Mittag-Leffler Function in different areas of life

Abstract

In this work, Mittag-Leffler (ML) function that is encountered in different solutions of fractional kinetic equation and generalizes exponential function has been handled. With the help of ML function, growth dynamics of population, foreign trade volume and greenhouse gas emission have been investigated. With this manner, graphical representations for these processes have been compared with real data. The harmony between theory and real data has been investigated with the help of order of fractional derivative.

Keywords

• Mittag-Leffler Function
• Fractional Calculus
• Memory Effect

Yaşamın Farklı Alanlarında Mittag-Leffler Fonksiyonu

Abstract

Bu çalışmada kesirsel kinetik denklemin farklı çözümlerinde karşımıza çıkan ve eksponansiyel fonksiyonu genelleyen Mittag-Leffler (ML) fonksiyonu ele alınmaktadır. ML fonksiyonu yardımıyla nüfus, sera gazı salınımı ve dış ticaret hacmi süreçlerindeki artma dinamikleri incelenmektedir. Bu amaçla bu süreçler için grafiksel gösterimler yapılmakta ve gerçek veriler ile karşılaştırılmaktadır. Kesirsel türev mertebesi yardımıyla gerçek değerler ile teorik hesaplamaların arasındaki uyum incelenmektedir.

Keywords

• Mittag-Leffler fonksiyonu
• Kesirsel matematik
• Bellek etkisi

References

1. Boğar, E. ve Boğar, Z. Ö., (2017). Türkiye’nin Sektörel CO2 Gazı Salınımlarının Yapay Sinir Ağları ile Tahmini, Akademia Disiplinlerarası Bilimsel Araştırmalar Dergisi, 3 (2), 12-24.
2. Bowen, W.D., McMillan, J and Mohn, R., (2003). Sustained exponential population growth of grey seals at Sable Island, Nova Scotia, ICES Journal of Marine Science, 60, 6, 1265–1274.
3. Büyükkılıç, F. and Demirhan, D., (2009). Cumulative growth with fibonacci approach, golden section and physics, Chaos, Solitons and Fractals, 42, 24–32.
4. Büyükkılıç, F., Ok Bayrakdar, 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.
5. Büyükkılıç. F., Ok Bayrakdar, Z. and Demirhan, D.,( 2015). Investigation of Cumulative Growth Process via Fibonacci Method and Fractional Calculus, Applied Mathematics and Computation, 265, 237-244.
6. Çalık, A. E., Ertik, H., Öder, B. and Şirin, H., (2013). A fractional calculus approach to investigate the alpha decay processes, International Journal of Modern Physics E, 22 (7), 1350049.
7. 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.
8. 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.

9. Çalık, A.E. ve Şirin, H., (2017). Türkiye’deki elektrik enerji ihtiyacının matematiksel bir modellemesi, Sakarya Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 21(6), 1475-1482.
10. Çalık, A.E. ve Şirin, H., (2018). Farklı tipte üretilen harçların yarı-değer kalınlıklarının teorik olarak incelenmesi, Anadolu Üniversitesi Bilim ve Teknoloji Dergisi B- Teorik Bilimler, 6(1), 36-44.
11. Çalık, A.E., Şirin, H. and Şen, M., (2020). Experimental and fractional analysis of half-value thicknesses of polyethylene absorber, Revista Mexicane de Fisica, 66 (2), 232-238.
12. Çeşmeli, M. Ş. ve Pençe, İ., (2020). Makine Öğrenimi Yöntemleri ile Türkiye için Sera Gazı Emisyonu Tahmini, Academic Platform Journal of Engineering and Science, 8-2, 332-348.
13. 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.
14. Haubold, H. J. and Mathai, A.M., (2000). The fractional kinetic equation and thermonuclear functions, Astrophysics and Space Science 327, 53–63.
15. Herrmann, R., (2011). Fractional Calculus: An Introduction for Physicists, World Scientific.
16. Hilfer, R., (2000). Applications of Fractional Calculus In Physics, World Scientific.
17. Hilfer, R., (2003). On fractional relaxation, Fractals, Vol. 11, Supplementary Issue, 251-257.
18. Holland, R.R., Ellis, C. A., Geller, B. M., Plante, D.A. and Secker-Walker, R.H., (1999). Life Expectancy Estimation with Breast Cancer: Bias of the Declining Exponential Function and an Alternative to Its Use, Medical Decision Making, 19, 4, 385-393.
19. İskender, C., (2018). Türkiye Nüfus Büyümesi ve Tahminleri: Matematiksel Büyüme Modelleri ve İstatistiksel Analiz ile Kuramsal ve Uygulamalı Bir Yaklaşım, Ekonometri ve İstatistik e-Dergisi, 14(28): 75–141.
20. Keeling, M. J., (2000). Simple stochastic models and their power-law type behaviour. Theoretical Population Biology 58, 21-31.
21. Keiding, N., (1975). Extinction and Exponential Growth in Random Environments, Theoretical Population Biology, 8 (1), 49-63.
22. Keitt, T. H. and Stanley, H. E., (1998). Dynamics of North American breeding bird populations. Nature 393, 257 -260.
23. Keitt, T. H., Amaral, L. A. N., Buldyrev, S. V. and Stanley, H. E., (2002). Scaling in the growth of geographically subdivided populations: invariant patterns from a continent-wide biological survey. Philosophical Transactions B, 357, 627-633.
24. Krane, K. S. (1988). Introductory Nuclear Physics, John Wiley and Sons.
25. Marquet, P. A., Quiñones, R. A., Abades, S., Labra, F., Tognelli, M., Arim and M, Rivadeneira, M., (2005). Scaling and power-laws in ecological systems, Journal of Experimental Biology, 208: 1749-1769.
26. Martinez, A.S., Gonzales, R.S. and Espindola A.L., (2009). Generalized exponential function and discrete growth models, Physica A: Statistical Mechanics and its Applications, 388, 2922-2930.
27. Miller K. S. and Ross B., (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons Inc.
28. Nivanen, L., Mehaute, A. and Wang Q.A., (2003). Generalized algebra within a nonextensive statistics, Reports on Mathematical Physics, 52, 437-444.
29. Oldham K. B. and Spanier, J., (2006). The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, Dover Publications.
30. Pabuçcu, H. ve Bayramoğlu, T., (2016). Yapay Sinir Ağları ile CO2 emisyonu tahmini: Türkiye örneği, Gazi Üniversitesi İktisadi ve İdari Bilimler Fakültesi Dergisi, 18 (3), 762-778.
31. Podlubny I., (1999). Fractional Differential Equations, Academic Press.
32. Ricketts, J. H. and Head, G. A., (1999). A five-parameter logistic equation for investigating asymmetry of curvature in baroreflex studies. American Journal of Physiology - Regulatory, Integrative and Comparative Physiology, 277 (2), 441‒454.
33. Sabatelli, L., Keating, S., Dudley, J. And Richmond, P., (2002). Waiting time distributions in financial markets, The European Physical Journal B, 27, 273-275.
34. Saxena, R., Mathai, A. and Haubold, H., (2002). On fractional kinetic equations, Astrophysics and Space Science, 282, 281–287.
35. Shimojo, M., (2014). An Application of Bondi K–Factor to the Preliminary Investigation into Some Natural Phenomena, Journal of the Faculty of Agriculture, Kyushu University, 59 (2), 301-303.
36. Shimojo, M., Nakano Y., (2013). An investigation into relationships between exponential functions and some natural phenomena, Journal of the Faculty of Agriculture, Kyushu University, 58 (1), 51-53.
37. Sibonga, J.D., Evans, H.J., Sung, H.G., Spector, E.R., Lang, T.F., Oganov, V.S., Bakulin, A.V., Shackelford, L.C., LeBlanc, A.D., (2007). Recovery of spaceflight-induced bone loss: Bone mineral density after long-duration missions as fitted with an exponential function, Bone, 41 (6), 973-978.
38. Silva, C. A, Prange, R. E. and Yakovenko V. M., (2004). Exponential distribution of financial returns at mesoscopic time lags: a new stylized fact, Physica A 344, 227–235.
39. Stanley, M. H. R., Amaral, L. A. N., Buldyrev, S. V., Havlin, S., Leschhorn, H., Maass, P., Salinger, M. A. & Stanley, H. E. (1996). Scaling behavior in the growth of companies. Nature 379,804 -806.
40. 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.
41. 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.
42. Şen, M., Çalık, A.E. and Şirin, H., (2019). Theoretical investigation of tenth-value thicknesses of aluminium absorbers, AIP Conference Proceedings, 2178, 030058.
43. Şirin, H. & Çalık, A.E. (2019). Newton’un Soğuma Kanunu: Kesirsel Bir Yaklaşım, Afyon Kocatepe Üniversitesi Fen ve Mühendislik Bilimleri Dergisi, 19, 60-66.
44. Vacher, H.L., (2000). The Exponential Function, Journal of Geoscience Education, 48 (1), 70-77.
45. Vandermeer, J., (2010). How Populations Grow: The Exponential and Logistic Equations, Nature Education Knowledge 3(10),15.
46. Verma, K.M., Asad, A. and Chatterjee S. (2020). COVID‑19 Pandemic: Power Law Spread and Flattening of the Curve, Transactions of the Indian National Academy of Engineerin, 5, 103-108.
47. Yurdakul, E. M. (2014). Türkiye’de İthalatın Gelişimi ve İthalatın Yapay Sinir Ağları Yöntemi İle Tahmin Edilebilirliğine Yönelik Bir Analiz, Doktora Tezi, Sosyal Bilimler Enstitüsü, Adnan Menderes Üniversitesi, Aydın.