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Year 2019, Volume: 7 Issue: 1, 128 - 135, 15.04.2019

Abstract

References

  • [1] R. P. Agarwal, M. Bohner and D. O’Regan, Time scale boundary value problems on infinite intervals, J. Comput. Appl. Math., 141 (2002), 27-34.
  • [2] B. P. Allahverdiev and H. Tuna, An expansion theorem for q-Sturm-Liouville operators on the whole line, Turk J Math, 42, (2018), 1060-1071.
  • [3] B. P. Allahverdiev and H. Tuna, Spectral expansion for the singular Dirac system with impulsive conditions, Turk J Math, 42, (2018), 2527 – 2545.
  • [4] D. R. Anderson, G. Sh. Guseinov and J. Hoffacker, Higher-order self-adjoint boundary-value problems on time scales, J. Comput. Appl. Math., 194 (2) (2006) ; 309-342.
  • [5] F. Atici Merdivenci and G. Sh. Guseinov, On Green’s functions and positive solutions for boundary value problems on time scales, J. Comput. Appl. Math., 141 (1-2) (2002); 75-99.
  • [6] M. Bohner and A. Peterson, Dynamic Equations on Time Scales, Birkh¨auser, Boston, 2001.
  • [7] M. Bohner and A. Peterson, (Eds.), Advances in Dynamic Equations on Time Scales, Birkh¨auser, Boston, 2003.
  • [8] T. Gulsen and E. Yilmaz, Spectral theory of Dirac system on time scales, Applicable Analysis, 96(16), (2017), 2684–2694.
  • [9] G. Sh. Guseinov, Self-adjoint boundary value problems on time scales and symmetric Green’s functions, Turkish J. Math., 29 (4); (2005) ; 365􀀀380.
  • [10] G. Sh. Guseinov, Eigenfunction expansions for a Sturm-Liouville problem on time scales. Int. J. Difference Equ. 2 (2007), no. 1, 93–104.
  • [11] G. Sh. Guseinov, An expansion theorem for a Sturm-Liouville operator on semi-unbounded time scales. Adv. Dyn. Syst. Appl. 3 (2008), no. 1, 147–160.
  • [12] S. Hilger, Analysis on measure chains-a unified approach to continuous and discrete calculus, Results Math. 18; (1990); 18􀀀56:
  • [13] G. Hovhannisyan, On Dirac equation on a time scale, Journal of Math. Physics, 52, no.10, 102701, 2011.
  • [14] A. N. Kolmogorov and S. V. Fomin, Introductory Real Analysis. Translated by R.A. Silverman, Dover Publications, New York, 1970.
  • [15] V. Lakshmikantham, S. Sivasundaram and B. Kaymakcalan, Dynamic Systems on Measure Chains, Kluwer Academic Publishers, Dordrecht, 1996.
  • [16] B. M. Levitan and I. S. Sargsjan, Sturm-Liouville and Dirac Operators. Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht, 1991 (translated from the Russian).
  • [17] B. P. Rynne, L2 spaces and boundary value problems on time-scales, J. Math. Anal. Appl. 328; (2007); 1217􀀀1236.
  • [18] B. Thaller, The Dirac Equation, Springer, 1992.
  • [19] E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations. Part I. Second Edition Clarendon Press, Oxford, 1962.
  • [20] J. Weidmann, Spectral Theory of Ordinary Differential Operators, Lecture Notes in Mathematics, 1258, Springer, Berlin 1987.

Eigenfunction Expansion in the Singular Case for Dirac Systems on Time Scales

Year 2019, Volume: 7 Issue: 1, 128 - 135, 15.04.2019

Abstract

In this work, we prove the existence of a spectral function for one dimensional singular Dirac operator on time scales. Further, we establish a Parseval equality and expansion formula in eigenfunctions by terms of the spectral function.



References

  • [1] R. P. Agarwal, M. Bohner and D. O’Regan, Time scale boundary value problems on infinite intervals, J. Comput. Appl. Math., 141 (2002), 27-34.
  • [2] B. P. Allahverdiev and H. Tuna, An expansion theorem for q-Sturm-Liouville operators on the whole line, Turk J Math, 42, (2018), 1060-1071.
  • [3] B. P. Allahverdiev and H. Tuna, Spectral expansion for the singular Dirac system with impulsive conditions, Turk J Math, 42, (2018), 2527 – 2545.
  • [4] D. R. Anderson, G. Sh. Guseinov and J. Hoffacker, Higher-order self-adjoint boundary-value problems on time scales, J. Comput. Appl. Math., 194 (2) (2006) ; 309-342.
  • [5] F. Atici Merdivenci and G. Sh. Guseinov, On Green’s functions and positive solutions for boundary value problems on time scales, J. Comput. Appl. Math., 141 (1-2) (2002); 75-99.
  • [6] M. Bohner and A. Peterson, Dynamic Equations on Time Scales, Birkh¨auser, Boston, 2001.
  • [7] M. Bohner and A. Peterson, (Eds.), Advances in Dynamic Equations on Time Scales, Birkh¨auser, Boston, 2003.
  • [8] T. Gulsen and E. Yilmaz, Spectral theory of Dirac system on time scales, Applicable Analysis, 96(16), (2017), 2684–2694.
  • [9] G. Sh. Guseinov, Self-adjoint boundary value problems on time scales and symmetric Green’s functions, Turkish J. Math., 29 (4); (2005) ; 365􀀀380.
  • [10] G. Sh. Guseinov, Eigenfunction expansions for a Sturm-Liouville problem on time scales. Int. J. Difference Equ. 2 (2007), no. 1, 93–104.
  • [11] G. Sh. Guseinov, An expansion theorem for a Sturm-Liouville operator on semi-unbounded time scales. Adv. Dyn. Syst. Appl. 3 (2008), no. 1, 147–160.
  • [12] S. Hilger, Analysis on measure chains-a unified approach to continuous and discrete calculus, Results Math. 18; (1990); 18􀀀56:
  • [13] G. Hovhannisyan, On Dirac equation on a time scale, Journal of Math. Physics, 52, no.10, 102701, 2011.
  • [14] A. N. Kolmogorov and S. V. Fomin, Introductory Real Analysis. Translated by R.A. Silverman, Dover Publications, New York, 1970.
  • [15] V. Lakshmikantham, S. Sivasundaram and B. Kaymakcalan, Dynamic Systems on Measure Chains, Kluwer Academic Publishers, Dordrecht, 1996.
  • [16] B. M. Levitan and I. S. Sargsjan, Sturm-Liouville and Dirac Operators. Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht, 1991 (translated from the Russian).
  • [17] B. P. Rynne, L2 spaces and boundary value problems on time-scales, J. Math. Anal. Appl. 328; (2007); 1217􀀀1236.
  • [18] B. Thaller, The Dirac Equation, Springer, 1992.
  • [19] E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations. Part I. Second Edition Clarendon Press, Oxford, 1962.
  • [20] J. Weidmann, Spectral Theory of Ordinary Differential Operators, Lecture Notes in Mathematics, 1258, Springer, Berlin 1987.
There are 20 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Bilender P. Allahverdiev This is me

Hüseyin Tuna

Publication Date April 15, 2019
Submission Date August 3, 2018
Acceptance Date December 20, 2018
Published in Issue Year 2019 Volume: 7 Issue: 1

Cite

APA Allahverdiev, B. P., & Tuna, H. (2019). Eigenfunction Expansion in the Singular Case for Dirac Systems on Time Scales. Konuralp Journal of Mathematics, 7(1), 128-135.
AMA Allahverdiev BP, Tuna H. Eigenfunction Expansion in the Singular Case for Dirac Systems on Time Scales. Konuralp J. Math. April 2019;7(1):128-135.
Chicago Allahverdiev, Bilender P., and Hüseyin Tuna. “Eigenfunction Expansion in the Singular Case for Dirac Systems on Time Scales”. Konuralp Journal of Mathematics 7, no. 1 (April 2019): 128-35.
EndNote Allahverdiev BP, Tuna H (April 1, 2019) Eigenfunction Expansion in the Singular Case for Dirac Systems on Time Scales. Konuralp Journal of Mathematics 7 1 128–135.
IEEE B. P. Allahverdiev and H. Tuna, “Eigenfunction Expansion in the Singular Case for Dirac Systems on Time Scales”, Konuralp J. Math., vol. 7, no. 1, pp. 128–135, 2019.
ISNAD Allahverdiev, Bilender P. - Tuna, Hüseyin. “Eigenfunction Expansion in the Singular Case for Dirac Systems on Time Scales”. Konuralp Journal of Mathematics 7/1 (April 2019), 128-135.
JAMA Allahverdiev BP, Tuna H. Eigenfunction Expansion in the Singular Case for Dirac Systems on Time Scales. Konuralp J. Math. 2019;7:128–135.
MLA Allahverdiev, Bilender P. and Hüseyin Tuna. “Eigenfunction Expansion in the Singular Case for Dirac Systems on Time Scales”. Konuralp Journal of Mathematics, vol. 7, no. 1, 2019, pp. 128-35.
Vancouver Allahverdiev BP, Tuna H. Eigenfunction Expansion in the Singular Case for Dirac Systems on Time Scales. Konuralp J. Math. 2019;7(1):128-35.
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