Impulsive Diffusion Equation on Time Scales

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Tuba Gulsen, Shaida Saber Mawlood Sian, Emrah Yilmaz, Hikmet Koyunbakan, Impulsive Diffusion Equation on Time Scales, Int. J. Anal. Appl., 16 (1) (2018), 137-148.

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Tuba Gulsen, Shaida Saber Mawlood Sian, Emrah Yilmaz, Hikmet Koyunbakan

Abstract

Application of boundary value problems (BVP's) on an arbitrary time scale T is a fairly new and important subject in mathematics. In this study, we deal with an eigenvalue problem for impulsive diffusion equation with boundary conditions on T. We generalize some noteworthy results about spectral theory of classical diffusion equation into T. Also, some eigenfunction estimates of the impulsive diffusion eigenvalue problem are established on T.

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References

  1. S. Hilger, Ein Maßkettenkalkül vnit Anwendung auf Zentruvnsvnannigfaltigkeiten,
  2. [Ph.D. Thesis], Universitüt Würzburg, 1988.
  3. B. Aulbach, S. Hilger, A unified approach to continuous and discrete dynamics, qualitative theory of differential equations, Szeged (1988); Colloq. Math. Soc. János Bolyai, North-Holland, Amsterdam, 53 (1990), 37-56.
  4. M. Bohner and A. Peterson, Dynamic equations on time scales: an introduction with applications, Boston (MA), Birkhäuser, Boston Inc, 2001.
  5. F. M. Atici, G. Sh. Guseinov, On Green's functions and positive solutions for boundary value problems on time scales, J. Comput. Appl. Math. 141 (2002), 75-99.
  6. E. Bairamov, Y. Aygar and T. Koprubasi, The spectrum of eigenparameter-dependent discrete Sturm-Liouville equations, J. Comput. Appl. Math. 235 (16) (2011), 4519-4523.
  7. Y. Aygar and M. Bohner, On the spectrum of eigenparameter-dependent quantum difference equations, Appl. Math. Inf. Sci. 9 (4) (2015), 1725-1729.
  8. G. Sh. Guseinov, Self-adjoint boundary value problems on time scales and symmetric Green's functions, Turk. J. Math. 29 (2005), 365-380.
  9. C. J. Chyan, J. M. Davis, J. Henderson, W. K. C. Yin, Eigenvalue comparisons for differential equations on a measure chain, Electron. J. Diff. Equ. 35 (1998), 1-7.
  10. R. P. Agarwal, M. Bohner and P. J. Y. Wong, Sturm-Liouville eigenvalue problems on time scales, Appl. Math. Comput. 99 (1999), 153-166.
  11. R. P. Agarwal, M. Bohner, D. O'Regan, Time scale boundary value problems on infinite intervals, J. Comput. Appl. Math. 141 (2002), 27-34.
  12. G. Sh. Guseinov, Eigenfunction expansions for a Sturm-Liouville problem on time scales, Int. J. Difference Equ. 2 (2007), 93-104.
  13. A. Huseynov and E. Bairamov, On expansions in eigenfunctions for second order dynamic equations on time scales, Nonlinear Dyn. Syst. Theory 9 (2009), 77-88.
  14. Y. Zhang and L. Ma, Solvability of Sturm-Liouville problems on time scales at resonance, J. Comput. Appl. Math. 233 (2010), 1785-1797.
  15. Q. G. Zhang, H. R. Sun, Variational approach for Sturm-Liouville boundary value problems on time scales, J. Appl. Math. Comput. 36 (1-2) (2011), 219-232.
  16. L. Erbe, R. Mert and A. Peterson, Spectral parameter power series for Sturm-Liouville equations on time scales, Appl. Math. Comput. 218 (2012), 7671-7678.
  17. E. Yilmaz, H. Koyunbakan and U. Ic, Some spectral properties of diffusion equation on time scales. Contemp. Anal. Appl. Math. 3 (2015), 238-246.
  18. Ë™ I. Yaslan, Existence of positive solutions for second-order impulsive boundary value problems on time scales, Mediterr. J. Math. 13 (4) (2016), 1613-1624.
  19. B. P. Allahverdiev, A. Eryilmaz and H. Tuna, Dissipative Sturm-Liouville operators with a spectral parameter in the boundary condition on bounded time scales, Electron. J. Diff. Equ. 95 (2017), 1-13.
  20. T. Gulsen and E. Yilmaz, Spectral theory of Dirac system on time scales, Appl. Anal. 96 (2017), 2684-2694.
  21. Y. Cakmak and S. Isik, Half Inverse problem for the impulsive diffusion operator with discontinuous coefficient, Filomat 30 (1) (2016), 157-168.
  22. M. Jaulent and C. Jean, The inverse s-wave scattering problem for a class of potentials depending on energy, Commun. Math. Phys. 28 (3) (1972), 177-220.
  23. A. Wazwaz, Partial differential equations: Methods and applications, Balkema Publishers, Leiden, 2002.
  24. F. G. Maksudov, M. M. Guseinov, A quadratic pencil of operators in the presence of a continuous spectrum, (Russian) Dokl. Akad. Nauk Azerbaidzhan SSR 35 (1) (1979), 9-13.
  25. M. G. Gasymov and G. Sh. Guseinov, Determination of a diffusion operator from the spectral data, Dokl. Akad. Nauk Azerbaijan SSSR 37 (2) (1981), 19-23.
  26. A. Fragela, Quadratic pencils of differential operators with integral boundary conditions, (Russian) Differential equations and their applications (Russian), 50-52, Moskov. Gos. Univ., Moscow, 1984.
  27. F. G. Maksudov, G. Sh. Guseinov, On the solution of the inverse scattering problem for the quadratic bundle of the one-dimensional Schrödinger operators on the whole axis, (Russian) Dokl. Akad. Nauk SSSR 289 (1) (1986), 42-46.
  28. E. Bairamov, ¨ O. C ¸akar and A. O. C ¸elebi, Quadratic pencil of Schrödinger operators with spectral singularities: discrete spectrum and principal functions, J. Math. Anal. Appl. 216 (1) (1997), 303-320.
  29. H. Koyunbakan, A new inverse problem for the diffusion operator, Appl. Math. Lett. 19 (10) (2006), 995-999.
  30. H. Koyunbakan, E. S. Panakhov, A uniqueness theorem for inverse nodal problem, Inverse Probl. Sci. Eng. 5 (6) (2007), 517-524.
  31. H. Koyunbakan and E. Yilmaz, Reconstruction of the potential function and its derivatives for the diffusion operator, Z. Nat. forsch. A: Phys. Sci. 63 (3-4) (2008), 127-130.
  32. C. F. Yang, Reconstruction of the diffusion operator from nodal data, Z. Nat. forsch. A: Phys. Sci. 65 (1-2) (2010), 100-106.
  33. C. F. Yang and A. Zettl, Half inverse problems for quadratic pencils of Sturm-Liouville operators, Taiwanese J. Math. 16 (5) (2012), 1829-1846.
  34. R. Hryniv and N. Pronska, Inverse spectral problems for energy-dependent Sturm-Liouville equations, Inverse Probl. 28 (8) (2012), 085008.
  35. S. A. Buterin and C. T. Shieh, Inverse nodal problem for differential pencils, Appl. Math. Lett. 22 (2009), 1240-1247.
  36. A. B. Yakhshimuratov, O. R. Allaberganov, The inverse problem for a quadratic pencil of Sturm-Liouville operators with a periodic potential on a half-axis, (Russian) Uzbek. Mat. Zh. 3 (2006), 96-107.
  37. I. M. Guseinov and I. M. Nabiev, The inverse spectral problem for pencils of differential operators, Sb. Math. 198 (11) (2007), 1579-1598.
  38. H. Koyunbakan, Inverse problem for a quadratic pencil of Sturm-Liouville operator, J. Math. Anal. Appl. 378 (2) (2011), 549-554.
  39. Y. P. Wang, The inverse problem for differential pencils with eigenparameter dependent boundary conditions from interior spectral data, Appl. Math. Lett. 25 (7) (2012), 1061-1067.
  40. R. Kh. Amirov, A. Nabiev, Inverse problems for the quadratic pencil of the Sturm-Liouville equations with impulse, Abstr. Appl. Anal. 2013 (2013), Art. ID 361989, 10 pp
  41. L. K. Sharma, P. V. Luhanga and S. Chimidza, Potentials for the Klein-Gordon and Dirac equations, Chiang Mai J. Sci. 38 (4) (2011), 514-526.
  42. K. Chadan, D. Colton, L. Paivarinta and W. Rundell, An introduction to inverse scattering and inverse spectral problems, SIAM, Philadelphia, PA, 1997.
  43. A. D. Orujov, On the spectrum of the quadratic pencil of differential operators with periodic coefficients on the semi-axis, Bound. Value Probl. 2015 (2015), Art. ID 117, 16 pp.