DIRECT AND INVERSE PROBLEMS FOR DIFFUSION OPERATOR WITH DISCONTINUITY POINTS

Year 2019, Volume: 9 Issue: 1, 9 - 21, 01.03.2019

A. Ergun R. Kh. Amirov

Abstract

In this study, the diusion operator with discontinuity points has been considered. Under certain initial and jump conditions, integral equations have been derived for solutions and integral representation have been presented. Some important spectral properties of eigenvalue and eigenfunctions have been obtained. Reconstruction of the diusion operator with discontinuity points problem have been proved by Weyl function, spectral datas and two sectra.

Keywords

Integral equation, Sturm-Liouville, Diusion operator, inverse problems.

References

• [1] Marchenko, V. A., (1986), Sturm-Liouville Operators and Applications, AMS Chelsea Publishing, Basel.
• [2] Levitan, B. M. and Gasymov, M. G., (1964), Determination of a differential equation by two spectra. Uspekhi Mathematicheskikh Nauk, 19, 2, 3-63.
• [3] Jaulent, M . and Jean, C., (1976), The Inverse Problem for the one-dimensional Schr¨odinger equation with an energy-dependent potential. Annales de L’Institut Henri Poincare A, 25, 2, 105-137.
• [4] Yurko, V. A., (2000), Introduction to the Theory of Inverse Spectral Problems, Fizmatlit, MOscow, Russia, 2007, English translation, Inverse Spectral Problems dor Differantial Operators and Their Applications, Gordon and Breach, Amsterdam, The Netherlands.
• [5] Levitan, B. M., (1978), Inverse Sturm-Liouville Problems, Nauka, Moscow, Russia.
• [6] Gasymov M. G. and Guseinov, G. S., (1981), Determination of a diffusion operator from spectral data, Akademiya Nauk Azerbaidzhanskoi SSR. Doklady, 37, 2, 19-23.
• [7] Yurko, V. A., (2000), An inverse problem for differential operator pencils, Methematicheskikh Sbornik, 191, 10, 137-160.
• [8] Nabiev, M. I., (2004), The inverse spectral problem for he diffusion operator on an interval. Mathematicheskaya Fizika, Analiz, Geometriya, 11, 3, 302-313
• [9] Guseinov, G. S., (1986), Inverse spectral problems for a quadratic pencil of Sturm-Liouville operators on a finite interval. Spectral Theory of Operators and Its Applications, 51-101, Elm, Baku, Azerbaijan.
• [10] Freiling G. and Yurko V., (2001), Inverse Sturm-Liouville problems and their applications, Nova Science, Huntington, NY, USA.
• [11] Yurko, V. A., (2000), Inverse spectral problems for differential operators and their applications, vol. 2 of Analytical Methods and Special Functions, Gordon and Breach, Amsterdam, The Netherlands.
• [12] Hald, O. H., (1984), Discontinuous inverse eigenvalue problems, Communications on Pure and Applied Mathematics, 37, 5, 539–577.
• [13] Amirov, R. Kh., (2006), On Sturm-Liouville operators with discontinuity conditions inside an interval.Journal of Mathematical Analysis and Applications, 317, 1, 163-176.
• [14] AmirovR. Kh. and Yurko, V. A., (2001), On differential operators with a singularity and discontinuity conditions inside an interval.Ukrainian Mathematical Journal, 53,11,1443–1457.
• [15] FreilingG.and Yurko, V., (2002), Inverse spectral problems for singular non-selfadjoint differential operators with discontinuities in an interior point. Inverse Problems, 18, 3, 757-773.
• [16] Levin, B. Ya., (1971), Entire Functions, MGV, Moscow, Russia.
• [17] Jdanovich, B. F. (1960), Formulae for the zeros of Drichlet polynomials and quasi-polynomials. Doklady Akademii Nauk SSSR, 135, 8, 1046-1049.
• [18] Kreyn M. and Levin, B. Ya., (1949), On entire almost periodic functions of exponential type, Doklady Akademii Nauk SSSR, 64, 285-287.
• [19] Buterin S. A., Yurko V.A., (2006), Inverse spectral problem for pencils of differential operators on a finite interval. Vesn. Bashkir. Univ, 4, 8-12.
• [20] Koyunbakan H., Panakhov E. S., (2007), Half-inverse problem for diffusion operators on the finite interval.Journal of Mathematical Analysis and Applications, 326, 1024-1030.
• [21] Yang C. F.,(2010), Reconstruction of the diffusion operator from nodal data Zeitschrift fur Naturforschung, 65, 100-106.
• [22] Amirov R. Kh., Ergun A., (2015), ˙Iki noktada s¨ureksizlik kosullarına sahip difuzyon denkleminin cozumleri icin integral gosterilim, Cumhuriyet Universty Science Journal, doi:10.17776/csj.06773.