Self-Adjoint Sturm-Liouville Dynamic Problem via Proportional Derivative

Year 2023, , 2945 - 2957, 01.12.2023

Tuba Gülşen , Mehmet Acar

https://doi.org/10.21597/jist.1313391

Abstract

The concept of a conformable derivative on time scales is a relatively new development in the field of fractional calculus. Traditional fractional calculus deals with derivatives and integrals of non-integer order on continuous time domains. However, time scale calculus extends these concepts to more general time domains that include both continuous and discrete points. The conformable derivative on time scales has several properties that make it advantageous in certain applications. For example, it satisfies a chain rule and has a simple relationship with the conformable integral, which facilitates the development of differential equations involving fractional order dynamics. It also allows for the analysis of systems with both continuous and discrete data points, making it suitable for modeling and control applications in various fields, including physics, engineering, and finance. In this study, the Sturm-Liouville problem and its properties are examined on an arbitrary time scale using the proportional derivative, a more general form of the fractional derivative. Important spectral properties such as self-adjointness, Green formula, Lagrange identity, Abel formula, and orthogonality of eigenfunctions for this problem are expressed in proportional derivatives on an arbitrary time scale.

Keywords

Time Scale Calculus, Proportional derivative, Spectral theory, Sturm-Liouville equation

References

• Abdeljewad, T. (2015). On conformable fractional calculus. Journal of Computational and Applied Mathematics, 279, 57–66.
• Agarwal, R., Bohner, M., O'Regan, D., & Peterson, A. (2002). Dynamic equations on time scales: a survey. Journal of Computational and Applied Mathematics, 141(1-2), 1-26.
• Anderson, D. R., & Georgiev, S. G. (2020). Conformable Dynamic Equations on Time Scales. Chapman and Hall/CRC.
• Anderson, D. R., & Ulness, D. J. (2015). Newly defined conformable derivatives. Advances in Dynamical Systems and Applications, 10(2), 109-137.
• Aulbach, B., & Hilger. S. (1990). A unified approach to continuous and discrete Dynamics. in: Qualitative Theory of Differential Equations (Szeged, 1988), 37–56, Colloq. Math. Soc. János Bolyai, 53 North-Holland, Amsterdam.
• Benkhettou, N., Brito da Cruz, A. M. C., & Torres, D. F. M. (2015). A fractional calculus on arbitrary time scales: Fractional differentiation and fractional integration. Signal Processing, 107, 230– 237.
• Benkhettou, N., Hassani, S., & Torres, D. F. M. (2016). A conformable fractional calculus on arbitrary time scales. Journal of King Saud University (Science), 28(1), 93-98.
• Bohner, M., & Peterson, A. (2001). Dynamic equations on time scales, An introduction with applications. Boston, MA: Birkhauser.
• Bohner, M., & Peterson, A. (2004). Advances in Dynamic Equations on Time Scales. Boston: Birkhauser.
• Bohner, M., & Svetlin, G. (2016). Multivariable dynamic calculus on time scales. Springer.
• Gulsen, T., Yilmaz, E., & Goktas, S. (2017). Conformable fractional Dirac system on time scales. Journal of Inequalities and Applications, 2017(1), 161.
• Gülşen, T., Yilmaz, E., & Kemaloğlu, H. (2018). Conformable fractional Sturm-Liouville equation and some existence results on time scales. Turkish Journal of Mathematics, 42(3), 1348-1360.
• Hilger, S. (1990). Analysis on measure chains a unified approach to continuous and discrete calculus. Results in mathematics, 18(1).
• Katugampola, U. (2014). A new fractional derivative with classical properties, arXiv:1410.6535v2.
• Khalil, R., Horani, M. Al., Yousef, A., & Sababheh, M. (2014). A new definition of fractional derivative. Journal of Computational and Applied Mathematics, 264, 57–66.
• Li, Y., Ang, K. H., Chong, G. C. (2006). PID control system analysis and design. IEEE Control Systems Magazine, 26(1), 32-41.
• Ortigueira, M. D., & Machado, J. T. (2015). What is a fractional derivative?. Journal of computational Physics, 293, 4-13.
• Segi Rahmat, M. R. (2019). A new definition of conformable fractional derivative on arbitrary time scales. Advances in Difference Equations, 2019 (1), 1-16.
• Yilmaz, E., Gulsen, T., & Panakhov, E. S. (2022). Existence Results for a Conformable Type Dirac System on Time Scales in Quantum Physics, Applied and Computational Mathematics an International Journal, 21(3), 279-291.