On Some Spectral Properties of Discrete Sturm-Liouville Problem

Year 2023, , 61 - 69, 29.03.2023

Ayşe Çiğdem Yar , Emrah Yılmaz , Tuba Gulsen

https://doi.org/10.33401/fujma.1242330

Abstract

Time scale theory helps us to combine differential equations with difference equations. Especially in models such as biology, medicine, and economics, since the independent variable is handled discrete, it requires us to analyze in discrete clusters. In these cases, the difference equations defined in $\mathbb{Z}$ are considered. Boundary value problems (BVP's) are used to solve and model problems in many physical areas. In this study, we examined spectral features of the discrete Sturm-Liouville problem. We have given some examples to make the subject understandable. The discrete Sturm-Liouville problem is solved by using the discrete Laplace transform. In the classical case, the discrete Laplace transform is preferred because it is a very useful method in differential equations and it is thought that the discrete Laplace transform will show similar properties. The other method obtained for the solution of this problem is the solutions obtained according to the states of the characteristic equation and $\lambda$ parameter. In this solution, discrete Wronskian and Cramer methods are used.

Keywords

Discrete analysis, Discrete Sturm-Liouville equation, Laplace Transform

References

• [1] S. Hilger, Ein Masskettenkalk¨ul mit Anwendung auf Zentrumsmannigfaltigkeiten, Ph.D. Thesis, Universit¨at Würzburg, 1988.
• [2] M. Bohner, A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, (MA): Birkhauser Boston, Boston, 2001.
• [3] M. Bohner, S. Georgiev, Multivariable Dynamic Calculus on Time Scales, Springer, Berlin, 2016.
• [4] F. M. Atıcı, G.Sh. Guseniov, On Green’s functions and positive solutions for boundary value problems on time scales, J. Comput. Appl. Math., 141(1-2) (2002), 75-99.
• [5] G. Sh. Guseinov, Integration on time scales, J. Math. Anal. Appl., 285(1) (2003), 107-127.
• [6] C. J. Chyan, J. M. Davis, J. Henderson, W.K.C. Yin, Eigenvalue Comparisons for Differential Equations on a Measure Chain, Electron. J. Differential Equations, 1998(35) (1998), 1-7.
• [7] R. P. Agarwal, M. Bohner, P.J.Y. Wong, Sturm-Liouville eigenvalue problems on time scales, Appl. Math. Comput. 99(2-3) (1999), 153-166.
• [8] G. D. Birkhoff, General theory of linear difference equations, Trans. Amer. Math. Soc., 12 (1911), 243-284.
• [9] M.A. Evgrafov, The asymptotic behavior of solutions of difference equations, Proc. USSR Acad. Sci., 121(1) (1958), 26-29.
• [10] S. Elaydi, An Introduction to Difference Equations, Springer-Verlag, New York, 1999.
• [11] C. Goodrich, A.C. Peterson, Discrete Fractional Calculus, Springer, New York, 2015.
• [12] H. Koyunbakan, Reconstruction of potential in Discrete Sturm–Liouville Problem, Qual. Theory Dyn. Syst., 21(13) (2002), 1-7.
• [13] A. Jirari, Second order Sturm-Liouville difference equations and orthogonal polynomials, Mem. Amer. Math. Soc., 113(542) (1995), 1-138.
• [14] G. Shi, H. Wu, Spectral theory of Sturm-Liouville difference operators, Linear Algebra Appl., 430(2-3) (2009), 830-846.
• [15] D. B. Hinton, R.T. Lewis, Spectral Analysis of second-order difference equations, J. Math. Anal. Appl., 63(2) (1978), 421-438.
• [16] M. R. S. Kulebovic, O. Merino, Discrete dynamical systems and difference equations with Mathematica, Chapman-Hall, 344 (2002).
• [17] W. G. Kelley, A.C. Peterson, Difference Equations an Introduction with Applications, Academic Press, Cambridge, 2000.
• [18] L.H. Kauffman, H.P. Noyes, Discrete physics and the Dirac equation, Phys. Lett. A, 218(3-6) (1996), 139-146.
• [19] M. Bohner, T. Cuchta, The Bessel difference equation, Proc. Amer. Math. Soc., 145(4) (2017), 1567-1580.
• [20] E. Bairamov, S¸ . Solmaz, Spectrum and scattering function of the impulsive discrete Dirac systems, Turk. J. Math., 42(6) (2018), 3182-3194.
• [21] E. Bairamov, Y. Aygar, D. Karslıo˘glu, Scattering analysis and spectrum of discrete Schr¨odinger equations with transmission conditions, Filomat, 31(17) (2017), 5391-5399.
• [22] E. Bairamov, Y. Aygar, S. Cebesoy, Spectral analysis of a selfadjoint matrix-valued discrete operator on the whole axis, J. Nonlinear Sci. Appl., 9(6) (2016), 4257-4262.
• [23] A. Akbulut, E. Bairamov, Discrete spectrum of a general quadratic pencil of Schrodinger equations, Indian J. Pure Appl. Math., 37(5) (2006), 307-316.
• [24] R. Ameen, H. K¨ose, F. Jarad, On the Discrete Laplace transform, Results in Nonlinear Anal., 2(2) (2019), 61-70.
• [25] T. K¨opr¨ubas¸ı, The cubic eigenparameter dependent discrete Dirac equations with principal functions, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat, 68(2) (2019), 1742-1760.
• [26] N. Cos¸kun, Non-selfadjoint finite system of Discrete Sturm-Liouville operators with hyperbolic eigenparameter, C¸ ankaya Univ. J. Sci. Eng., 19(2) (2022), 62-69.
• [27] G. Mutlu, Spectrum of discrete Sturm-Liouville equation with self-adjoint operator coefficients on the half-line, J. Instit. Sci. Technol., 11(4) (2021), 3055-3062.
• [28] A. A. Nabiev, M. G¨urdal, On the solution of an infinite system of discrete equations, Turkish J. Math. Comput. Sci., 12(2) (2020), 157-160.
• [29] Y. Aygar, Investigation of spectral analysis of matrix quantum difference equations with spectral singularities, Hacet. J. Math. Stat., 45(4) (2016), 999-1005.