## THERMAL SCIENCE

International Scientific Journal

### Thermal Science - Online First

online first only
### Fractional derivatives of constant and variable orders applied to anomalous relaxation models in heat-transfer problems

**ABSTRACT**

In this paper, we address a class of the fractional derivatives of constant and variable orders for the first time. Fractional-order relaxation equations of constants and variable orders in the sense of Caputo type are modeled from mathematical view of point. The comparative results of the anomalous relaxation among the various fractional derivatives are also given. They are very efficient in description of the complex phenomenon arising in heat transfer.

**KEYWORDS**

PAPER SUBMITTED: 2016-12-16

PAPER REVISED: 2016-12-17

PAPER ACCEPTED: 2016-12-24

PUBLISHED ONLINE: 2017-01-14

- West, B. J., Fractional Calculus View of Complexity: Tomorrow's Science, CRC Press, Boca Raton, 2015
- Mainardi, F., Fractional Calculus and Waves in Linear Viscoelasticity: an Introduction to Mathematical Models, World Scientific, 2010
- Herrmann, R., Fractional Calculus: An Introduction for Physicists, World Scientific, 2014
- He, J. H., A New Fractal Derivation, Thermal Science, 15(2011), Suppl. 1, pp. 145-147
- Yang, X. J., et al., Local Fractional Integral Transforms and Their Applications, Academic Press, 2015
- Yang, X. J., et al., On Local Factional Operators View of Computational Complexity： Diffusion and Relaxation Defined on Cantor Sets, Thermal Science, 20 (2016), Suppl. 3, pp. 145-147
- Yang, X. J., et al., On a Fractal LC-electric Circuit Modeled by Local Fractional Calculus, Communications in Nonlinear Science and Numerical Simulation, 47 (2017), pp.200-206
- Mainardi, F., Gorenflo, R., Time-fractional Derivatives in Relaxation Processes: a Tutorial Survey, Fractional Calculus and Applied Analysis, 10(2007), 3, pp.269-308
- Mainardi, F., Spada, G., Creep, Relaxation and Viscosity Properties for Basic Fractional Models in Rheology, The European Physical Journal Special Topics, 193(2011),1, pp.133-160
- Kilbas, A. A., et al., Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006
- Samko, S. G., Ross, B., Integration and Differentiation to a Variable Fractional Order, Integral Transforms and Special Functions, 1 (1993), 4, pp.277-300
- Kilbas, A. A., et al., Fractional Integral and Derivatives: Theory and Applications, Gordon and Breach, Switzerland,1993
- Lorenzo, C. F., Hartley, T. T., Variable Order and Distributed Order Fractional Operators, Nonlinear dynamics, 29(2002), 1-4, pp.57-98
- Coimbra, C. F. Mechanics with Variable Order Differential Operators, Annalen der Physik, 12 (2003), 11-12, pp.692-703
- Ramirez, L.E.S., Coimbra, C. F. M., A Variable Order Constitutive Relation for Viscoelasticity, Annalen der Physik, 16(2007), 7-8, pp.543-552
- Bhrawy, A. H., Zaky, M. A., Numerical Algorithm for the Variable-order Caputo Fractional Functional Differential Equation, Nonlinear Dynamics, 85(2016),3, pp 1815-1823
- 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
- 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, 20 (2016), 2, pp.765-770
- Hristov, J., Steady-State Heat Conduction in A Medium With Spatial Non-Singular Fading Memory: Derivation of Caputo-Fabrizio Space-fractional Derivative with Jeffrey's Kernel and Analytical Solutions, Thermal Science, 2016, DOI:10.2298/TSCI160229115H
- A. Atangana, On the New Fractional Derivative and Application to Nonlinear Fisher's Reaction-Diffusion Equation, Applied Mathematics and Computation, 273(2016), pp.948-956
- Gómez-Aguilar, J. F., et al., Analytical Solutions of the Electrical RLC Circuit via Liouville-Caputo Operators with Local and Non-local Kernels, Entropy, 18 (2016), 8, pp.402
- Alkahtani, B. S. T., Atangana, A., Controlling the Wave Movement on the Surface of Shallow Water with the Caputo-Fabrizio Derivative with Fractional Order, Chaos, Solitons & Fractals, 89 (2016), pp.539-546
- Losada, 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
- Caputo, M., Fabrizio, M., Applications of New Time and Spatial Fractional Derivatives with Exponential Kernels, Progress in Fractional Differentiation and Applications, 2 (2016), 2, pp.1-11
- Yang, X.-J., et al., A New Fractional Derivative without Singular Kernel: Application to the Modelling of the Steady Heat Flow, Thermal Science, 20(2016), 2, pp.753-756
- Yang, A. M., et al., On Steady Heat Flow Problem Involving Yang-Srivastava-Machado Fractional Derivative Without Singular Kernel, Thermal Science, 20 (2016), Suppl. 3, pp. 717-723
- Sun, H., et al., Relaxation and Diffusion Models with Non-singular Kernels, Physica A, 2016, DOI: 10.1016/j.physa.2016.10.066
- Atangana A., Dumitru, B., New Fractional Derivatives with Non-local and Non-singular Kernel: Theory and Application to Heat Transfer Model, Thermal Science, 20 (2016), 2, pp.763-769