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Eigenfunction expansion for impulsive singular Hahn–Dirac system

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Abstract

In this study, a singular impulsive Hahn–Dirac system is studied. A spectral function has been established for this type of system. With the help of this function, the Parseval equation and eigenfunction expansion were obtained.

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References

  1. Allahverdiev, B.P., Tuna, H.: The spectral expansion for the Hahn–Dirac system on the whole line. Turk. J. Math. 43, 1668–1687 (2019)

    Article  MathSciNet  Google Scholar 

  2. Allahverdiev, B.P., Tuna, H.: The Parseval equality and expansion formula for singular Hahn–Dirac system. In: Alparslan Gök, S., Aruğaslan Çinçin, D. (eds.) Emerging Applications of Differential Equations and Game Theory, pp. 209–235. IGI Global, Hershey (2020). https://doi.org/10.4018/978-1-7998-0134-4.ch010

  3. Allahverdiev, B.P., Tuna, H.: Spectral analysis of Hahn–Dirac system. Proyecciones (Antofagasta, On line) 40(6), 1547–1567 (2021)

    Article  MathSciNet  Google Scholar 

  4. Annaby, M.H., Hamza, A.E., Aldwoah, K.A.: Hahn difference operator and associated Jackson–Nörlund integrals. J. Optim. Theory Appl. 154, 133–153 (2012)

    Article  MathSciNet  Google Scholar 

  5. Annaby, M.H., Hamza, A.E., Makharesh, S.D.: A Sturm–Liouville theory for Hahn difference operator. In: Li, X., Nashed, Z. (eds.) Frontiers of Orthogonal Polynomials and \(q\)-Series, pp. 35–84. World Scientific, Singapore (2018)

    Chapter  Google Scholar 

  6. Amirov, Kh.R., Ozkan, A.S.: Discontinuous Sturm–Liouville problems with eigenvalue dependent boundary condition. Math. Phys. Anal. Geom. 17(3–4), 483–491 (2014)

    Article  MathSciNet  Google Scholar 

  7. Aydemir, K., Olğar, H., Mukhtarov, O.S.: The principal eigenvalue and the principal eigenfunction of a boundary-value-transmission problem. Turk. J. Math. Comput. Sci. 11(2), 97–100 (2019)

    Google Scholar 

  8. Aydemir, K., Olgar, H., Mukhtarov, O.S., Muhtarov, F.: Differential operator equations with interface conditions in modified direct sum spaces. Filomat 32(3), 921–931 (2018)

    Article  MathSciNet  Google Scholar 

  9. Aygar, Y., Bairamov, E.: Scattering theory of impulsive Sturm–Liouville equation in quantum calculus. Bull. Malays. Math. Sci. Soc. 42, 3247–3259 (2019). https://doi.org/10.1007/s40840-018-0657-2

    Article  MathSciNet  Google Scholar 

  10. Bohner, M., Cebesoy, S.: Spectral analysis of an impulsive quantum difference operator. Math. Methods Appl. Sci. 42, 5331–5339 (2019). https://doi.org/10.1002/mma.5348

    Article  MathSciNet  Google Scholar 

  11. Guseinov, G.S.: An expansion theorem for a Sturm–Liouville operator on semi-unbounded time scales. Adv. Dyn. Syst. Appl. 3(1), 147–160 (2008)

    MathSciNet  Google Scholar 

  12. Guseinov, G.S.: Eigenfunction expansions for a Sturm–Liouville problem on time scales. Int. J. Differ. Equ. 2(1), 93–104 (2007)

    MathSciNet  Google Scholar 

  13. Hahn, W.: Beitraäge zur theorie der Heineschen Reihen. Math. Nachr. 2, 340–379 (1949). ((in German))

    Article  MathSciNet  Google Scholar 

  14. Hahn, W.: Ein Beitrag zur theorie der orthogonalpolynome. Monatsh. Math. 95, 19–24 (1983)

    Article  MathSciNet  Google Scholar 

  15. Hira, F.: Dirac system associated with Hahn difference operator. Bull. Malays. Math. Sci. Soc. 43, 3481–3497 (2020)

    Article  MathSciNet  Google Scholar 

  16. Huseynov, A., Bairamov, E.: On expansions in eigenfunctions for second order dynamic equations on time scales. Nonlinear Dyn. Syst. Theory 9(1), 77–88 (2009)

    MathSciNet  Google Scholar 

  17. Karahan, D., Mamedov, K.R.: On a \(q\)-boundary value problem with discontinuity conditions. Vestn. Yuzhno-Ural. Gos. Un-ta. Ser. Matem. Mekh. Fiz. 13(4), 5–12 (2021)

    Google Scholar 

  18. Karahan, D., Mamedov, K.R.: On a \(q\)-analogue of the Sturm–Liouville operator with discontinuity conditions. Vestn. Samar. Gos. Tekh. Univ. Ser. Fiz.-Mat. Nauk. 26(3), 407–418 (2022)

    Article  Google Scholar 

  19. Karahan, D., Mamedov, K.R.: Sampling theory associated with \(q\)-Sturm–Liouville operator with discontinuity conditions. J. Contemp. Appl. Math. 10(2), 40–48 (2020)

    Google Scholar 

  20. Koyunbakan, H., Panakhov, E.S.: Solution of a discontinuous inverse nodal problem on a finite interval. Math. Comput. Model. 44(1–2), 204–209 (2006)

    Article  MathSciNet  Google Scholar 

  21. Levitan, B.M., Sargsjan, I.S.: Sturm–Liouville and Dirac Operators. Mathematics and Its Applications (Soviet Series), Kluwer Academic Publishers Group, Dordrecht (1991). ((translated from the Russian))

    Book  Google Scholar 

  22. Naimark, M.A.: Linear Differential Operators, 2nd edn. Nauka, Moscow (1969). (English transl. of 1st. edn., 1,2, New York (1968))

    Google Scholar 

  23. Ozkan, A.S., Amirov, R.K.: An interior inverse problem for the impulsive Dirac operator. Tamkang J. Math. 42(3), 259–263 (2011)

    Article  MathSciNet  Google Scholar 

  24. Titchmarsh, E.C.: Eigenfunction Expansions Associated with Second-Order Differential Equations, Part I, 2nd edn. Clarendon Press, Oxford (1962)

    Google Scholar 

  25. Wang, Y.P., Koyunbakan, H.: On the Hochstadt–Lieberman theorem for discontinuous boundary-valued problems. Acta Math. Sin. Engl. Ser. 30(6), 985–992 (2014)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Khazar University, AZ1096, Baku, Azerbaijan

    Bilender P. Allahverdiev & Hamlet A. Isayev

  2. Research Center of Econophysics, UNEC-Azerbaijan State University of Economics, Baku, Azerbaijan

    Bilender P. Allahverdiev & Hüseyin Tuna

  3. Department of Mathematics, Mehmet Akif Ersoy University, 15030, Burdur, Turkey

    Hüseyin Tuna

Authors
  1. Bilender P. Allahverdiev
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  2. Hüseyin Tuna
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  3. Hamlet A. Isayev
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Correspondence to Hüseyin Tuna.

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Allahverdiev, B.P., Tuna, H. & Isayev, H.A. Eigenfunction expansion for impulsive singular Hahn–Dirac system. Rend. Circ. Mat. Palermo, II. Ser (2024). https://doi.org/10.1007/s12215-024-01028-0

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  • DOI: https://doi.org/10.1007/s12215-024-01028-0

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