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Integral Representation for Solution of Discontinuous Diffusion Operator with Jump Conditions

Year 2018, Volume: 39 Issue: 4, 842 - 863, 24.12.2018

Abdullah Ergün

https://doi.org/10.17776/csj.443898
Cited By: 4

Abstract

In this study, the diffusion operator with discontinuity function and
the jump conditions is considered. Under certain initial and discontinuity
conditions, integral equations have been derived for the solutions. Integral
representations, which is too useful for this type equation, have been
presented.

Keywords

Diffusion Operator Jump condition Sturm-Liouville Equation

References

  • [1]. Jdanovich,B. F. Formulae for the zeros of Drichlet polynomials and quasi-polynomials, Doklady Akad. Nauk SSSR. 135 (1960), 1046–1049.
  • [2]. Levitan,B. M. and Gasymov,M. G. Determination of a differantial equation by two spctra, Uspekhi Mathematicheskikh Nauk. 19 (1964), 3-63.
  • [3]. Levitan,B. M. and Sargsyan ,I. S. Introduction to Spectral Theory. Amer. Math. Soc., (1975).
  • [4]. Levitan,B. M. Inverse Sturm-Liouville Problems. Nauka, Moscow, Russia, (1987).
  • [5]. Naimark, M. A. Linear Differantial Operators. London, Toronto, Harrap, (1968).
  • [6]. Yang,C. F. Reconstruction of the diffusion operator from nodal data. Zeitschrift für Naturforschung, 65 (2010), 100–106.
  • [7]. Borg,G. Eine Umkehrung der Sturm-Lİouvilleschen Eigenwertaufgabe. Acta Mathematica, 78 (1946), 1-96.
  • [8]. Freiling,G. and Yurko,V. Inverse spectral problems for singular non-selfadjoint differential operators with discontinuities in an interior point. Inverse Problems, 18 (2002), 757–773.
  • [9]. Guseinov,G. S. On the spectral analysis of a quadratic pencil of Sturm-Liouville operators. Dodlady Akad. Nauk SSSR, 285 (1985), 1292-1296.
  • [10]. Guseinov,G. S. Inverse spectral problems for a quadratic pencil of Sturm-Lİouville operators on a finite interval. Spectral Theory of Operators and Its Applications. Elm, Baku, Azerbaijan (1986) 51-101.
  • [11]. Koyunbakan,H. and Panakhov,E. S. Half-inverse problem for diffusion operators on the finite interval. J. of Math. Anal. and App., 326 (2007), 1024–1030.
  • [12]. McLaughin ,J. R. Analyical methods for recovering coefficients in differantial equations from spectral data. SIAM 28 (1986) , 53-72.
  • [13]. Aydemir,K. and Muhtaroglu,O. Asymptotic distribution of eigenvalues and eigenfunctions for a multi point discontinuous Sturm Liouville Problem. Electronic Journal Of Differential Equations, 131 (2016), 1-14.
  • [14]. Anderson,L. Inverse eigenvalue problems with discontiuous coefficients. Inverse Problems 4 (1998), 353-397.
  • [15]. Gasymov,M. G. and Guseinov,G. S. Determination of a diffusion operatör from spectral data. Akad. Nauk Azerbaidzhanskoi SSR, 37 (1981), 19-23.
  • [16]. Nabiev,M. I. Inverse periodic Problem for a diffusion operatör. Transactions of Acad. of science of Azerbaijan, 23 (2003), 125-130.
  • [17]. Nabiev,M. I. The inverse spectral problem for he diffusion operatör on an interval. Mathematicheskaya Fizika, Analiz, Geometriya, 11 (2004), 302-313.
  • [18]. Krein,M. and Levin,B. Ya. On entire almost periodic functions of exponential type. Doklady Akad. Nauk SSSR, 64 (1949), 285–287.
  • [19]. Guliyev,N. J. Inverse eigenvalue problems for Sturm–Liouville equations with spectral parameter linearly contained in one of the boundary conditions. Inverse Problems, 21:4 (2005), 1315–1330
  • [20]. Hald,O. H. Discontinuous inverse eigenvalue problems. Comm. on Pure and Appl. Math., 37 (1984), 539–577.
  • [21]. Carlson,R. An inverse spectral problem for Sturm-Liouville operators with discontinuous coefficients. Proceedings of the Amer. Math. Soc., 120 (1994), 475-484.
  • [22]. Amirov,R. Kh. and Yurko,V. A. On differential operators with a singularity and discontinuity conditions inside an interval. Ukrainian Math. J., 53 (2001), 1443–1457.
  • [23]. Amirov,R. Kh. and Nabiev,A. A. Inverse problems for quadratic pencil of the Sturm-Liouville Equations with Impulse. J. of Abst. and Anal. (2013), doi:10.1155/2013/361989.
  • [24]. Amirov,R. Kh. and Ergun,A. İki noktada süreksizlik koşullarına sahip difüzyon denkleminin çözümleri için integral gösterilim. Cumhuriyet Sci. J. 36-5 (2015), 71-85. DOI: 10.17776/csj.46267
  • [25]. Buterin,S. A. and Yurko,V. A. Inverse spectral problem for pencils of differential operators on a finite interval. Vesn. Bashkir. Univ. 4 (2006), 8–12.
  • [26]. Ambartsumyan,V. A. Über eine frage der eigenwerttheorie. Zeitschrift für Physik,53 (1929), 690-695.
  • [27]. Marchenko,V. A. Sturm-Liouville Operators and Applications. AMS Chelsea Publishing, (1986).
  • [28]. Yurko,V. A. Inverse spectral problems for linear differential operators and their applications. Gordon and Breach, New York, (2000).
  • [29]. Yurko,V. A. Introduction to the Theory of Inverse Spectral Problems. Fizmatlit, Moscow, Russia, (2007).
  • [30]. Fulton,C. T.Singular eigenvalue problems with eigenvalue parameter contained in the boundary conditions, Proc. R. Soc. Edinburgh, A87, (1980),1-34.

Sıçrama Şartlarına Sahip Süreksiz Difüzyon Operatörünün Çözümleri için Integral Gösterilim

Year 2018, Volume: 39 Issue: 4, 842 - 863, 24.12.2018

Abdullah Ergün

https://doi.org/10.17776/csj.443898
Cited By: 4

Abstract

Bu makalede süreksizlik fonksiyonuna ve sıçrama şartlarına sahip
difüzyon operatörü incelenmiştir. Belirli başlangıç ve süreksizlik koşulları
altında çözümler için integral denklemler üretilmiştir. Bu tip difüzyon
operatörler için oldukça kullanışlı olan integral gösterilimler elde
edilmiştir.

Keywords

Difüzyon Operatör Sıçrama şartı Sturm- Liouville denklemi

References

  • [1]. Jdanovich,B. F. Formulae for the zeros of Drichlet polynomials and quasi-polynomials, Doklady Akad. Nauk SSSR. 135 (1960), 1046–1049.
  • [2]. Levitan,B. M. and Gasymov,M. G. Determination of a differantial equation by two spctra, Uspekhi Mathematicheskikh Nauk. 19 (1964), 3-63.
  • [3]. Levitan,B. M. and Sargsyan ,I. S. Introduction to Spectral Theory. Amer. Math. Soc., (1975).
  • [4]. Levitan,B. M. Inverse Sturm-Liouville Problems. Nauka, Moscow, Russia, (1987).
  • [5]. Naimark, M. A. Linear Differantial Operators. London, Toronto, Harrap, (1968).
  • [6]. Yang,C. F. Reconstruction of the diffusion operator from nodal data. Zeitschrift für Naturforschung, 65 (2010), 100–106.
  • [7]. Borg,G. Eine Umkehrung der Sturm-Lİouvilleschen Eigenwertaufgabe. Acta Mathematica, 78 (1946), 1-96.
  • [8]. Freiling,G. and Yurko,V. Inverse spectral problems for singular non-selfadjoint differential operators with discontinuities in an interior point. Inverse Problems, 18 (2002), 757–773.
  • [9]. Guseinov,G. S. On the spectral analysis of a quadratic pencil of Sturm-Liouville operators. Dodlady Akad. Nauk SSSR, 285 (1985), 1292-1296.
  • [10]. Guseinov,G. S. Inverse spectral problems for a quadratic pencil of Sturm-Lİouville operators on a finite interval. Spectral Theory of Operators and Its Applications. Elm, Baku, Azerbaijan (1986) 51-101.
  • [11]. Koyunbakan,H. and Panakhov,E. S. Half-inverse problem for diffusion operators on the finite interval. J. of Math. Anal. and App., 326 (2007), 1024–1030.
  • [12]. McLaughin ,J. R. Analyical methods for recovering coefficients in differantial equations from spectral data. SIAM 28 (1986) , 53-72.
  • [13]. Aydemir,K. and Muhtaroglu,O. Asymptotic distribution of eigenvalues and eigenfunctions for a multi point discontinuous Sturm Liouville Problem. Electronic Journal Of Differential Equations, 131 (2016), 1-14.
  • [14]. Anderson,L. Inverse eigenvalue problems with discontiuous coefficients. Inverse Problems 4 (1998), 353-397.
  • [15]. Gasymov,M. G. and Guseinov,G. S. Determination of a diffusion operatör from spectral data. Akad. Nauk Azerbaidzhanskoi SSR, 37 (1981), 19-23.
  • [16]. Nabiev,M. I. Inverse periodic Problem for a diffusion operatör. Transactions of Acad. of science of Azerbaijan, 23 (2003), 125-130.
  • [17]. Nabiev,M. I. The inverse spectral problem for he diffusion operatör on an interval. Mathematicheskaya Fizika, Analiz, Geometriya, 11 (2004), 302-313.
  • [18]. Krein,M. and Levin,B. Ya. On entire almost periodic functions of exponential type. Doklady Akad. Nauk SSSR, 64 (1949), 285–287.
  • [19]. Guliyev,N. J. Inverse eigenvalue problems for Sturm–Liouville equations with spectral parameter linearly contained in one of the boundary conditions. Inverse Problems, 21:4 (2005), 1315–1330
  • [20]. Hald,O. H. Discontinuous inverse eigenvalue problems. Comm. on Pure and Appl. Math., 37 (1984), 539–577.
  • [21]. Carlson,R. An inverse spectral problem for Sturm-Liouville operators with discontinuous coefficients. Proceedings of the Amer. Math. Soc., 120 (1994), 475-484.
  • [22]. Amirov,R. Kh. and Yurko,V. A. On differential operators with a singularity and discontinuity conditions inside an interval. Ukrainian Math. J., 53 (2001), 1443–1457.
  • [23]. Amirov,R. Kh. and Nabiev,A. A. Inverse problems for quadratic pencil of the Sturm-Liouville Equations with Impulse. J. of Abst. and Anal. (2013), doi:10.1155/2013/361989.
  • [24]. Amirov,R. Kh. and Ergun,A. İki noktada süreksizlik koşullarına sahip difüzyon denkleminin çözümleri için integral gösterilim. Cumhuriyet Sci. J. 36-5 (2015), 71-85. DOI: 10.17776/csj.46267
  • [25]. Buterin,S. A. and Yurko,V. A. Inverse spectral problem for pencils of differential operators on a finite interval. Vesn. Bashkir. Univ. 4 (2006), 8–12.
  • [26]. Ambartsumyan,V. A. Über eine frage der eigenwerttheorie. Zeitschrift für Physik,53 (1929), 690-695.
  • [27]. Marchenko,V. A. Sturm-Liouville Operators and Applications. AMS Chelsea Publishing, (1986).
  • [28]. Yurko,V. A. Inverse spectral problems for linear differential operators and their applications. Gordon and Breach, New York, (2000).
  • [29]. Yurko,V. A. Introduction to the Theory of Inverse Spectral Problems. Fizmatlit, Moscow, Russia, (2007).
  • [30]. Fulton,C. T.Singular eigenvalue problems with eigenvalue parameter contained in the boundary conditions, Proc. R. Soc. Edinburgh, A87, (1980),1-34.
There are 30 citations in total.

Details

Primary Language English
Journal Section Natural Sciences
Authors

Abdullah Ergün

Publication Date December 24, 2018
Submission Date July 14, 2018
Acceptance Date October 23, 2018
Published in Issue Year 2018Volume: 39 Issue: 4

Cite

APA Ergün, A. (2018). Integral Representation for Solution of Discontinuous Diffusion Operator with Jump Conditions. Cumhuriyet Science Journal, 39(4), 842-863. https://doi.org/10.17776/csj.443898

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