## THERMAL SCIENCE

International Scientific Journal

### FRACTIONAL MAXWELL FLUID WITH FRACTIONAL DERIVATIVE WITHOUT SINGULAR KERNEL

**ABSTRACT**

In this paper we propose a new model for the fractional Maxwell fluid within fractional Caputo-Fabrizio derivative operator. We present the fractional Maxwell fluid in the differential form for the first time. The analytical results for the proposed model with the fractional Losada-Nieto integral operator are given to illustrate the efficiency of the fractional order operators to the line viscoelasticity.

**KEYWORDS**

PAPER SUBMITTED: 2015-12-01

PAPER REVISED: 2016-01-18

PAPER ACCEPTED: 2016-02-23

PUBLISHED ONLINE: 2016-09-24

**THERMAL SCIENCE** YEAR

**2016**, VOLUME

**20**, ISSUE

**Supplement 3**, PAGES [S871 - S877]

- Blair, G. S., The Role of Psychophysics in Rheology, Journal of Colloid Science, 2 (1947), 1, pp. 21-32
- Debnath, L., Recent Applications of Fractional Calculus to Science and Engineering, International Journal of Mathematics and Mathematical Sciences, 2003 (2003), 54, pp. 3413-3442
- Rogosin, S., Mainardi, F., George William Scott Blair - the Pioneer of Fractional Calculus in Rheology, Communications in Applied and Industrial Mathematics, 6 (2014), 1, e681
- Gerasimov, A. N., A Generalization of Linear Laws of Deformation and Its Application to Problems of Internal Friction (in Russian), Akad. Nauk SSSR. Prikl. Mat. Meh., 12 (1948), pp. 251-260
- Kai-Xin, H., et al., Mechanical Analogies of Fractional Elements, Chinese Physics Letters, 26 (2009), 10, ID 108301
- Caputo, M., Mainardi, F., Linear Models of Dissipation in Anelastic Solids, Rivista del Nuovo Cimento (Ser. II), 1 (1971), 2, pp. 161-198
- Mainardi, F., An Historical Perspective on Fractional Calculus in Linear Viscoelasticity, Fractional Calculus and Applied Analysis, 15 (2012), 4, pp. 712-717
- Carpinteri, A., Mainardi, F., Fractals and Fractional Calculus in Continuum Mechanics, Springer Verlag, New York, USA, 1997
- Kilbas, A. A., et al., Theory and Applications of Fractional Differential Equations, Elsevier Science Limited, Amsterdam, The Netherlands, 2006
- Yang, X.-J., et al., Local Fractional Integral Transforms and Their Applications, Academic Press, New York, USA, 2015
- Caputo, M., Fabrizio, M., A New Definition of Fractional Derivative without Singular Kernel, Progress in Fractional Differentiation and Applications, 1 (2015), 2, pp. 73-85
- Lozada, J., Nieto, J. J. Properties of a New Fractional Derivative without Singular Kernel, Progress in Fractional Differentiation and Applications, 1 (2015), 2, pp. 87-92
- Alsaedi, A., et al., Fractional Electrical Circuits, Advances in Mechanical Engineering, 7 (2015), 12, pp. 1-7
- Atangana, A., Nieto, J. J., Numerical Solution for the Model of RLC Circuit via the Fractional Derivative without Singular Kernel, Advances in Mechanical Engineering, 7 (2015), 10, pp. 1-7
- Gomez-Aguilar, J. F., et al., Modeling of a Mass-Spring-Damper System by Fractional Derivatives with and without a Singular Kernel, Entropy, 17 (2015), 9, pp. 6289-6303
- Mainardi, F., Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, World Scientific, Singapore, 2010