Introduction of fuzzy q-fractional derivative and its properties
Subject Areas : Statistics
Naser
Mikael Vand
1
(Department of Mathematics, Islamic Azad University, Ardabil Branch, Ardabil, Iran.)
Zahra
Noeiaghdam
2
(Faculty of Mathematical and Computer Sciences, Shahid University, Tehran, Iran)
Keywords: q - انتگرال ریمن-لیوول کسری فازی. , تفاضل هاکوهارای تعمیم یافته, حساب کوانتومی, q-مشتق فازی, q- مشتق کاپوتوی کسری فازی,
Abstract :
The quantum calculus or q-calculus begins with F. H. Jackson in the early twentieth century, but only recently it has aroused interest, due to high demand of mathematics that is modeling quantum computing and it has been an important subject for applied sciences . The quantum calculus is one of the applied and inter disciplinary sciences, which is more important than the classical calculus because in the standard calculus the definition of the derivative depends on the existence of limit but the quantum derivative in quantum calculus works without the definition of limit and for this reason the work with a quantum calculus is numerically faster and easier than the standard calculus. In this paper, fuzzy quantum derivative, fuzzy quantum fractional derivative in Caputo sense by using generalized Hukuhara difference and fuzzy quantum fractional integral of the Riemann-Liouville type are introduced, then the related theorems and properties are provided in details.These results occur in many applications as physics, quantum theory, number theory, statistical mechanics, etc.
[1] Jackson, F. H. (1908). On -functions and certain difference operator, Trans Roy Soc Edin, 46, 253-281.
2- احمدی، الهام؛ احمدی، نازنین (1399) روشی جدید برای حل معادلات دیفرانسیل فازی از مرتبه nام با استفاده از چندجملهای درونیاب، پژوهشهای نوین در ریاضی.
3- الهویرنلو، توفیق؛احمدی، الهام؛ احمدی، نازنین (1396) جواب تقریبی معادلات دیفرانسیل فازی مرتبه اول تحت مشتق تعمیم یافته، پژوهشهای نوین در ریاضی.
4- پرندین، نورالدین (1398) حل عددی معادلات دیفرانسیل فازی مرتبه n با استفاده از روش آدامز- بشفورث، پژوهش های نوین در ریاضی.
5- وثوقی، حسین؛ عباسبندی، سعید (1396) درونیابی توابع مشتق پذیر تعمیم یافته هاکوهارا از مرتبه دوم، پژوهشهای نوین در ریاضی.
[6] Bede, B. and Stefanini, L. (2013). Generalized differentiability of fuzzy-valued functions, Fuzzy Set and Systems, 230, 119-141.
[7] Ma, M., Friedman, M. and Kandel, A. (1999). A new fuzzy arithmetic, Fuzzy set Syst, 108, 83-90.
[8] Mikaeilvand, N., Noeiaghdam, Z. (2012). The general solution of Fuzzy Linear Systems, Middle-East Journal of Scientific Research, 11 (1), 128-133.
[9] Mikaeilvand, N., Noeiaghdam, Z. (2015). The general solutions of fuzzy linear matrix equations, Journal of Mathematical Extension, 9, 1-13.
[10] Noeiaghdam, Z. and Mikaeilvand, N. (2012). Least Squares Solutions of Inconsistent Fuzzy Linear Matrix Equations, International Journal of Industrial Mathematics, 365-374.
[11] Stefanini, L. and Bede, B. (2009). Generalized Hukuhara differentiability of interval-valued functions and interval differential equations, Nonlinear Analysis, 71, 1311-1328.
[12] Puri,M.L. and Ralescu, D.A. (1986). Fuzzy random variables, Math Anal Appl, 114, 409-422.
[13] Ahmad, B., Nieto, J. J., Alsaedi, A. and Al-Hutami, H. (2014). Existence of solutions for nonlinear fractional -difference integral equations with two fractional orders and nonlocal four-point boundary conditions, Journal of the Franklin Institute, 351 (5), 2890-2909.
[14] Ahmad, B. and Nieto, J. J. (2013). Basic Theory of Nonlinear Third-Order -Difference Equations and Inclusions, Math. Model. Anal, 18, 122-135.
[15] Abdeljawad, T. and Baleanu, D. (2011). Caputo -fractional initial value problems and a -analogue mittag-leffler function, Communications in Nonlinear Science and Numerical Simulation, 16 (12), 4682-4688.
[16] Kac, V. and Cheung, P. (2001). Quantum Calculus-Universitext, Springer-Verlag, New York, Berlin, Heidelberg.
[17] Li, X., Han, Z., Sun, Sh. and Sun, L. (2016). Eigenvalue problems of fractional -difference equations withgeneralized p-Laplacian, Applied Mathematics Letters, 57, 46-53.
[18] Rajkovic, P. M., Marinkovic, S. D. and Stankovic, M. S. (2007). Fractional Integrals and Derivatives in -Calculus, Applicable Analysis and Discrete Mathematics, 1, 311-323.
