Newton’s Law of Cooling: A Fractional Approach

Yıl 2019, Cilt: 19 Sayı: 1, 60 - 66, 28.05.2019

Hüseyin Şirin Abdullah Engin Çalık

https://doi.org/10.35414/akufemubid.378360

Cited By: 2

Öz

In this work, the
differential equation describing Newton's law of cooling has been redefined and
solved using Caputo fractional derivative. 
Unlike similar fractional solutions in the literature, this new solution
does not contain any time parameter.  
The obtained results have been compared with the experimental and the
standard ones. Order of fractional derivative which is most suitable for the
experimental results has been obtained.

Anahtar Kelimeler

Newton’s Law of Cooling, Fractional Calculus, Fractional Derivative Order

Newton’un Soğuma Kanunu: Kesirsel Bir Yaklaşım

Öz

Bu çalışmada Newton’un
soğuma kanununu tasvir eden diferansiyel denklem Caputo kesirsel türevi
kullanılarak yeniden tanımlanmış ve çözülmüştür. Bu yeni çözüm literatürdeki
benzer kesirsel çözümlerden farklı olarak herhangi bir zaman parametresi
içermemektedir. Elde edilen sonuçlar deneysel ve standart sonuçlarla
karşılaştırılmıştır. Deneysel sonuçlara en uygun olan kesirsel türev mertebesi
elde edilmiştir.

Anahtar Kelimeler

Newton’un Soğuma Kanunu, Kesirsel Matematik, Kesirsel Türev Mertebesi

Kaynakça

• Almeida, R., 2017. What is the best fractional derivative to fit data?. Applicable Analysis and Discrete Mathematics, 11, 358-368.
• Baehr H. D. and Stephan K., 2006. Heat and Mass Transfer. Springer-Verlag, Berlin, 668p.
• Büyükkılıç, F., Bayrakdar Ok, Z. and Demirhan, D., 2016. Investigation of the cumulative diminution process using the Fibonacci method and fractional calculus. Physica A Statistical Mechanics and Its Applications, 444, 336-344. Carpinteri, A. and Mainardi, F., 1997. Fractals and Fractional Calculus in Continuum Mechanics. Springer Verlag, 348p.
• Çalık, A. E., Ertik, H., Öder, B. and Şirin, H., 2013. A fractional calculus approach to ınvestigate the alpha decay processes. International Journal of Modern Physics E, 22 (7), 1350049-13 pages.
• Çalık, A. E., Şirin, H., Ertik, H., Öder, B. and Şen, M., 2014. Half-lives of spherical proton emitters within the framework of fractional calculus. International Journal of Modern Physics E, 23 (9), 1450044, 11 pages.
• Çalık, A.E., Şirin, H., Ertik, H. and Şen, M., 2016. Analysis of charge variation in fractional order LC electrical circuit. Revista Mexicane de Fisica, 62, 437-441.
• Çalık, A. E. and Şirin, H., 2017. Türkiye’deki elektrik enerji ihtiyacının matematiksel bir modellemesi. Sakarya Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 21 (6), 1475-1482.
• Ertik, H., Çalık, A. E., Şirin, H., Şen, M. and Öder, B., 2015. Investigation of electrical RC circuit within the framework of fractional calculus. Revista Mexicane de Fisica, 61, 58-63.
• Godinez, F. A., Navarrete, M., Chavez, O. A., Merlin A. and Valdes J.R., 2015. Two fractal versions of Newton’s Law of Cooling. Progress in Fractional Differentiation and Applications, 1, 133-143.
• Gómez-Aguilar, J. F. and Razo-,Hernández J. R., 2014. Fractional Newton cooling law. Investigación y Ciencia, 61, 12-18.
• Gonzalez-Hernandez G.J. and Medellin-Verduzco C., 2017. An experimental setup for teaching Newton’s Law of Cooling. International Journal of Humanities and Social Science Invention, 6 (1), 24-27.
• Jiji, L. M., 2003. Heat Conduction. Springer-Verlag, Berlin, 418p.
• Miller, K.S. and Ross, B., 1993. An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley and Sons Inc., New York, 384p.
• Oldham, K. B. and Spainer, J., 1974. The Fractional Calculus. Academic Press, San Diego, 234p.
• Podlubny, I., 1999. Fractional Differential Equations. Academic Press, San Diego, 368p.
• Sokolov I. M., Klafter J. and Blumen A., 2002. Fractional Kinetics. Physics Today, 53, 48-54.
• Şen, M. and Çalık, A. E., 2014. Calculation of half-value thickness for aluminum absorbers by means of fractional calculus. Annals of Nuclear Energy, 63, 46-50.
• Şen, M., Çalık, A. E. and Ertik H., 2014. Determination of half-value thickness of aluminum foils for different beta sources by using fractional calculus. Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms, 335, 78-84.
• Sirin, H., Buyukkilic, F., Ertik, H. and Demirhan, D., 2010. The influence of fractality on the time evolution of the diffusion process. Physica A-Statistical Mechanics and Its Applications, 389(10), 2007-2013.
• Sirin, H., Buyukkilic, F., Ertik, H. and Demirhan, D., 2011. The effect of time fractality on the transition coefficients: Historical Stern-Gerlach experiment revisited. Chaos Solitions and Fractals, 44, 43-47.