INVERSE NODAL PROBLEM FORp LAPLACIAN DIFFUSION EQUATION WITH POLYNOMIALLY DEPENDENT SPECTRAL PARAMETER

Year 2016, Volume: 65 Issue: 2, 23 - 36, 01.08.2016

Tuba Gulsen Emrah Yılmaz

https://doi.org/10.1501/Commua1_0000000756

Cited By: 6

Abstract

In this study, solution of inverse nodal problem for one-dimensional
p-Laplacian diffusion equation is extended to the case that boundary condition depends on polynomial eigen parameter. To find the spectral datas as
eigen values and nodal parameters of this problem, we used a modified Prefer
substitution. Then, reconstruction formula of the potential function is also
given by using nodal lenghts. Furthermore, this method is similar to used in
[1], our results are more general.

Keywords

Inverse Nodal Problem, Prüfer Substitution, Diğusion equation

References

• Koyunbakan, H. 2013. Inverse nodal problem for p Laplacian energy-dependent Sturm- Liouville equation, Boundary Value Problems 2013:272 (Erratum: Inverse nodal problem for p Laplacian energy-dependent Sturm-Liouville equation, Boundary Value Problems, 2014:222 (2014).
• Yang, C.F. and Yang, X. 2011. Ambarzumyan’s theorem with eigenparameter in the boundary conditions, Acta Mathematica Scientia 31(4), 1561-1568.
• Jaulent, M. and Jean, C. 1972. The inverse s-wave scattering problem for a class of potentials depending on energy, Commun.Math. Phys. 28(3), 177-220.
• Gasymov, M. G. and Guseinov, G.S. 1981. Determination of a diğusion operator from spectral data, Dokl. Akad. Nauk Azerb. SSR 37(2), 19-23.
• Guseinov, G. S. 1985. On the spectral analysis of a quadratic pencil of Sturm-Liouville oper- ators, Sov. Math. Dokl. 32, 1292-1296.
• Hryniv, R. and Pronska, N. 2012. Inverse spectral problems for energy dependent Sturm- Liouville equations, Inverse Probl. 28(8), 085008.
• Nabiev, I. M. 2007. The inverse quasiperiodic problem for a diğusion operator, Dokl. Math. 76(1), 527-529.
• Wang, Y.P., Yang, C.F. and Huang, Z.Y. 2011. Half inverse problem for a quadratic pencil of Schrödinger operators, Acta Math. Sci. 31(6), 1708-1717.
• Koyunbakan, H. and Yilmaz, E. 2008. Reconstruction of the potential function and its deriv- atives for the diğusion operator, Z. Naturforch A 63(3-4), 127-130.
• Buterin, S. A. and Shieh, C. T. 2008. Inverse nodal problem for diğerential pencils, Applied Mathematics Letters 22(8), 1240-1247.
• Koyunbakan, H. 2011. Inverse problem for a quadratic pencil of Sturm-Liouville operator, J. Math. Anal. Appl. 378(2), 549-554.
• Guseinov, I. M. and Nabiev, I. M. 2000. A class of inverse problems for a quadratic pencil of Sturm-Liouville operators, Diğ erential Equations 36(3), 471-473.
• Maksudov, F. G. and Guseinov, G. Sh.1987. On the solution of the inverse scattering prob- lem for a quadratic pencil of one dimensional Schrodinger operators on whole axis, Soviet Mathematics Doklady 34, 34-38.
• Bairamov, E., Cakar O. and Celebi, A. O. 1997. Quadratic pencil of Schrodinger operators with spectral singularities: discrete spectrum and principal functions, Journal of Mathemat- ical Analysis and Applications 216(1), 303-320.
• Yang, C. F. and Zettl, A. 2012. Half inverse problems for quadratic pencils of Sturm-Liouville operators, Taiwanese Journal of Mathematics 16(5), 1829-1846.
• Amirov, Rauf Kh. and Nabiev, A. A. 2013. Inverse problems for the quadratic pencil of the Sturm-Liouville equations with Impulse, Abstract and Applied Analysis Volume 2013, Article ID 361989.
• Wang, Y. P. 2012. The inverse problem for diğerential pencils with eigenparameter dependent boundary conditions from interior spectral data, Applied Mathematics Letters 25(7) 1061- 1067.
• Sat, M. and Panakhov, E. S. 2014. Spectral problem for diğusion operator, Applicable Analy- sis 93(6) 1178-1186.
• Chadan, K., Colton, D., Paivarinta, L. and Rundell, W. 1997. An introduction to inverse scattering and inverse spectral problems, Society for Industrial and Applied Mathematics.
• Aygar, Y. and Bohner M. 2015. On the spectrum of eigenparameter-dependent quantum diğerence equations, Applied Mathematics &Information Sciences, 9(4), 1725-1729.
• McLaughlin, J. R. 1988. Inverse spectral theory using nodal points as data-a uniqueness result, J. Diğ erential Equations 73(2), 354-362.
• Hald, O. H. and McLaughlin, J.R. 1989. Solution of inverse nodal problems, Inverse Probl. 5(3), 307-347.
• Yurko, V. A. 2008. Inverse nodal problems for Sturm-Liouville operators on star-type graphs, J. Inv. Ill Posed Probl. 16(7), 715–722.
• Law, C. K. and Yang, C. F. 1999. Reconstruction of the potential function and its derivatives using nodal data, Inverse Probl. 14(2), 299-312.
• Yang, C. F. and Yang, X. P. 2011. Inverse nodal problem for the Sturm–Liouville equation with polynomially dependent on the eigenparameter, Inverse Prob. Sci. Eng. 19(7), 951–961 (2011).
• Browne, P. J. and Sleeman, B. D. 1996. Inverse nodal problems for Sturm–Liouville equations with eigenparameter dependent boundary conditions, Inverse Probl. 12(4), 377–381.
• Ozkan, A. S. and Keskin, B. 2015. Inverse nodal problems for Sturm-Liouville equation with eigenparameter dependent boundary and jump conditions, Inverse Problems in Science and Engineering 23(8), 1306-1312.
• Chen, H. Y. 2009. On generalized trigonometric functions, Master of Science, National Sun Yat-sen University, Kaohsiung, Taiwan.
• Law, C. K., Lian, W. C. and Wang, W. C. 2009. The inverse nodal problem and the Am- barzumyan problem for the p Laplacian, Proc. Roy. Soc. Edinburgh Sect. A Math. 139(6), 1261-1273.
• Wang, W. C., Cheng, Y. H. and Lian, W. C. 2011. Inverse nodal problems for the p Laplacian with eigenparameter dependent boundary conditions, Math. Comput. Modelling 54 (11-12), 2718-2724.
• Wang, W. C. 2010. Direct and inverse problems for one dimensional p Laplacian operators, National Sun Yat-sen University, PhD Thesis.
• Elbert, A. 1987. On the half-linear second order diğerential equations, Acta Math. Hungar. 49(3-4), 487–508.
• Binding, P. and Drábek, P. 2003. Sturm–Liouville theory for the p Laplacian, Studia Sci. Math. Hungar. 40(4), 375–396.
• Binding, P. A. and Rynne, B. P. 2008. Variational and non-variational eigenvalues of the p
• Laplacian, J. Diğ erential Equations 244(1), 24-39.
• Brown, B. M. and Reichel ,W. 2004. Eigenvalues of the radially symmetric p Laplacian in Rn, J. Lond. Math. Soc. 69, 657-675.
• Walter, W. 1998. Sturm-Liouville theory for the radial poperator. Math. Z. 227(1), 175- 185.
• Yantır, A. 2004. Oscillation theory for second order diğerential equations and dynamic equa- tions on time scales, Master of Science, Izmir institue of Technology, Izmir.
• Current address : Tuba GULSEN : Firat University, Department of Mathematics, 23119, Elazıg TURKEY
• E-mail address : [e-mail]
• Current address : Emrah YILMAZ : Firat University, Department of Mathematics, 23119, Elazıg TURKEY
• E-mail address : [e-mail]