## THERMAL SCIENCE

International Scientific Journal

### Thermal Science - Online First

online first only
### Solutions of Cattaneo-Hristov model of elastic heat diffusion with Caputo-Fabrizio and Atangana-Baleanu fractional derivatives

**ABSTRACT**

Recently Hristov using the concept of a relaxation kernel with no singularity developed a new model of elastic heat diffusion equation based on the Caputo-Fabrizio fractional derivative as an extended version of Cattaneo model of heat diffusion equation. In the present article, we solve exactly the Cattaneo-Hristov Model (CHM) by and extend it by the concept of a derivative with non-local and non- singular kernel by using the new Atangana-Baleanu derivative. The Cattaneo-Hristov model with the extended derivative is solved analytically with the Laplace transform, and numerically using the Crank-Nicholson scheme.

**KEYWORDS**

PAPER SUBMITTED: 2016-02-09

PAPER REVISED: 2016-04-13

PAPER ACCEPTED: 2016-04-20

PUBLISHED ONLINE: 2016-05-08

- Cattaneo, C, On the conduction of heat (In Italian), Atti Sem. Mat. Fis. Universit´a Modena, 3 (1948),1, pp. 83-101.
- Joseph, D.D., Preciozi, Heat waves, Rev.Mod. Phys., 61 (1989), 1, pp. 41-73
- Hristov,J. A Note on the Integral Approach to Non-Linear Heat Conduction with Jeffrey's Fading Memory, Thermal Science, 17 (2013),3,pp. ,733-737
- Dahlquist, F.W., Lovely, P.,Koshland, D.E. Jr: Qualitative analysis of bacterial migrationin chemotaxis. Nature, New Biol.236(1972), pp. 120-123.
- Gurtin, M.E., Pipkin, A.C.:A general theory of heat conduction with finite wave speed.Arch. Rational Mech. Anal. 31(1968), pp.113-126.
- C.I. Christov, P.M. Jordan, Heat conduction paradox involving second-sound propagation in moving media, Phys. Rev. Lett. 94 (2005),pp.154301-1-154301-4.
- H. Ziegler, C. Wehrli, The derivation of constitutive relations from the free energy and the dissipation functions, in: T.Y. Wu, J.W. Hutchinson (Eds.), Adv.Appl. Mech., vol. 25, Academic Press, New York, 1987, pp. 183-238
- D.G.B. Edelen, On the existence of symmetry relations and dissipative potentials, Arch. Rat. Mech. Anal. 51 (1973), pp218-227.
- . Hristov J., Transient heat diffusion with a non-singular fading memory: From the Cattaneo constitutive equation with Jeffrey's kernel to the Caputo-Fabrizio time-fractional derivative, thermal science, in press doi: 10.2298/TSCI160112019H
- Hristov J., Double Integral-Balance Method to the Fractional Subdiffusion Equation: Approximate solutions, optimization problems to be resolved and numerical simulations, J. Vibration and Control, in press, DOI: 10.1177/1077546315622773
- Caputo, M., Fabrizio, M., A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl. ,1 (2015), 2, pp. 73-85.
- Caputo, M., Fabrizio, M., Applications of new time and spatial fractional derivatives with exponential kernels, Progr. Fract. Differ. Appl., 2 (2016), 2, pp. 1-11.
- Losada, J., Nieto, J. J., Properties of a New Fractional Derivative without Singular Kernel, Progr. Fract. Differ. Appl. , 1 (2015), 2, pp. 87-92.
- Xiao-Jun Yang, Dumitru Baleanu, Mihailo P. Lazarević and Milan S. Cajić , Fractal boundary value problems for integral and differential equations with local fractional operators, Thermal Science, 19(2015), pp. 959-966
- He C-H, He, J-H,, Double trials method for nonlinear problems arising in heat transfer, Thermal science ,15(2011), pp. 153 - 155
- Fu-Juan Liu, Zheng-Biao Li, Sheng Zhang, and Hong-Yan Liu, He's fractional derivative for heat conduction in a fractal medium arising in silkworm cocoon hierarchy, Thermal Science, 19(2011)4, pp.1155 - 1159
- Abdon Atangana, Dumitru Baleanu New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Thermal science, 20 (2016)(2),pp. 757-763
- A. Atangana, I. Koca, Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos, Solitons and Fractals, dx.doi.org/10.1016/j.chaos.2016.02.012