METHOD OF VARIATION OF PARAMETERS FOR THE THIRD-ORDER LINEAR PROPORTIONAL DYNAMIC EQUATIONS

Year 2023, Volume: 6 Issue: 2, 135 - 144, 31.12.2023

Tuba Gülşen Mehmet Acar

https://doi.org/10.55930/jonas.1321901

Abstract

The differentiation and integration of an integer order are known as fractional calculus. It is possible to think of the proportional derivative as a generalization of a congruent fractional derivative, which is one of the types of fractional calculus. In this article, utilizing the proportional derivative and its characteristics on the time scale, the method of variation of parameters for the third-order linear nonhomogeneous differential equations is given. Then, an example is provided to illustrate how to apply the provided approach.

Keywords

Time scales, variation of parameters, proportional derivative

Thanks

This research is part of the second author’s master’s thesis, which is carried out at Firat University, Türkiye.

References

• 1. Abdeljewad, T. (2015). On conformable fractional calculus. Journal of Computational and Applied Mathematics, 279, 57–66.
• 2. 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.
• 3. Anderson, D.R. & Georgiev, S.G. (2020). Conformable Dynamic Equations on Time Scales. Chapman and Hall/CRC.
• 4. Anderson, D.R. & Ulness, D.J. (2015). Newly defined conformable derivatives. Advances in Dynamical Systems and Applications, 10(2), 109-137.
• 5. 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.
• 6. 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.
• 7. 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.
• 8. Bohner, M. & Peterson, A. (2001). Dynamic equations on time scales, An introduction with applications. Boston, MA: Birkhauser.
• 9. Bohner, M. & Peterson, A. (2004). Advances in Dynamic Equations on Time Scales. Boston: Birkhauser.
• 10. Bohner, M. & Svetlin, G. (2016). Multivariable dynamic calculus on time scales. Cham: Springer.
• 11. Gulsen, T., Yilmaz, E. & Goktas, S. (2017). Conformable fractional Dirac system on time scales. Journal of Inequalities and Applications, 2017(1), 161.
• 12. 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.
• 13. Hilger, S. (1990). Analysis on measure chains a unified approach to continuous and discrete calculus. Results in Mathematics, 18(1).
• 14. Katugampola, U. (2014). A new fractional derivative with classical properties, arXiv:1410.6535v2.
• 15. 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.
• 16. Li, Y., Ang, K. H. & Chong, G. C. (2006). PID control system analysis and design. IEEE Control Systems Magazine, 26(1), 32-41.
• 17. Ortigueira, M. D. & Machado, J. T. (2015). What is a fractional derivative?. Journal of Computational Physics, 293, 4-13.
• 18. Segi Rahmat, M. R. (2019). A new definition of conformable fractional derivative on arbitrary time scales. Advances in Difference Equations, 2019 (1), 1-16.
• 19. 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.