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On a lipschitz stability problem for p-Laplacian Bessel equation

Year 2017, Volume: 66 Issue: 2, 253 - 262, 01.08.2017
https://doi.org/10.1501/Commua1_0000000816

Abstract

In this study, we are enunciative of some asymptotic expansionsand reconstruction formulas for inverse nodal problem of p-Laplacian Besselequation.Dirichlet boundary conditions is solved. And, it is also proved that the spaceof all potential functions w is homeomorphic to the partition set of all asymptotically equivalent nodal sequences induced by an equivalence relation

References

  • Koyunbakan, H., Inverse nodal problem for p Laplacian energy-dependent Sturm-Liouville equation, Boundary Value Problems, 2013:272 (2013) (Erratum: Inverse nodal problem for p-Laplacian energy-dependent Sturm-Liouville equation, Boundary Value Problems, 2014:222 (2014).
  • Coskun, M., Gulsen, T., Koyunbakan, H., Inverse nodal problem for p Laplacian Bessel Equation (Submitted to the journal).
  • Hamadamen, M. H., Inverse nodal problem for p Laplacian diğerential operators, Master Thesis, The institute of Natural and Applied Sciences, Firat University, (2017).
  • Baumeister, J., Stable solution of inverse problems, Advanced Lectures in Mathematics, (1987).
  • Ambarzumyan, V.A., Uber eine Frage der Eigenwerttheorie, Zeitschrift für Physik, 53, 690– 695 (1929).
  • Bas, E., Panakhov, E. S., Yilmazer, R., The uniqueness theorem for hydrogen atom equation, TWMS Journal of Pure and Applied Mathematics, 4(1), 20-28 (2013).
  • Bondarenko, N., An inverse spectral problem for the matrix Sturm-Liouville operator with a Bessel-type singularity, International Journal of Diğerential Equations, Art. ID 647396, 4 pp. (2015).
  • Guldu, Y., Amirov, R. Kh., Topsakal, N., On impulsive Sturm-Liouville operators with sin- gularity and spectral parameter in boundary conditions, Ukrainian Mathematical Journal, 64(12), 1816–1838 (2013).
  • McLeod, J. B., The distribution of the eigenvalues for the Hydrogen atom and similar cases, Proceedings of the London Mathematical Society, 3(1), 139-158 (1961).
  • Masood, K., Messaoudi, S., Zaman, F. D., Initial inverse problem in heat equation with Bessel operator, International Journal of Heat and Mass Transfer, 45, 2959–2965 (2002).
  • Albeverio, S., Hryniv, R., Mykytyuk, Y., Inverse spectral problems for Bessel operators, Journal of Diğerential Equations, 241(1), 130–159 (2007).
  • Hryniv, R., Sacks, P., Numerical solution of the inverse spectral problem for Bessel operators, Journal of Computational and Applied Mathematics, 235(1), 120–136 (2010).
  • Alekseeva, V. S., Ananieva, A. Y., On extensions of the Bessel operator on a …nite interval and a half-line, Journal of Mathematical Sciences, 187(1), 1-8 (2012).
  • Sat, M., Panakhov, E. S., A uniqueness theorem for Bessel operator from interior spectral data, Abstract and Applied Analysis, Volume 2013, Article ID 713654, 6 pages.
  • Yilmaz, E., Koyunbakan, H., Some Ambarzumyan type theorems for Bessel operator on a …nite interval, Diğerential Equations and Dynamical Systems, (2016).
  • McLaughlin, J. R., Inverse spectral theory using nodal points as data-a uniqueness result, Journal of Diğerential Equations, 73(2), 354-362 (1988).
  • Shen, C. L., On the nodal sets of the eigenfunctions of the string equations, SIAM Journal on Mathematical Analysis, 19(6), 1419–1424 (1988).
  • Yurko, V. A., Inverse nodal problems for Sturm-Liouville operators on star-type graphs, Journal of Inverse and Ill-Posed Problems, 16(7), 715–722 (2008).
  • Hald, O. H., McLaughlin, J. R., Solutions of the inverse nodal problems, Inverse Problems, 5(3), 307-347 (1989).
  • Yang, C. F., Inverse nodal problems for the Sturm-Liouville operator with a constant delay, Journal of Diğerential Equations, 257(4), 1288-1306 (2014).
  • Koyunbakan, H., Yilmaz, E., Reconstruction of the potential function and its derivatives for the diğusion operator, Zeitschrift für Naturforschung A, 63(3-4), 127-130 (2008).
  • Yang, C. F., An inverse problem for a diğerential pencil using nodal points as data, Israel Journal of Mathematics, 204(1), 431-446 (2014).
  • Law, C. K., Yang, C. F., Reconstruction of the potential function and its derivatives using nodal data, Inverse Problems, 14, 299-312 (1999).
  • Buterin, S. A., Shieh, C. T., Inverse nodal problem for diğerential pencils, Applied Mathe- matics Letters, 22, no. 8, 1240–1247 (2009).
  • Law, C. K., Lian, W. C., Wang, W. C., The inverse nodal problem and the Ambarzumyan problem for the p Laplacian, Proceedings of the Royal Society of Edinburgh Section A Mathematics, 139(6), 1261-1273 (2009).
  • Chen, H. Y., On generalized trigonometric functions, Master of Science, National Sun Yat-sen University, Kaohsiung, Taiwan, (2009).
  • Elbert, A., On the half-linear second order diğerential equations, Acta Mathematica Hungar- ica, 49(3-4), 487–508 (1987).
  • Binding, P. A., Rynne, B. P., Variational and non-variational eigenvalues of the p
  • Laplacian, Journal of Diğerential Equations, 244(1), 24-39 (2008).
  • Brown, B. M., Reichel, W., Eigenvalues of the radially symmetric p Laplacian in Rn, Journal of the London Mathematical Society, 69(3), 657-675 (2004).
  • Walter, W., Sturm-Liouville theory for the radial poperator, Mathematische Zeitschrift, 227(1), 175-185 (1998).
  • Binding, P., Drábek, P., Sturm–Liouville theory for the p Laplacian, Studia Scientiarum Mathematicarum Hungarica, 40(4), 373–396 (2003).
  • Wang, W. C., Cheng, Y. H., Lian, W. C., Inverse nodal problems for the p-Laplacian with eigenparameter dependent boundary conditions, Mathematical and Computer Modelling, 54(11-12), 2718-2724 (2011).
  • Wang, W. C., Direct and inverse problems for one dimensional p Laplacian operators, Na- tional Sun Yat-Sen University, PhD Thesis, (2010).
  • Pinasco, J. P., Scarola, C. A., A nodal inverse problem for a quasi-linear ordinary diğerential equation in the half-line, Journal of Diğerential Equations, 261(2), 1000–1016 (2016).
  • Gulsen, T., Yilmaz, E., Inverse nodal problem for p Laplacian diğusion equation with poly- nomially dependent spectral parameter, Communications, Series A1; Mathematics and Sta- tistics, 65(2), 23-36, (2016).
  • Gulsen, T., Yilmaz, E., Koyunbakan, H., Inverse nodal problem for p-Laplacian Dirac system, Mathematical methods in applied sciences, Doi: 10.1002/mma.4141, (2016).
  • Yantir, A., Oscillation theory for second order diğerential equations and dynamic equations on time scales, Master of Science, Izmir institue of Technology, Izmir, (2004).
  • Law, C. K., Tsay, J., On the well-posedness of the inverse nodal problem, Inverse Problems, 17(5), 1493-1512 (2001).
  • Marchenko, V. A., Maslov, K. V., Stability of the problem of recovering the Sturm-Liouville operator from the spectral function, Matematicheskii Sbornik, 123(4), 475-502 (1970).
  • McLaughlin, J. R., Stability theorems for two inverse spectral problems, Inverse Problems, 4(2), 529-540 (1988).
  • Yilmaz, E., Koyunbakan, H., On the high order Lipschitz stability of inverse nodal problem for string equation, Dynamics of Continuous, Discrete and Impulsive Systems. Series A. Mathematical Analysis, 21, 79-88 (2014).
  • Yilmaz, E., Lipschitz stability of inverse nodal problem for energy-dependent Sturm-Liouville equation, New Trends in Mathematical Sciences, 3(1), 46-61 (2015).
  • Current address : Tuba GULSEN: Firat University, Department of Mathematics, 23119, Elazıg TURKEY
  • E-mail address : tubagulsen87@hotmail.com
  • Current address : Emrah YILMAZ (Corresponding author): Firat University, Department of Mathematics, 23119, Elazıg TURKEY
  • E-mail address : emrah231983@gmail.com
  • Current address : E. S. PANAKHOV: Baku State University, Institute of Applied Mathematics, Baku AZARBAIJAN
  • E-mail address : epenahov@hotmail.com
Year 2017, Volume: 66 Issue: 2, 253 - 262, 01.08.2017
https://doi.org/10.1501/Commua1_0000000816

Abstract

References

  • Koyunbakan, H., Inverse nodal problem for p Laplacian energy-dependent Sturm-Liouville equation, Boundary Value Problems, 2013:272 (2013) (Erratum: Inverse nodal problem for p-Laplacian energy-dependent Sturm-Liouville equation, Boundary Value Problems, 2014:222 (2014).
  • Coskun, M., Gulsen, T., Koyunbakan, H., Inverse nodal problem for p Laplacian Bessel Equation (Submitted to the journal).
  • Hamadamen, M. H., Inverse nodal problem for p Laplacian diğerential operators, Master Thesis, The institute of Natural and Applied Sciences, Firat University, (2017).
  • Baumeister, J., Stable solution of inverse problems, Advanced Lectures in Mathematics, (1987).
  • Ambarzumyan, V.A., Uber eine Frage der Eigenwerttheorie, Zeitschrift für Physik, 53, 690– 695 (1929).
  • Bas, E., Panakhov, E. S., Yilmazer, R., The uniqueness theorem for hydrogen atom equation, TWMS Journal of Pure and Applied Mathematics, 4(1), 20-28 (2013).
  • Bondarenko, N., An inverse spectral problem for the matrix Sturm-Liouville operator with a Bessel-type singularity, International Journal of Diğerential Equations, Art. ID 647396, 4 pp. (2015).
  • Guldu, Y., Amirov, R. Kh., Topsakal, N., On impulsive Sturm-Liouville operators with sin- gularity and spectral parameter in boundary conditions, Ukrainian Mathematical Journal, 64(12), 1816–1838 (2013).
  • McLeod, J. B., The distribution of the eigenvalues for the Hydrogen atom and similar cases, Proceedings of the London Mathematical Society, 3(1), 139-158 (1961).
  • Masood, K., Messaoudi, S., Zaman, F. D., Initial inverse problem in heat equation with Bessel operator, International Journal of Heat and Mass Transfer, 45, 2959–2965 (2002).
  • Albeverio, S., Hryniv, R., Mykytyuk, Y., Inverse spectral problems for Bessel operators, Journal of Diğerential Equations, 241(1), 130–159 (2007).
  • Hryniv, R., Sacks, P., Numerical solution of the inverse spectral problem for Bessel operators, Journal of Computational and Applied Mathematics, 235(1), 120–136 (2010).
  • Alekseeva, V. S., Ananieva, A. Y., On extensions of the Bessel operator on a …nite interval and a half-line, Journal of Mathematical Sciences, 187(1), 1-8 (2012).
  • Sat, M., Panakhov, E. S., A uniqueness theorem for Bessel operator from interior spectral data, Abstract and Applied Analysis, Volume 2013, Article ID 713654, 6 pages.
  • Yilmaz, E., Koyunbakan, H., Some Ambarzumyan type theorems for Bessel operator on a …nite interval, Diğerential Equations and Dynamical Systems, (2016).
  • McLaughlin, J. R., Inverse spectral theory using nodal points as data-a uniqueness result, Journal of Diğerential Equations, 73(2), 354-362 (1988).
  • Shen, C. L., On the nodal sets of the eigenfunctions of the string equations, SIAM Journal on Mathematical Analysis, 19(6), 1419–1424 (1988).
  • Yurko, V. A., Inverse nodal problems for Sturm-Liouville operators on star-type graphs, Journal of Inverse and Ill-Posed Problems, 16(7), 715–722 (2008).
  • Hald, O. H., McLaughlin, J. R., Solutions of the inverse nodal problems, Inverse Problems, 5(3), 307-347 (1989).
  • Yang, C. F., Inverse nodal problems for the Sturm-Liouville operator with a constant delay, Journal of Diğerential Equations, 257(4), 1288-1306 (2014).
  • Koyunbakan, H., Yilmaz, E., Reconstruction of the potential function and its derivatives for the diğusion operator, Zeitschrift für Naturforschung A, 63(3-4), 127-130 (2008).
  • Yang, C. F., An inverse problem for a diğerential pencil using nodal points as data, Israel Journal of Mathematics, 204(1), 431-446 (2014).
  • Law, C. K., Yang, C. F., Reconstruction of the potential function and its derivatives using nodal data, Inverse Problems, 14, 299-312 (1999).
  • Buterin, S. A., Shieh, C. T., Inverse nodal problem for diğerential pencils, Applied Mathe- matics Letters, 22, no. 8, 1240–1247 (2009).
  • Law, C. K., Lian, W. C., Wang, W. C., The inverse nodal problem and the Ambarzumyan problem for the p Laplacian, Proceedings of the Royal Society of Edinburgh Section A Mathematics, 139(6), 1261-1273 (2009).
  • Chen, H. Y., On generalized trigonometric functions, Master of Science, National Sun Yat-sen University, Kaohsiung, Taiwan, (2009).
  • Elbert, A., On the half-linear second order diğerential equations, Acta Mathematica Hungar- ica, 49(3-4), 487–508 (1987).
  • Binding, P. A., Rynne, B. P., Variational and non-variational eigenvalues of the p
  • Laplacian, Journal of Diğerential Equations, 244(1), 24-39 (2008).
  • Brown, B. M., Reichel, W., Eigenvalues of the radially symmetric p Laplacian in Rn, Journal of the London Mathematical Society, 69(3), 657-675 (2004).
  • Walter, W., Sturm-Liouville theory for the radial poperator, Mathematische Zeitschrift, 227(1), 175-185 (1998).
  • Binding, P., Drábek, P., Sturm–Liouville theory for the p Laplacian, Studia Scientiarum Mathematicarum Hungarica, 40(4), 373–396 (2003).
  • Wang, W. C., Cheng, Y. H., Lian, W. C., Inverse nodal problems for the p-Laplacian with eigenparameter dependent boundary conditions, Mathematical and Computer Modelling, 54(11-12), 2718-2724 (2011).
  • Wang, W. C., Direct and inverse problems for one dimensional p Laplacian operators, Na- tional Sun Yat-Sen University, PhD Thesis, (2010).
  • Pinasco, J. P., Scarola, C. A., A nodal inverse problem for a quasi-linear ordinary diğerential equation in the half-line, Journal of Diğerential Equations, 261(2), 1000–1016 (2016).
  • Gulsen, T., Yilmaz, E., Inverse nodal problem for p Laplacian diğusion equation with poly- nomially dependent spectral parameter, Communications, Series A1; Mathematics and Sta- tistics, 65(2), 23-36, (2016).
  • Gulsen, T., Yilmaz, E., Koyunbakan, H., Inverse nodal problem for p-Laplacian Dirac system, Mathematical methods in applied sciences, Doi: 10.1002/mma.4141, (2016).
  • Yantir, A., Oscillation theory for second order diğerential equations and dynamic equations on time scales, Master of Science, Izmir institue of Technology, Izmir, (2004).
  • Law, C. K., Tsay, J., On the well-posedness of the inverse nodal problem, Inverse Problems, 17(5), 1493-1512 (2001).
  • Marchenko, V. A., Maslov, K. V., Stability of the problem of recovering the Sturm-Liouville operator from the spectral function, Matematicheskii Sbornik, 123(4), 475-502 (1970).
  • McLaughlin, J. R., Stability theorems for two inverse spectral problems, Inverse Problems, 4(2), 529-540 (1988).
  • Yilmaz, E., Koyunbakan, H., On the high order Lipschitz stability of inverse nodal problem for string equation, Dynamics of Continuous, Discrete and Impulsive Systems. Series A. Mathematical Analysis, 21, 79-88 (2014).
  • Yilmaz, E., Lipschitz stability of inverse nodal problem for energy-dependent Sturm-Liouville equation, New Trends in Mathematical Sciences, 3(1), 46-61 (2015).
  • Current address : Tuba GULSEN: Firat University, Department of Mathematics, 23119, Elazıg TURKEY
  • E-mail address : tubagulsen87@hotmail.com
  • Current address : Emrah YILMAZ (Corresponding author): Firat University, Department of Mathematics, 23119, Elazıg TURKEY
  • E-mail address : emrah231983@gmail.com
  • Current address : E. S. PANAKHOV: Baku State University, Institute of Applied Mathematics, Baku AZARBAIJAN
  • E-mail address : epenahov@hotmail.com
There are 49 citations in total.

Details

Primary Language English
Journal Section Research Articles
Authors

Tuba Gulsen This is me

Emrah Yılmaz This is me

S. Panakhov E. This is me

Publication Date August 1, 2017
Published in Issue Year 2017 Volume: 66 Issue: 2

Cite

APA Gulsen, T., Yılmaz, E., & Panakhov E., S. (2017). On a lipschitz stability problem for p-Laplacian Bessel equation. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 66(2), 253-262. https://doi.org/10.1501/Commua1_0000000816
AMA Gulsen T, Yılmaz E, Panakhov E. S. On a lipschitz stability problem for p-Laplacian Bessel equation. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. August 2017;66(2):253-262. doi:10.1501/Commua1_0000000816
Chicago Gulsen, Tuba, Emrah Yılmaz, and S. Panakhov E. “On a Lipschitz Stability Problem for P-Laplacian Bessel Equation”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 66, no. 2 (August 2017): 253-62. https://doi.org/10.1501/Commua1_0000000816.
EndNote Gulsen T, Yılmaz E, Panakhov E. S (August 1, 2017) On a lipschitz stability problem for p-Laplacian Bessel equation. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 66 2 253–262.
IEEE T. Gulsen, E. Yılmaz, and S. Panakhov E., “On a lipschitz stability problem for p-Laplacian Bessel equation”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 66, no. 2, pp. 253–262, 2017, doi: 10.1501/Commua1_0000000816.
ISNAD Gulsen, Tuba et al. “On a Lipschitz Stability Problem for P-Laplacian Bessel Equation”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 66/2 (August 2017), 253-262. https://doi.org/10.1501/Commua1_0000000816.
JAMA Gulsen T, Yılmaz E, Panakhov E. S. On a lipschitz stability problem for p-Laplacian Bessel equation. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2017;66:253–262.
MLA Gulsen, Tuba et al. “On a Lipschitz Stability Problem for P-Laplacian Bessel Equation”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 66, no. 2, 2017, pp. 253-62, doi:10.1501/Commua1_0000000816.
Vancouver Gulsen T, Yılmaz E, Panakhov E. S. On a lipschitz stability problem for p-Laplacian Bessel equation. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2017;66(2):253-62.

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

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