theory, 66(4) (2010), 539-551.
  • Gulsen, T., Yilmaz, E., Koyunbakan, H., Inverse nodal problem for p-laplacian Dirac system, Mathematical Methods in the Applied Sciences, 40(2017), 2329-2335.
  • Yang, C. F., Pivovarchik, V. N., Inverse nodal problem for Dirac system with spectral parameter in boundary conditions, Complex Analysis and Operator Theory, 7(4) (2013), 1211-1230.
  • Marchenko, V. A., Maslov, K. V., Stability of the problem of recovering the Sturm-Liouville operator from the spectral function, Mathematics of the USSR Sbornik, 81(4) (1970), 525-551.
  • McLaughlin, J. R., Stability theorems for two inverse spectral problems, Inverse Problems, 4(2) (1988), 529-540.
Toplam 40 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Uygulamalı Matematik
Bölüm Research Article
Yazarlar

Emrah Yılmaz 0000-0002-7822-9193

Hikmet Kemaloğlu 0000-0002-7664-1467

Yayımlanma Tarihi 30 Haziran 2021
Gönderilme Tarihi 6 Mayıs 2020
Kabul Tarihi 24 Ocak 2021
Yayımlandığı Sayı Yıl 2021 Cilt: 70 Sayı: 1

Kaynak Göster

APA Yılmaz, E., & Kemaloğlu, H. (2021). On the Lipschitz stability of inverse nodal problem for Dirac system. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 70(1), 341-356. https://doi.org/10.31801/cfsuasmas.733215
AMA Yılmaz E, Kemaloğlu H. On the Lipschitz stability of inverse nodal problem for Dirac system. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. Haziran 2021;70(1):341-356. doi:10.31801/cfsuasmas.733215
Chicago Yılmaz, Emrah, ve Hikmet Kemaloğlu. “On the Lipschitz Stability of Inverse Nodal Problem for Dirac System”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 70, sy. 1 (Haziran 2021): 341-56. https://doi.org/10.31801/cfsuasmas.733215.
EndNote Yılmaz E, Kemaloğlu H (01 Haziran 2021) On the Lipschitz stability of inverse nodal problem for Dirac system. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 70 1 341–356.
IEEE E. Yılmaz ve H. Kemaloğlu, “On the Lipschitz stability of inverse nodal problem for Dirac system”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., c. 70, sy. 1, ss. 341–356, 2021, doi: 10.31801/cfsuasmas.733215.
ISNAD Yılmaz, Emrah - Kemaloğlu, Hikmet. “On the Lipschitz Stability of Inverse Nodal Problem for Dirac System”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 70/1 (Haziran 2021), 341-356. https://doi.org/10.31801/cfsuasmas.733215.
JAMA Yılmaz E, Kemaloğlu H. On the Lipschitz stability of inverse nodal problem for Dirac system. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2021;70:341–356.
MLA Yılmaz, Emrah ve Hikmet Kemaloğlu. “On the Lipschitz Stability of Inverse Nodal Problem for Dirac System”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, c. 70, sy. 1, 2021, ss. 341-56, doi:10.31801/cfsuasmas.733215.
Vancouver Yılmaz E, Kemaloğlu H. On the Lipschitz stability of inverse nodal problem for Dirac system. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2021;70(1):341-56.

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

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