International Journal of Nonlinear Analysis and Applications

Fuzzy q-Taylor Theorem

Articles in Press

Document Type : Research Paper

Authors

1 Department of Mathematics, Shahed University, Tehran, Iran

2 Faculty of Basic and Advanced Technologies in Biology, University of Science and Culture, Tehran, Iran

3 Quantum Technologies Research Center, (QTRC), Science and Research Branch, Islamic Azad University, Tehran 1477893855, Iran

10.22075/ijnaa.2021.23034.2465

Abstract

The main purpose of this work is to introduce and investigate fuzzy quantum calculus. Our idea begins with a general definition of fuzzy $q$-derivative on arbitrary time scales using the generalized Hukuhara difference. It compiled some basic facts in the fields of the fuzzy $q$-derivative and the fuzzy $q$-integral and proved them in detail. Proceed with this work, specifying the particular concept of fuzzy $q$-Taylor's expansion, especially for continuous and fuzzy valued functions which are non-differentiable in the classical (usual) concept, as the best tool for approximating functions and solving the fuzzy initial value $q$-problems. Eventually, some numerical examples of fuzzy $q$-Taylor's expansion of special functions and functions with switching points, are solved for illustration.

Keywords

[1] T. Allahviranloo, Z. Noeiaghdam, and S. Noeiaghdam, J.J. Nieto, A fuzzy method for solving fuzzy fractional differential equations based on the generalized fuzzy Taylor expansion, Mathematics 8 (2020), no. 12, 2166.
[2] T. Allahviranloo, Fuzzy Fractional Differential Operators and Equations, Springer, Cham, 2020.
[3] T. Allahviranloo, Uncertain Information and Linear Systems, Springer, Cham, 2019.
[4] T.V. Amit, K. Sheerin, and J.J. Gopal, A note on the convergence of fuzzy transformed finite difference methods, J. Appl. Math. Comput. 63 (2020), 143–170.
[5] M.H. Annaby and Z.S. Mansour, q-Taylor and interpolation series for Jackson q-difference operators, J. Math. Anal. Appl. 344 (2008), 472–483.
[6] A. Aral, V. Gupta, and R.P. Agarwal, Applications of q-Calculus in Operator Theory, Springer, Science and Business Media, New York, 2013.
[7] F.M. Atici and P.W. Eloe, Fractional q-calculus on a time scale, J. Nonlinear Math. Phys. 14 (2007), no. 3, 333–344.
[8] T. Ernst, A new notation for q-Calculus a new q-Taylor’s formula, U.U.D.M. Report 1999:25, Department of Mathematics, Uppsala University, 1999.
[9] H. Fakhri and S.E. Mousavi Gharalari, Approach of the continuous q-Hermite polynomials to x-representation of q-oscillator algebra and its coherent states, International J. Geom. Meth. Mod. Phys. 17 (2020), no. 2, 2050021.
[10] H. Fakhri and M. Sayyah-Fard, Triplet q-cat states of the Biedenharn-Macfarlane q-oscillator with q > 1, Quantum Inf. Process. 19 (2020), 19.
[11] H. Fakhri and V. Sayyah-Fard, q-coherent states associated with the noncommutative complex plane Cq2 for the Biedenharn-Macfarlane q-oscillator, Ann. Phys. 387 (2017), 14–28.
[12] M.E.H. Ismail and D. Stanton, q-Taylor theorems, polynomial expansions, and interpolation of entire functions, J. Approx. Theory 123 (2003), 125–146.
[13] F.H. Jackson, On q-functions and certain difference operator, Trans. Roy. Soc. Edin. 46 (1909), 253–281.
[14] F.H. Jackson, On a q-definite integrals, Quart. J. Pure Appl. Math. 41 (1910), 193–203.
[15] S.C. Jing and H.Y. Fan, q-Taylor’s formula with its q-remainder, Commun. Theore. Phys. 23 (1995), no. 1, 117–120.
[16] V. Kac and P. Cheung, Quantum Calculus, Springer-Verlag, New York, Berlin, Heidelberg, 2001.
[17] R. Koekoek, P.A. Lesky, and R.F. Swarttouw, Hypergeometric Orthogonal Polynomials and Their q-Analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, Heidelberg, 2010.
[18] B. Lopez, J.M. Marco, and J. Parcet, Taylor series for the Askey-Wilson operator and classical summation formulas, Proc. Amer. Math. Soc. 134 (2006), no. 8, 2259–2270.
[19] J.M. Marco and J. Parcet, A new approach to the theory of classical hypergeometric polynomials, Trans. Amer. Math. Soc. 358 (2006), no. 1, 183–214.
[20] N. Mikaeilvand, Z. Noeiaghdam, S. Noeiaghdam, and J.J. Nieto, A novel technique to solve the fuzzy system of equations, Mathematics 8 (2020), no. 5, 850.
[21] Z. Noeiaghdam, T. Allahviranloo, and J.J. Nieto, q-Fractional differential equations with uncertainty, Soft Com[1]put. 23 (2019), 9507–9524.
[22] Z. Noeiaghdam, M. Rahmani, and T. Allahviranloo, Introduction of the numerical methods in quantum calculus with uncertainty, J. Math. Model. 9 (2021), no. 2, 303–322.
[23] Z.M. Odibat and N.T. Shawagfeh, Generalized Taylor’s formula, Appl. Math. Comput. 186 (2007), 286–293.
[24] M. Xin-You and W.U. Yu-Qian, Dynamical analysis of a fuzzy phytoplankton-zooplankton model with refuge, fishery protection and hervesting, J. Appl. Math. Comput. 63 (2020), 361–389.
International Journal of Nonlinear Analysis and Applications

Articles in Press, Corrected Proof
Available Online from 04 November 2024
  • Receive Date: 02 February 2021
  • Accept Date: 20 April 2021