Isaac Scientific Publishing

Advances in Analysis

Inverse Nodal Problems for Impulsive Sturm-Liouville Equation with Boundary Conditions Depending on the Parameter

Download PDF (445.7 KB) PP. 151 - 156 Pub. Date: May 3, 2017

DOI: 10.22606/aan.2017.23002

Author(s)

  • Baki Keskin
    Department of Mathematics, Faculty of Science, Cumhuriyet University, 58140, Sivas, Turkey
  • A. Sinan Ozkan*

    Department of Mathematics, Faculty of Science, Cumhuriyet University, 58140, Sivas, Turkey

Abstract

In this work, the Sturm–Liouville problem with boundary conditions depending rationally on the spectral parameter is studied. We give a uniqueness theorem and algorithm to reconstruct the potential of the problem from nodal points (zeros of eigenfunctions).

Keywords

Sturm-Liouville equation, inverse nodal problem, parameter dependent boundary condition, discontinuity condition.

References

[1] P.A. Binding, P.J. Browne and K. Seddighi, Sturm–Liouville problems with eigenparameter dependent boundary conditions, Proc. Edinburgh Math. Soc., 37(2), (1993), 57-72.

[2] P.A. Binding, P.J. Browne and B.A. Watson, Inverse spectral problems for Sturm–Liouville equations with eigenparameter dependent boundary conditions, J. London Math. Soc., 62, (2000), 161-182.

[3] P.A. Binding, P.J. Browne, B.A. Watson, Sturm–Liouville problems with boundary conditions rationally dependent on the eigenparameter, I, Proc.Edinburgh Math.Soc., 45, (2002), 631–645.

[4] P.A. Binding, P.J. Browne, B.A. Watson, Sturm–Liouville problems with boundary conditions rationally dependent on the eigenparameter, II, Journal of Computational and Applied Mathematics, 148, (2002), 147–168.

[5] P.J. Browne and B.D. Sleeman, Inverse nodal problem for Sturm–Liouville equation with eigenparameter depend boundary conditions, Inverse Problems 12 (1996), pp. 377–381.

[6] Y.H. Cheng, C-K. Law and J. Tsay, Remarks on a new inverse nodal problem, J. Math. Anal. Appl. 248 (2000), pp. 145–155.

[7] G. Freiling and V.A. Yurko, Inverse Sturm–Liouville Problems and their Applications, Nova Science, New York, 2001.

[8] G. Freiling and V.A. Yurko, Inverse problems for Sturm–Liouville equations with boundary conditions polynomially dependent on the spectral parameter, Inverse Problems, 26, (2010), p. 055003 (17pp.).

[9] C.T. Fulton, Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions, Proc. R. Soc. Edinburgh, A77, (1977), 293-308.

[10] I.M. Gelfand and B.M. Levitan, On the determination of a differential equation from its spectral function, Amer. Math Soc. Trans. 1 (1951), pp. 253–304.

[11] N.J. Guliyev, Inverse eigenvalue problems for Sturm-Liouville equations with spectral parameter linearly contained in one of the boundary condition, Inverse Problems, 21, (2005), 1315-1330.

[12] O.H. Hald, Discontinuous inverse eigenvalue problems, Comm. Pure Appl. Math., 37, (1984), 539-577.

[13] O.H. Hald and J.R. McLaughlin, Solutions of inverse nodal problems, Inv. Prob. 5 (1989), pp. 307–347.

[14] H. Hochstadt and B. Lieberman, An Inverse Sturm-Liouville Problem with Mixed Given Data, SIAM J. Appl. Math. 34 (1978), 676–680.

[15] J.R. McLaughlin, Inverse spectral theory using nodal points as data – a uniqueness result, J. Diff. Eq. 73 (1988), pp. 354–362.

[16] R. Mennicken, H. Schmid and A.A. Shkalikov, On the eigenvalue accumulation of Sturm-Liouville problems depending nonlinearly on the spectral parameter, Math. Nachr., 189, (1998), 157-170.

[17] A.S. Ozkan, B. Keskin, Inverse nodal problems for Sturm–Liouville equation with eigenparameter-dependent boundary and jump conditions, Inverse Problems in Science and Engineering, 23(8), (2015), 1306-1312.

[18] Chung-Tsun Shieh and V. A. Yurko, Inverse nodal and inverse spectral problems for discontinuous boundary value problems, J. Math. Anal. Appl. 347 (2008) 266-272.

[19] A.A. Shkalikov, Boundary value problems for ordinary differential equations with a parameter in the boundary conditions. J. Sov. Math., 33, (1986), 1311-1342, Translation from Tr. Semin. Im. I.G. Petrovskogo, 9, (1983), 190-229.

[20] Xeu-Feng Yang, A solution of the nodal problem, Inverse Problems, 13, (1997) 203-213.

[21] Chuan-Fu Yang and Xiao-Ping Yang Inverse nodal problems for the Sturm-Liouville equation with polynomially dependent on the eigenparameter, Inverse Problems in Science and Engineering, 19(7), (2011), 951-961.

[22] Chuan-Fu Yang, Inverse nodal problems of discontinuous Sturm–Liouville operator, J. Differential Equations, 254, (2013) 1992–2014.

[23] V.A. Yurko, Boundary value problems with a parameter in the boundary conditions, Izv. Akad. Nauk Armyan. SSR, Ser. Mat., 19(5), (1984), 398–409, English translation in Soviet J. Contemporary Math. Anal., 19(5), (1984), 62-73.