Çarpımsal Uyumlu Kesirli Diferansiyel Denklemler

Yıl 2022, Cilt: 17 Sayı: 1, 99 - 108, 20.03.2022

Sertaç Göktaş

https://doi.org/10.55525/tjst.1065429

Öz

Bu çalışmada, çarpımsal uyumlu kesirli diferansiyel denklemler sunulmuştur. Çarpımsal uyumlu kesirli analiz üzerinde Wronskian kavramı, lineer bağımlılık-bağımsızlık kavramları tanımlanarak bunlar arasında bazı teoremler ve sonuçlar verilmiştir. Son olarak, çarpımsal uyumlu kesirli diferansiyel denklemlerin genel çözümlerinin bulunması üzerine bazı metotlar verilerek bazı örnekler çözülmüştür.

Anahtar Kelimeler

Uyumlu kesirli türev, çarpımsal uyumlu kesirli türev, non-Newtonian analiz, parametrelerin değişimi.

Multiplicative Conformable Fractional Differential Equations

Öz

In this study, multiplicative conformable fractional differential equations are presented. Wronskian concept, linear dependence-independence concepts are defined on multiplicative conformable fractional calculus and some theorems and results are given among them. Finally, some examples are solved by giving some methods for finding general solutions of multiplicative conformable fractional differential equations.

Anahtar Kelimeler

Conformable fractional derivative, multiplicative conformable fractional derivative, non-Newtonian calculus, variation of parameters.

Kaynakça

• [1] Grossman M. An introduction to non-Newtonian calculus. Int J Math Educ Sci Technol 1979; 10(4): 525–528.
• [2] Grossman M, Katz R. Non-Newtonian calculus, Pigeon Cove, MA: Lee Press, 1972.
• [3] Bashirov AE, Kurpınar EM, Özyapıcı A. Multiplicative calculus and its applications. J Math Anal Appl 2008; 337(1): 36-48.
• [4] Stanley D. A multiplicative calculus. Primus 1999; IX (4): 310-326.
• [5] Bashirov AE, Misirli E, Tandoğdu Y, Özyapıcı, A. On modeling with multiplicative differential equations. Appl Math J Chinese Univ Ser A 2011; 26(4): 425-438.
• [6] Bashirov AE, Riza M. On complex multiplicative differentiation. TWMS J of Apl & Eng Math 2011; 1(1): 75-85.
• [7] Gurefe Y, Kadak U, Misirli E, Kurdi A. A new look at the classical sequence spaces by using multiplicative calculus. UPB Sci Bul Ser A. 2016; 78(2): 9-20.
• [8] Kadak U, Gurefe Y. A generalization on weighted means and convex functions with respect to the Non-Newtonian calculus. Int J Anal 2016; 5416751: 1-10.
• [9] Uzer A. Multiplicative type complex calculus as an alternative to the classical calculus. Comput Math Appl 2010; 60(10): 2725-2737.
• [10] Yalcin N, Celik E. Solution of multiplicative homogeneous linear differential equations with constant exponentials. New Trend Math Sci 2018; 6(2): 58–67.
• [11] Yalcin N, Dedeturk M. Solutions of multiplicative ordinary differential equations via the multiplicative differential transform method. AIMS Mathematics 2021; 6(4): 3393-3409.
• [12] Yilmaz E. Multiplicative Bessel equation and its spectral properties, Ric. Mat. 2021; 1-17. https://doi.org/10.1007/s11587-021-00674-1.
• [13] Bashirov AE, Bashirova G. Dynamics of literary texts and diffusion. OJCMT 2011; 1(3): 60-82.
• [14] Filip D, Piatecki C, A non-newtonian examination of the theory of exogenous economic growth. Math. AEterna 2014; 4(2): 101–117.
• [15] Florack L, Assen Hv. Multiplicative calculus in biomedical image analysis. J Math Imaging Vis 2012; 42(1): 64–75.
• [16] Baleanu D, Guvenc ZB, Machado JAT. New Trends in Nanotechnology and Fractional Calculus Applications. New York (NY): Springer; 2010.
• [17] Kilbas AA, Srivastava HM, Trujillo JJ. Theory and Applications of Fractional Differential Equations (Vol. 204) Elsevier, 2006.
• [18] Miller KS, An Introduction to Fractional Calculus and Fractional Differential Equations. New York (NY): J Wiley and Sons; 1993.
• [19] Oldham KB, Spanier J. The Fractional Calculus. New York (NY): Academic Press; 1974.
• [20] Podlubny I. Fractional Differential Equations, New York: Academic Press; 1999.
• [21] Abdeljawad T. On conformable fractional calculus. J Comput Appl Math 2015; 279: 57-66.
• [22] Khalil R, Al Horani M, Yousef A, Sababheh M. A new definition of fractional derivative. J Comput Appl Math 2014; 264: 65–70.
• [23] Atangana A, Baleanu D, Alsaedi A. New properties of conformable derivative. Open Math 2015; 13(1): 889-898.
• [24] Benkhettou N, da Cruz AMB, Torres DFM, A fractional calculus on arbitrary time scales: fractional differentiation and fractional integration. Signal Processing 2015, 107: 230–237.
• [25] Miller KS. An Introduction to Fractional Calculus and Fractional Differential Equations. NewYork (NY): J Wiley and Sons; 1993.
• [26] Gulsen T, Yilmaz E, Goktas S. Conformable fractional Dirac system on time scales. J Inequal Appl 2017; 2017(1): 1-10.
• [27] Gulsen T, Yilmaz E, Kemaloglu H. Conformable fractional Sturm-Liouville equation and some existence results on time scales. Turk J Math 2018; 42(3): 1348- 1360.
• [28] Abdeljawad T, Grossman M. On geometric fractional calculus. J. Semigroup Theory Appl. 2016; 2016(2): 1-14.
• [29] Ilie M, Biazar J, Ayati Z. General solution of second order fractional differential equations. Int J Appl Math Research 2018; 7(2): 56-61.
• [30] Al Horani M, Hammad MA, Khalil R. Variation of parameters for local fractional nonhomogenous linear differential equations. J. Math. Computer Sci 2016;16: 147-153.
• [31] Gurefe Y. Multiplicative differential equations and its applications. Ph.D. Thesis, 2013; Ege University, pp.91.