Numerical solutions of Fourier's law involving fractional derivatives with bi-order

Document Type : Article


1 CONACyT-Centro Nacional de Investigacion y Desarrollo Tecnologico. Tecnologico Nacional de Mexico

2 Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences. University of the Free State, Bloemfontein 9300, South Africa.

3 Centro Nacional de Investigacion y Desarrollo Tecnologico.Tecnologico Nacional de Mexico


In this paper, we present an alternative representation of the fractional spacetime
Fourier's law equation using the concept of derivative with two fractional
orders and . The new de nitions are based on the concept of power law
and the generalized Mittag-Leer function, where, the rst fractional order
is included in the power law function and the second fractional order is the
generalized Mittag-Leer function. The new approach is capable of considering
media with two di erent layers, scales and properties. The generalization of
this equation exhibit di erent cases of anomalous behavior and Non-Fourier
heat conduction processes. Numerical solutions using an iterative scheme were


Main Subjects

1. Liu, F.J. \He's fractional derivative for heat conduction
in a fractal medium arising in silkworm cocoon hierarchy",
Thermal Science, 19, pp. 1155-1159 (2015).
2. Oldham, K.B. and Spanier, J. \The fractional calculus",
Academic Press, New York (1974).
3. Yao, J.J., Kumar, A., and Kumar, S. \A fractional
model to describe the Brownian motion of particles
and its analytical solution", Advances in Mechanical
Engineering, 7(12), pp. 1-11 (2015).
4. Gomez-Aguilar, J.F., Razo-Hernandez, R., and Granados
-Lieberman, D. \A physical interpretation of fractional
calculus in observables terms: analysis of the
fractional time constant and the transitory response",
Rev. Mex. Fs. 60, pp. 32-38 (2014).
5. Bia, P., Caratelli, D., Mescia, L., Cicchetti, R.,
Maione, G., and Prudenzano, F. \A novel FDTD formulation
based on fractional derivatives for dispersive
Havriliak-Negami media", Signal Processing, 107, pp.
312-318 (2015).
6. Mescia, L., Bia, P., and Caratelli, D. \Fractional
derivative based FDTD modeling of transient wave
propagation in Havriliak-Negami media", IEEE Transactions
on Microwave Theory and Techniques, 62(9),
pp. 1920-1929 (2014).
7. Ghaziani, R.K., Alidousti, J., and Eshkaftaki, A.B.
\Stability and dynamics of a fractional order Leslie-
Gower prey-predator model", Applied Mathematical
Modelling, 40(3), pp. 2075-2086 (2016).
8. Caratelli, D., Mescia, L., Bia, P., and Stukach,
O.V. \Fractional-calculus-based FDTD algorithm for
ultrawideband electromagnetic characterization of arbitrary
dispersive dielectric materials", IEEE Transactions
on Antennas and Propagation, 64(8), pp. 3533-
3544 (2016).
9. Kumar, S. \A new analytical modelling for fractional
telegraph equation via Laplace transform", Applied
Mathematical Modelling, 38(13), pp. 3154-3163 (2014).
10. Zingales, M. \Fractional-order theory of heat transport
in rigid bodies", Communications in Nonlinear
Science and Numerical Simulation, 19(11), pp. 3938-
3953 (2014).
11. Mainardi, F., Luchko, Y., and Pagnini, G. \The fundamental
solution of the space-time fractional di usion
equation", arXiv Preprint cond-mat/0702419 (2007).
12. Qi, H.T. and Jiang, X.Y. \Solutions of the spacetime
fractional Cattaneo di usion equation", Physica
A. 390, pp. 1876-1883 (2011).
13. Liu, L., Zheng, L., Liu, F., and Zhang, X. \Heat
conduction with fractional Cattaneo-Christov upperconvective
ux model", International Journal
of Thermal Sciences, 112, pp. 421-426 (2017).
14. Ezzat, M.A., El-Bary, A.A., and Fayik, M.A. \Fractional
Fourier law with three-phase lag of thermoelasticity",
Mechanics of Advanced Materials and Structures,
20(8), pp. 593-602 (2013).
15. Zhao, J., Zheng, L., Zhang, X., and Liu, F. \Convection
heat and mass transfer of fractional MHD
uid in a porous medium with Soret and
Dufour e ects". International Journal of Heat and
Mass Transfer, 103, pp. 203-210 (2016).
16. Narayan, O. and Ramaswamy, S. \Anomalous heat
conduction in one-dimensional momentum-conserving
systems", Physical review letters, 89(20), p. 200601
17. Zheng, L., Liu, Y., and Zhang, X. \Slip e ects
on MHD
ow of a generalized Oldroyd-B
uid with
fractional derivative", Nonlinear Analysis: Real World
Applications, 13(2), pp. 513-523 (2012).
18. Xiaojun, Y. and Baleanu, D. \Fractal heat conduction
problem solved by local fractional variation iteration
method", Thermal Science, 17(2), pp. 625-628 (2013).
19. Povstenko, Y.Z. \Fractional radial di usion in a cylinder",
Journal of Molecular Liquids, 137(1), pp. 46-50
20. Abouelregal, A.E. \Fractional heat conduction equation
for an in nitely generalized, thermoelastic, long
solid cylinder", International Journal for Computational
Methods in Engineering Science and Mechanics,
17(5-6), pp. 374-381 (2016).
21. Jiang, X. and Xu, M. \The time fractional heat conduction
equation in the general orthogonal curvilinear
coordinate and the cylindrical coordinate systems",
Physica A: Statistical Mechanics and its Applications,
389(17), pp. 3368-3374 (2010).
22. Atangana, A. \Derivative with two fractional orders:
A new avenue of investigation toward revolution in
fractional calculus", The European Physical Journal
Plus, 131(10), pp. 1-13 (2016)
23. Liu, Y., Fang, Z., Li, H., and He, S. \A mixed nite
element method for a time-fractional fourth-order partial
di erential equation", Applied Mathematics and
Computation, 243, pp. 703-717 (2014).
24. Atangana, A. and Baleanu, D. \New fractional derivatives
with nonlocal and non-singular kernel: Theory
and application to heat transfer model", Therm Sci.
20(2), pp. 763-769 (2016).
25. Gomez-Aguilar, J.F., Rosales-Garca, J.J., Bernal-
Alvarado, J.J., Cordova-Fraga, T., and Guzman-
Cabrera, R. \Fractional mechanical oscillators". Rev.
Mex. Fs, 58(4), pp. 348-352 (2012).
J.F. Gomez-Aguilar et al./Scientia Iranica, Transactions B: Mechanical Engineering 25 (2018) 2175{2185 2185
26. Mainardi, F. \An historical perspective on fractional
calculus in linear viscoelasticity", Fractional Calculus
and Applied Analysis, 15(4), pp. 712-717 (2012).
27. Povstenko, Y. \Fractional heat conduction and related
theories of thermoelasticity", In Fractional Thermoelasticity,
Springer International Publishing, pp. 13-33