On the solution of a Sturm-Liouville problem by using Laplace transform on time scales

Year 2021, Volume: 42 Issue: 1, 132 - 140, 29.03.2021

Emrah Yılmaz Sertaç Göktaş

https://doi.org/10.17776/csj.831443

Cited By: 2

Abstract

In this study, we solve a Sturm-Liouville problem on time scales with constant graininess by using Laplace transform which is one of the finest representatives of integral transformation used in applied mathematics. Eigenfunctions on the time scale were obtained in different cases with the Laplace transform. Thus, it was seen that the Laplace transform is an effective method on time scales. The results that will contribute to the spectral theory were obtained on the time scale with the examples discussed. It is very interesting that the results obtained differ as the time scale changes and this transformation can be applied to other types of problems. The problems that were established and solved enabled the subject to be understood on the time scale.

Keywords

Time scales, Sturm-Liouville problem, Laplace transform

References

• [1] Schiff J. L., The Laplace transform: theory and applications (Undergraduate Texts in Mathematics Springer: New York: Springer, (1999).
• [2] Hilger S., Special functions, Laplace and Fourier transform on measure chains, Dynamic Systems and Applications, 8 (3-4) (1999) 471-488.
• [3] Bohner M., Peterson A., Laplace transform and Z-transform: Unification and Extension, Methods and Applications of Analysis, 9 (1) (2002) 151-158.
• [4] Ameen R., Kose H., Jarad F., On the discrete Laplace transform, Results in Nonlinear Analysis, 2 (2) (2019) 61-70.
• [5] Bohner M., Guseinov G. Sh., The convolution on time scales, Abstract and Applied Analysis, (2007) 1-24.
• [6] Bohner M., Guseinov G. Sh., The Laplace transform on isolated time scales, Computers and Mathematics with Applications, 60(6) (2010) 1536-1547.
• [7] Bohner M. Guseinov G. Sh., The h-Laplace and q-Laplace transforms, Journal of Mathematical Analysis and Applications, 365 (1) (2010) 75-92.
• [8] Bohner M., Guseinov G. Sh., Karpuz B., Properties of the Laplace transform on time scales with arbitrary graininess, Integral Transforms and Special Functions, 22 (11) (2011) 785-800.
• [9] Bohner M., Guseinov G. Sh., Peterson A., Further properties of the Laplace transform on time scales with arbitrary graininess, Integral Transforms and Special Functions, 24 (4) (2013) 289-301.
• [10] Davis J. M., Gravagne I. A., Jackson B. J. , Marks II, R. J. and Ramos A. A., The Laplace transform on time scales revisited, Journal of Mathematical Analysis and Applications, 332 (2) (2007) 1291-1307.
• [11] Davis J. M., Gravagne I. A., Marks II, R. J., Convergence of unilateral Laplace transforms on time scales, Circuits, Systems and Signal Processing, 29 (5) (2010) 971-997.
• [12] Davis J. M., Gravagne I. A., Marks II, R. J., Bilateral Laplace Transforms on Time Scales: Convergence, Convolution, and the Characterization of Stationary Stochastic Time Series, Circuits, Systems and Signal Processing, 29 (1) (2010) 1141-1165.
• [13] Ortigueira M. D., Torres D. F., Trujillo J. J., Exponentials and Laplace transforms on nonuniform time scales, Communications in Nonlinear Science and Numerical Simulation, 39 (2016) 252-270.
• [14] Rahmat M. R. S., The (q; h)-Laplace transform on discrete time scales, Computers & Mathematics with Applications, 62 (2011) 272-281.
• [15] Hilger S., Ein Masskettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten. Ph. D. thesis Universitat Würzburg, (1988).
• [16] Hilger S., Analysis on measure chains-a unified approach to continuous and discrete Calculus, Results in Mathematics, 18 (1-2) (1990) 18-56.
• [17] Bohner M., Peterson A., Dynamic equations on time scales, an introduction with applications, Boston (MA):Birkauser, Boston, (2001).
• [18] Bohner M., Peterson A., Advances in dynamic equations on time scales, Boston (MA): Birkauser, Boston, (2003).
• [19] Bairamov E., Aygar Y., Karslıoglu D., Scattering analysis and spectrum of discrete Schrödinger equations with transmission conditions, Filomat, 31 (17) (2017) 5391-5399.
• [20] Chatzarakis G. E., Jadlovska. I., Difference equations with several non-monotone deviating arguments: Iterative oscillation tests, Dynamic Systems and Applications, 27 (2) (2018) 271-298.
• [21] Huseynov A., Bairamov, E., On expansions in eigenfunctions for second order dynamic equations on time scales, Nonlinear Dynamics and Systems Theory , 9 (1) (2009) 77-88.
• [22] Keskin B., Ozkan A.S., Inverse Nodal Problems for Impulsive Sturm-Liouville Equation with Boundary Conditions Depending on the Parameter, Advances in Analysis, 2 (2017).
• [23] Ozkan A. S., Adalar I., Half-inverse Sturm-Liouville problem on a time scale, Inverse Problems, 36 (2020) 1-8.
• [24] Ozkan A. S., Keskin B., Inverse nodal problems for Sturm-Liouville equation with eigenparameter-dependent boundary and jump conditions, Inverse Problem in Science and Engineering, 23 (2015) 1306-1312.
• [25] Zettl, A., Sturm-Liouville Theory, American Mathematical Society, (2005).
• [26] Levitan B. M., Sargsyan I. S., Introduction to Spectral Theory, American Mathematical Society, Rhode Island, (1975).