Falling Body Motion in Time Scale Calculus

Year 2024, , 210 - 224, 28.03.2024

Neslihan Nesliye Pelen , Zeynep Kayar

https://doi.org/10.54287/gujsa.1427944

Abstract

The falling body problem for different time scales, such as ℝ, ℤ, hℤ, qℕ0, ℙc,d is the subject of this study. To deal with this problem, we use time-scale calculus. Time scale dynamic equations are used to define the falling body problem. The exponential time scale function is used for the solutions of these problems. The solutions of the falling body problem in each of these time scales are found. Moreover, we also test our mathematical results with numerical simulations.

Keywords

Time Scale Calculus, Falling Body Motion, Delta Derivative, H-Calculus, Q-Calculus

References

• Akın, E. & Bohner, M. (2003). Miscellaneous Dynamic Equations. Methods and Applications of Analysis, 10(1), pp.11-30. https://dx.doi.org/10.4310/MAA.2003.v10.n1.a2
• Akın, E., Pelen, N. N., Tiryaki I. U., & Yalcin, F. (2020). Parameter identification for gompertz and logistic dynamic equations. Plos One, 15(4): e0230582. https://doi.org/10.1371/journal.pone.0230582
• Alanazi, A. M., Ebaid, A., Alhawiti W. M., & Muhiuddin, G. (2020). The Falling Body Problem in Quantum Calculus. Front. Phys., 8, 43. https://doi.org/10.3389/fphy.2020.00043
• Anderson, D. R. (2005). Time-scale integral inequalities. J. Inequal. Pure Appl. Math., 6(3), 66.
• Bohner, M., & Peterson, A. (2001). Dynamic Equations on Time Scale: An Introduction with Applications. Birkhauser, Boston, Inc., Boston, MA. https://doi.org/10.1007/978-1-4612-0201-1
• Elaydi, S. (2005). An Introduction to Difference Equations. Springer SBM. https://doi.org/10.1007/0-387-27602-5
• Hilger, S. (1988). Ein Maßkettenkalk ̈ul mit Anwendung auf Zentrumsmannigfaltigkeiten. PhD Thesis. Universitat Würzburg
• Jackson, F. H. (1910). On a q-definite integrals. The Quarterly Journal of Pure and Applied Mathematics, 41, 193-203.
• Kayar, Z., Kaymakçalan, B., & Pelen, N. N. (2022). Diamond alpha Bennett-Leindler type dynamic inequalities and their applications. Mathematical Methods in the Applied Sciences, 45(5), 2797-2819. https://doi.org/10.1002/mma.7955
• Kayar, Z., & Kaymakçalan B. (2022a). Applications of the novel diamond alpha Hardy–Copson type dynamic inequalities to half linear difference equations. Journal of Difference Equations and Applications, 28(4), 457-484. https://doi.org/10.1080/10236198.2022.2042522
• Kayar Z., & Kaymakçalan, B. (2022b). Some new extended nabla and delta Hardy-Copson type inequalities and their applications in oscillation theory. Bulletin of the Iranian Mathematical Society, 48, 2407-2439. https://doi.org/10.1007/s41980-021-00651-2
• Thornton, S. T., & Marion, J. B. (2004). Classical dynamics of particles and systems, Thomson Brooks/Cole 24.