Investigation of Newton's Law of Cooling on Time Scales with Proportional Derivative

Merve Çolak; Emrah Yılmaz

doi:10.33434/cams.1698737

EN

Investigation of Newton's Law of Cooling on Time Scales with Proportional Derivative

Abstract

In this study, the proportional type Newton’s law of cooling is discussed, and solutions are obtained on some common time scales. This model, which plays an important role in physics, is examined in both proportional and delta derivative cases on time scales. Then, the obtained solutions are interpreted graphically, and the contributions of time scale calculus with proportional derivative to the theory are revealed. The aim of the study is to examine the effect of the delta and the proportional derivative on the model on time scales.

Keywords

Newton, Proportional derivative, Time scales

References

[1] K. Stephan, Heat and Mass Transfer, Springer-Verlag, Berlin, 2006.
[2] L. M. Jiji, Heat Conduction, Springer-Verlag, Berlin, 2009.
[3] I. Newton, The Principia, translated by A. Motte, Prometheus Books, Amherst, 1995.
[4] R. Almeida, What is the best fractional derivative to fit data?, Appl. Anal. Discrete Math., 11(2) (2017), 358–368. https://doi.org/10.2298/AADM170428002A
[5] T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57-66. https://doi.org/10.1016/j.cam.2014.10.016
[6] O. S. Iyiola, E. R. Nwaeze, Some new results on the new conformable fractional calculus with application using D’Alembert approach, Progr. Fract. Differ. Appl., 2(2) (2016), 115-122. http://dx.doi.org/10.18576/pfda/020204
[7] R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65-70. https://doi.org/10.1016/j.cam.2014.01.002
[8] K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, 1993.
[9] K. B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.
[10] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.

[11] C. Ünlü, Kesirli türevli diferansiyel denklemler ve çözüm yöntemleri, Ph.D. Thesis, Istanbul University, 2014.
[12] F. A. Godinez, M. Navarrete, O. A. Chavez, A. Merlin, J. R. Valdes, Two fractal versions of Newton’s Law of Cooling, Progr. Fract. Differ. Appl., 1(2) (2015), 133-143.
[13] K. C. Cheng, Some observations on the origins of Newton’s Law of Cooling and its influences on thermofluid science, Appl. Mech. Rev., 62(6) (2009), Article ID 060803. https://doi.org/10.1115/1.3090832
[14] J. F. Gomez-Aguilar, J. R. Razo-Hernandez, Fractional Newton cooling law, Investigacion Ciencia, 61 (2014), 12-18.
[15] B. Jackson, Partial dynamic equations on time scales, J. Comput. Appl. Math., 186(2) (2006), 391–415. https://doi.org/10.1016/j.cam.2005.02.011
[16] G. Sh. Guseinov, Integration on time scales, J. Math. Anal. Appl., 285(1) (2003), 107-127. https://doi.org/10.1016/ S0022-247X(03)00361-5
[17] S. Hilger, Analysis on measure chains-a unified approach to continuous and discrete calculus, Results Math., 18 (1990), 18-56. https://doi.org/10.1007/BF03323153
[18] M. Bohner, S. G. Georgiev, Multivariable Dynamic Calculus on Time Scales, Springer, Cham, 2016.
[19] M. Bohner, A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, 2001.
[20] J. Mahaffy, Calculus: A Modeling Approach for the Life Sciences, Volume II: Integral Calculus and Differential Equations, Pearson Custom Publishing, Boston, 2005.
[21] D. R. Anderson, D. J. Ulness, Newly defined conformable derivatives, Adv. Dyn. Syst. Appl., 10(2) (2015), 109–137.
[22] D. R. Anderson, S. G. Georgiev, Conformable Dynamic Equations on Time Scales, CRC Press, Boca Raton, 2020.
[23] G. J. Gonzalez-Hernandez, C. Medellin-Verduzco, An experimental setup for teaching Newton’s law of cooling, International Journal of Humanities and Social Science Invention, 6(1) (2017), 24–27.
[24] J. A. Ruffner, Reinterpretation of the genesis of Newton’s law of cooling, Arch. Hist. Exact Sci., 2(2) (1964), 138–152.
[25] S. Maruyama, S. Moriya, Newton’s law of cooling: Follow-up and exploration, Int. J. Heat Mass Transf., 164 (2021), Article ID 120544. https://doi.org/10.1016/j.ijheatmasstransfer.2020.120544
[26] W. S. Chung, Fractional Newton mechanics with conformable fractional derivative, J. Comput. Appl. Math., 290 (2015), 150-158. https://doi.org/10.1016/j.cam.2015.04.049
[27] H. Şirin, A. E. Çalık, Newton’un soğuma kanunu: Kesirsel bir yaklasşım, AKU J. Sci. Eng., 19(1) (2019), 60-66. https://doi.org/10.35414/akufemubid.378360
[28] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.

Details

Primary Language

English

Subjects

Ordinary Differential Equations, Difference Equations and Dynamical Systems

Journal Section

Research Article

Authors

Merve Çolak
0009-0006-3978-5505
Türkiye

Emrah Yılmaz ^*
0000-0002-7822-9193
Türkiye

Early Pub Date

September 23, 2025

Publication Date

September 23, 2025

Submission Date

May 13, 2025

Acceptance Date

September 20, 2025

Published in Issue

Year 2025 Volume: 8 Number: 3

DOI

https://doi.org/10.33434/cams.1698737

IZ

https://izlik.org/JA55HD63DT

APA

Çolak, M., & Yılmaz, E. (2025). Investigation of Newton’s Law of Cooling on Time Scales with Proportional Derivative. Communications in Advanced Mathematical Sciences, 8(3), 160-172. https://doi.org/10.33434/cams.1698737

AMA

1.Çolak M, Yılmaz E. Investigation of Newton’s Law of Cooling on Time Scales with Proportional Derivative. Communications in Advanced Mathematical Sciences. 2025;8(3):160-172. doi:10.33434/cams.1698737

Chicago

Çolak, Merve, and Emrah Yılmaz. 2025. “Investigation of Newton’s Law of Cooling on Time Scales With Proportional Derivative”. Communications in Advanced Mathematical Sciences 8 (3): 160-72. https://doi.org/10.33434/cams.1698737.

EndNote

Çolak M, Yılmaz E (September 1, 2025) Investigation of Newton’s Law of Cooling on Time Scales with Proportional Derivative. Communications in Advanced Mathematical Sciences 8 3 160–172.

IEEE

[1]M. Çolak and E. Yılmaz, “Investigation of Newton’s Law of Cooling on Time Scales with Proportional Derivative”, Communications in Advanced Mathematical Sciences, vol. 8, no. 3, pp. 160–172, Sept. 2025, doi: 10.33434/cams.1698737.

ISNAD

Çolak, Merve - Yılmaz, Emrah. “Investigation of Newton’s Law of Cooling on Time Scales With Proportional Derivative”. Communications in Advanced Mathematical Sciences 8/3 (September 1, 2025): 160-172. https://doi.org/10.33434/cams.1698737.

JAMA

1.Çolak M, Yılmaz E. Investigation of Newton’s Law of Cooling on Time Scales with Proportional Derivative. Communications in Advanced Mathematical Sciences. 2025;8:160–172.

MLA

Çolak, Merve, and Emrah Yılmaz. “Investigation of Newton’s Law of Cooling on Time Scales With Proportional Derivative”. Communications in Advanced Mathematical Sciences, vol. 8, no. 3, Sept. 2025, pp. 160-72, doi:10.33434/cams.1698737.

Vancouver

1.Merve Çolak, Emrah Yılmaz. Investigation of Newton’s Law of Cooling on Time Scales with Proportional Derivative. Communications in Advanced Mathematical Sciences. 2025 Sep. 1;8(3):160-72. doi:10.33434/cams.1698737

The published articles in CAMS are licensed under a Creative Commons Attribution-NonCommercial 4.0 International License..

Abstract

Investigation of Newton's Law of Cooling on Time Scales with Proportional Derivative

Abstract

Keywords

References

Details

Primary Language

Subjects

Journal Section

Authors

Early Pub Date

Publication Date

Submission Date

Acceptance Date

Published in Issue

DOI

IZ

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