The fuzzy generalized Taylor’s expansion with application in fractional differential equations

Document Type: Original Manuscript

Authors

1 Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran

2 Imam Khomeini Int. University

Abstract

In this paper, the generalized Taylor’s expansion is presented for fuzzy-valued functions. To achieve this aim, fuzzy
fractional mean value theorem for integral, and some properties of Caputo generalized Hukuhara derivative are necessary
that we prove them in details. In application, the fractional Euler’s method is derived for solving fuzzy fractional
differential equations in the sense of Caputo differentiability. The effectiveness of the proposed method is verified by
three examples.


Keywords


[1] R.P. Agarwal, V. Lakshmikanthama, J. J. Nieto, On The Concept of Solution for Fractional Differential Equations
with Uncertainty
, Nonlinear Analysis, 72 (2010), 2859-2862.
[2] A. Ahmadian, M. Suleiman, S.Salahshour, D. Baleanu, A Jacobi Operational Matrix for Solving A Fuzzy Linear
Fractional Differential Equation
, Advances in Difference Equations, 2013 (2013), 1-29.
[3] R. Alikhani, F. Bahrami,
Global Solutions For Nonlinear Fuzzy Fractional Integral And Integrodifferential Equations,
Communications in Nonlinear Science and Numerical Simulation,
18 (2013), 2007-2017.
[4] T. Allahviranloo, A. Armand, Z. Gouyandeh,
Fuzzy fractional differential equations under generalized fuzzy Caputo
derivative
, Journal of Intelligent & Fuzzy Systems, 26 (2014), 1481-1490.
[5] T. Allahviranloo, A. Armand, Z. Gouyandeh, H. Ghadiri,
Existence and Uniqueness of Solutions for Fuzzy Fractional
Volterra-Fredholm Integro-Differential Equations
, Journal of Fuzzy Set Valued Analysis, 2013 (2013), 1-9.
[6] T. Allahviranloo, Z. Gouyandeh, A. Armand, A. Hasanoglu,
On Fuzzy Solutions For Heat Equation Based On
Generalized Hukuhara Differentiability
, Fuzzy Sets and Systems, 265 (2015), 1-23.
[7] T. Allahviranloo, S. Salahshour, S. Abbasbandy,
Explicit Solutions Of Fractional Differential Equations With Uncertainty, Soft Computting, 16 (2012), 297-302.
[8] G. A. Anastassiou,
Fuzzy Mathematics:Approximation Theory, Studies in Fuzziness and Soft Computing, 251 (2010),
1434-9922.
[9] A. Armand, T. Allahviranloo, Z. Gouyandeh,
General Solution Of Basset Equation With Caputo Generalized
Hukuhara Derivative
, Journal Of Applied Analysis and Computation, 6 (2016), 119-130.
[10] S. Arshad, V. Luplescu,
Fractional Differential Equation With Fuzzy Initial Conditon, Electronic Journal Of Differential Equations, 34 (2011), 1-8.
[11] B. Bede, S. G. Gal,
Generalizations Of The Differentiability Of Fuzzy-Number-Valued Functions With Applications
To Fuzzy Differential Equations
, Fuzzy Set and Systems, 151 (2005), 581-599.
[12] B. Bede, L. Stefanini,
Generalized Differentiability Of Fuzzy-Valued Functions, Fuzzy Sets And Systems, 230
(2013), 119-141.
[13] B. Bede, L. Stefanini,
Solution Of Fuzzy Differential Equations With Generalized Differentiability Using LuParametric Representation, Eusflat, 1 (2011), 785-790.
[14] K. Diethelm,
The Mean Value theorem And A Nagumo-Type Uniqueness theorem For Caputo’S Fractional Calculus,
Fractional Calculus and Applied Analysis,
15 (2012), 304-313.
[15] D. Dubois, H. Prade,
Fuzzy Numbers: An Overview, Analysis Of Fuzzy Information, l (1987), 3-39.
[16] R. Goetschel, W. Voxman,
Elementary Fuzzy Calculus, Fuzzy Sets and Systems, 24 (1986), 31-43.
[17] O. Kaleva, S. Seikkala,
On Fuzzy Metric Spaces, Fuzzy Sets and Systems, 12 (1984), 215-229.
[18] E. Khodadadi, E. Celik,
The Variational Iteration Method For Fuzzy Fractional Differential Equations With Uncertainty, Fixed Point Theory And Applications, 2013 (2013), 1-13.
[19] H. V Long, N. T.K Son, H. T. T Tam, J.C Yao, Ulam Stability for Fractional Partial Integro-Differential Equation
with Uncertainty
, Acta Mathematica Vietnamica, 42 (2017), 675-700.
[20] H. V Long, N. T.K Son, N. V Hoa,
Fuzzy Fractional Partial Differential Equations In Partially Ordered Metric
Spaces
, Iranian Journal of Fuzzy Systems, 14 (2017), 107-126.
[21] H. V Long, N. T.K Son, H. T. T Tam,
The Solvability of Fuzzy Fractional Partial Differential Equations Under
Caputo gH-Differentiability
, Fuzzy Sets and Systems, 309 (2017), 35-63.
[22] H. V Long, N. T.K Son, H. T. T Tam,
Global Existence of Solutions to Fuzzy Partial Hyperbolic Functional
Differential Equations with Generalized Hukuhara Derivatives
, Journal of Intelligent & Fuzzy Systems, 29 (2015),
939-954.
[23] V. Lupulescu,
Fractional Calculus For Interval-Valued Functions, Fuzzy Sets and Systems, 265 (2015), 63-85.
[24] M. Mazandarani, A. Vahidian Kamyad,
Modified fractional Euler method for solving Fuzzy Fractional Initial Value
Problem
, Communications in Nonlinear Science and Numerical Simulation, 18 (2013), 12-21.
[25] J. J. Nieto, Rosana Rodriguez-Lopez,
Some Results On Boundary Value Problems For Fuzzy Differential Equations
With Functional Dependence
, Fuzzy Set and Systems, 230 (2013), 92-118.
[26] C. V. Negoita, D. Ralescu,
Applications of Fuzzy Sets to Systems Analysis, Wiley, New York, 1975.
[27] I. Podlubny,
Fractional Differential Equations, Academic Press, 1999.
[28] S. Salahshour, T. Allahviranloo, S. Abbasbandy, D. Baleanu,
Existence And Uniqueness Results For Fractional
Differential Equations With Uncertainty
, Advances In Difference Equations, 2012 (2012), 1-12.
[29] S. Salahshour, T. Allahviranloo, S. Abbasbandy,
Solving Fuzzy Fractional Differential Equations By Fuzzy Laplace
Transforms
, Communications in Nonlinear Science and Numerical Simulation, 17 (2012), 1372-1381.
[30] L. Stefanini, B. Bede,
Generalized Hukuhara Differentiability Of Interval-Valued Functions And Interval Differential
Equations
, Nonlinear Analysis, 71 (2009), 1311-1328.