## THERMAL SCIENCE

International Scientific Journal

### Thermal Science - Online First

online first only
### Uncertain fractional operator with application arising in the steady heat flow

**ABSTRACT**

In the recent years much efforts were made to propose simple and well-behaved fractional operators to inherit the classical properties from the first order derivative and overcome the singularity problem of the kernel appearing for the existing fractional derivatives. Therefore, we propose in this research an interesting approach to acquire the interval solution of fractional interval differential equations (FIDEs) under a new fractional operator, that does not have the above defect with uncertain parameters. In fact, this scheme is developed to achieve the interval solution of the uncertain steady heat flow based on the FIDEs. An example is experienced to illustrate our approach and validate it.

**KEYWORDS**

PAPER SUBMITTED: 2018-01-10

PAPER REVISED: 2018-08-23

PAPER ACCEPTED: 2018-11-29

PUBLISHED ONLINE: 2018-12-16

- D. Baleanu, K. Diethelm, E. Scalas, J. J. Trujillo, Fractional Calculus: Models and Numerical Methods, World Scientific, (2012).
- I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, CA, 1999.
- A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, North-Holland Mathematics Studies, 204, (2006).
- W. Yan, Y.F. Zhang, J.G. Liu, M. Iqbal, A fractional Whitham-Broer-Kaup equation and its possible application to Tsunami prevention, Thermal Science, 21 (2017) 1847-1855.
- K.L. Wang, S.Y. Liu, He's fractional derivative for non-linear fractional heat transfer equation. Thermal Science, 20 (2016) 793-796.
- K.L. Wang, S.Y. Liu, He's fractional derivative and its application for fractional Fornberg-Whitham equation, Thermal Science, 21 (2016) 2049-2055.
- G. Feng, X.J. Yang, S.T.Mohyud-Din, On linear viscoelasticity within general fractional derivatives without singular kernel, Thermal Science, 21(2017) 335-342.
- X.J Yang, Fractional derivatives of constant and variable orders applied to anomalous relaxation models in heat-transfer problems, Thermal Science, 21 (2017), 1161-1171
- D. Benson, S.W. Wheatcraft, M.M. Meerschaert, The fractional-order governing equation of L´evy motion, Water Resour. Res. 36 (2000) 1413-1423
- R. Khalil, M. Al Horani, A. Yousef, M. Sababhehb, A new definition of fractional derivative, J. Comput. Appl. Math. 264 (2014) 65-70.
- E. M. Hussein, Fractional Order Thermoelastic Problem for an Infinitely Long Solid Circular Cylinder, Journal of Thermal Stresses, 38 (2015) 133-45.
- X. J. Yang, H. M. Srivastava, J. T. Machado, A new fractional derivative without singular kernel, Thermal Science, 20 (2016) 753-756.
- T. Abdeljawad, On conformable fractional calculus, J. Computational and Applied Mathematics 279 (2015) 57-66.
- H. Batarfi, J. Losada, J. J. Nieto, W. Shammakh, Three-Point BoundaryValue Problems for Conformable Fractional Differential Equations, Journal of Function Spaces Volume 2015, Article ID 706383, 6 pages.
- N. Benkhettou, S. Hassani, Delfim F.M. Torres, A conformable fractional calculus on arbitrary time scales, Journal of King Saud University-Science, (2015), DOI:10.1016/j.jksus.2015.05.003.
- S.Markov, Calculus for interval functions of a real variables, Computing 22 (1979) 325-337.
- V. Lupulescu, Fractional calculus for interval-valued functions, Fuzzy Sets and Systems 265 (2015) 63-85.
- M. T. Malinowski, Interval differential equations with a second type Hukuhara derivative, Applied Mathematics Letters 24 (2011) 2118-2123.
- M. T. Malinowski, Interval Cauchy problem with a second type Hukuhara derivative, Information Sciences 213 (2012) 94-105.
- L. Stefanini, A generalization of Hukuhara difference and division for interval and fuzzy arithmetic, Fuzzy Sets and Systems 161 (2010) 1564-1584.
- S. Salahshour, A. Ahmadian, S. Abbasbandy, D. Balean, M-fractional derivative under interval uncertainty: Theory, properties and applications, Chaos, Solitons & Fractals, 117 (2018) 84-93.
- A. Ahmadian, S. Salahshour, C.S. Chan, D. Baleanu, Numerical solutions of fuzzy differential equations by an efficient Runge-Kutta method with generalized differentiability, Fuzzy Sets and Systems, 331 (2018) 47-67.
- S. Salahshour, A. Ahmadian, N. Senu, D. Baleanu, P. Agarwal, On analytical solutions of the fractional differential equation with uncertainty: application to the Basset problem, Entropy, 17 (2015) 885-902.
- A. Ahmadian, C.S. Chang, S. Salahshour, Fuzzy approximate solutions to fractional differential equations under uncertainty: operational matrices approach, IEEE Trans. Fuzzy Syst. 25 (2017) 218-236.
- S. Salahshour, A. Ahmadian, F. Ismail, D. Baleanu, A fractional derivative with non-singular kernel for interval-valued functions under 155 uncertainty, Optik-International Journal for Light and Electron Optics, 130 (2017) 273-286.
- B. Bede, I. J. Rudas, A. L. Bencsik, First order linear fuzzy differential equations under generalized differentiability, Information sciences 177 (2007) 1648-1662.
- B. Bede, S.G. Gal, Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations, Fuzzy Sets and Systems 151 (2005) 581-599.
- S. Salahshour, A. Ahmadian, F. Ismail, D. Baleanu, N. Senu, A New fractional derivative for differential equation of fractional order under interval uncertainty, Advances in Mechanical Engineering 7 (2015), 1687814015619138.