Isaac Scientific Publishing

Advances in Analysis

On Singular Sturm-Liouville Problem with Impulse

Download PDF (555.4 KB) PP. 121 - 135 Pub. Date: October 25, 2016

DOI: 10.22606/aan.2016.12007

Author(s)

  • Rauf Kh AMIROV*
    Department of Mathematics, Faculty of Science, Cumhuriyet University, 58140 Sivas, Turkey
  • Selma GÜLYAZ
    Department of Mathematics, Faculty of Science, Cumhuriyet University, 58140 Sivas, Turkey

Abstract

Sturm-Liouville problem with boundary and discontinuity conditions was studied. For the solution inverse problem necessary and sufficient condition was obtained by the classical Gelfand-Levitan-Marchenko (GLM) type main integral equation and also algorithm was already given.

Keywords

Impulse conditions, inverse problem, kernel, integral equation

References

[1] Agranovich, Z.S. and Marchenko, V. A., The lnverse Problem Scaterring Theory (New York: Gordonand Breach) (1963).

[2] Akhmedova, E.N., Huseynov, H. M., On Inverse Problem Sturm-Liouville Operator with Discontinuous Coefficients, Trnasactions of Saratov University, vol.10, Mathematical, Mechanical and Informatical series no.1, (2010)

[3] Amirov, R. Kh., On Sturm-Liouville Operators with Discotinuity conditions inside an interval, J. Math. Anal. Appl. 317 (2006), pp. 163-176.

[4] Borg, G., Eine umkehrung der Sturm Liouvillesehen eigenwertaufgabe, Acta Math. 78 (1946), pp. 1-96.

[5] Coen, S., On the elastic profiles of a profiles of a layered medium from reflection data. Part I. Plane-wave sources J. Acoust. Soc. Am. 70 (1981), pp. 172-5.

[6] Conway, J.B., Functions of One Complex Variable. 2. Edition, vol. I, Springer-Verlag, New York, (1995).

[7] Faydaoglu, S., and Guseinov, G. Sh., An expansion result for a Sturm-Liouville eigenvalue problem with impulse, Turk. J. Math. 34, No. 3, (2010), pp. 355-366.

[8] Gelfand I.M. and Levitan,B.M., On the determination of a differential equation from its spectral function, Izv. Akad. Nauk SSR. Ser. Mat. 15 (1951), pp.309-360 (in Russian), English transl. in Amer. Math. Soc. Transl. Ser. 2 (1) (1955), pp.253-304.

[9] Guseinov, I. M. and Mammadova, L. I., Reconstruction of the Diffusion Equation with Singular Coefficients for Two Spectra, Doklady Mathematies, vol.9, no. 1, (2014), pp. 401-404.

[10] Hald, O.H., Discontinuous inverse eigenvalue problem, Commun. Pure Appl. Math. 37 (1984) 539-577.

[11] He, X. and Volkmer, H., Riesz bases of Sturm-Liouville equations, J. Fourier Anal. Appl. 7(2001), 297-307.

[12] Krueger, R. J., An inverse problem for an absorbing medium with multiple discontinuities Q. Appl. Math. 34 (1976), pp. 129-47.

[13] Krueger, R.J., Inverse problems for nonabsorbing media with discontinuous material properties, J. Math. Phys. 23 (1982), pp. 396-404.

[14] Levitan, B.M. (1984). Inverse Sturm-Liouville Problems, Moscow: Nauka, , (Engl. Transl.1987 (Utrecth: VNU Science Press)).

[15] Litvinenko, O.N. and Soshnikov, V.I., The theory of Heterogeneous Lines and Their Applications in Radio Engineering, Radio, Moscow, (1964) (in Russian).

[16] Marchenko, V.A.,Some Problems in the Theory of Second-Order Differential Operators, Dokl. Akad.,Nauk SSSR. 72 , (1950), pp. 457-560.

[17] Marchenko, V.A., Sturm Liouville Operators and Their Applications, Naukova Dumka, Kiev, 1977, English transl.: Birkh?user, (1986).

[18] Meschanov, V.P. and Feldstein, A.L., Automatic Design of Directional Couplers, Sviaz, Moscow, (1980).

[19] Newton, R. G., Inversion ofrcflection data for layered media: a review of exact methods Geophys. J. R.Aslron. Soc. 65 (1981), pp.191-215.

[20] Pöschel, J., Trubowitz, E., Inverse Spectral Theory, Academic Press, Orlando, (1987).

[21] Yurko, V.A., Boundary value Problems with Discontinuity Conditions in an Interior of the Interval,Differential Equations, vol. 36, No.8, (2000), pp. 1266-1269.

[22] Yurko, V.A., Method of Spectral Mappings in the Inverse Problem Theory, in: Inverse Ill-posed Probl. Ser., VSP, Utrecht, (2002).