THERMAL SCIENCE

International Scientific Journal

Thermal Science - Online First

Authors of this Paper

External Links

online first only

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

ABSTRACT
Starting from the Cattaneo constitutive relation with a Jeffrey’s kernel the derivation of a transient heat diffusion equation with relaxation term expressed through the Caputo-Fabrizio time fractional derivative has been developed. This approach allows seeing the physical background of the newly defined Caputo-Fabrizio time fractional derivative and demonstrates how other constitutive equations could be modified with non-singular fading memories.
KEYWORDS
PAPER SUBMITTED: 2016-02-29
PAPER REVISED: 2016-05-04
PAPER ACCEPTED: 2016-05-04
PUBLISHED ONLINE: 2016-05-21
DOI REFERENCE: https://doi.org/10.2298/TSCI160229115H
REFERENCES
  1. Caputo, M., Fabrizio, M., A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl. ,1 (2015), 2, pp. 73-85.
  2. Podlubny, I, Fractional Differential Equations, Academic Press, New York, 1999.
  3. 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.
  4. Losada, J., Nieto, J. J., Properties of a New Fractional Derivative without Singular Kernel, Progr. Fract. Differ. Appl. , 1 (2015), 2, pp. 87-92.
  5. Atangana, A.; Badr, S.T.A. Extension of the RLC electrical circuit to fractional derivative without singular kernel. Adv. Mech. Eng. 2015, 7, pp1-6.
  6. 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, doi: 10.1177/1687814015613758.
  7. Atangana, A., Badr, S.T.A. Analysis of the Keller-Segel model with a fractional derivative without singular kernel, Entropy, 17(2015),,pp. 4439-4453; doi:10.3390/e17064439
  8. Atangana, A, On the new fractional derivative and application to nonlinear Fisher's reaction-diffusion equation, App. Math. Comp., 273 (2016), pp. 948-956; dpi: 10.1016/j.amc.2015.10.021 .
  9. Alsaedi, A., Baleanu,D. Sina Etemad,S., Shahram Rezapour, S. On Coupled Systems of Time-Fractional Differential Problems by Using a New Fractional Derivative, J. Function Spaces, v. 2016, Article ID 4626940, doi: 10.1155/2016/4626940.
  10. Gómez-Aguilar, J.F. ,Yépez-Martínez. H., Calderón-Ramón, C., Cruz-Orduña, I. Fabricio Escobar-Jiménez, R., Hugo Olivares-Peregrino , V., Modeling of a Mass-Spring-Damper System by Fractional Derivatives with and without a Singular Kernel, Entropy, 17 (2005), pp. 6289-6303; doi:10.3390/e17096289
  11. Atangana, A, Badr, S.T.A., New model of groundwater flowing within a confine aquifer: application of Caputo-Fabrizio derivative, Arabian Journal of Geosciences, January 2016, 9:8, doi: 10.1007/s12517-015-2060-8.
  12. 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.xxx-xxx, in press; DOI:10.2298/TSCI160112019H
  13. Cattaneo, C, On the conduction of heat (In Italian), Atti Sem. Mat. Fis. Universit´a Modena, 3 (1948),1, pp. 83-101.
  14. Carillo, S., Some Remarks on Materials with Memory: Heat Conduction and Viscoelasticity, J. Nonlinear Math. Phys., 12 (2005), Suppl. 1, pp. 163-178.
  15. Ferreira, J.A., de Oliveira, P., Qualitative analysis of a delayed non-Fickian model, Applicable Analysis, 87(2008), 8, pp. 873-886.
  16. Araujo, A., Ferreira, J.A., Oliveira, P., The effect of memory terms in diffusion phenomena, J. Comp. Math., 24 (2000), 1, 91-102.
  17. Joseph, D.D., Preciozi, Heat waves, Rev.Mod. Phys., 61 (1989), 1, pp. 41-73
  18. Curtin, M. E, Pipkin, A.C., A general theory of heat conduction with finite wave speeds, Archives of Rational Mathematical Analysis, 31 (1968), 2, pp. 313-332.
  19. Yang, X.J., Srivastava, H.M., Machado, J.A.T. A new fractional derivative without singular kernel: application to the modelling of the steady heat flow, Thermal Science, 2016, in press, doi: 10.2298/TSCI151224222Y