THERMAL SCIENCE

International Scientific Journal

Abass H. Abdel Kader

,

Mohamed S. Abdel Latif

,

Dumitru Baleanu

STUDYING HEAT CONDUCTION IN A SPHERE CONSIDERING HYBRID FRACTIONAL DERIVATIVE OPERATOR

ABSTRACT
In this paper, the fractional heat equation in a sphere with hybrid fractional derivative operator is investigated. The heat conduction is considered in the case of central symmetry with heat absorption. The closed form solution in the form of three parameter Mittag-Leffler function is obtained for two Dirichlet boundary value problems. The joint finite sine Fourier-Laplace transform is used for solving these two problems. The dynamics of the heat transfer in the sphere is illustrated through some numerical examples and figures.
KEYWORDS
Heat conduction with absorption, Hybrid fractional derivative operator, Three parameter Mittag-Leffler function, Finite Fourier transform, laplace transform
PAPER SUBMITTED: 2020-05-24
PAPER REVISED: 2021-04-22
PAPER ACCEPTED: 2021-04-30
PUBLISHED ONLINE: 2021-12-04
DOI REFERENCE: https://doi.org/10.2298/TSCI200524332K
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2022, VOLUME 26, ISSUE Issue 2, PAGES [1675 - 1683]
REFERENCES
  1. Podlubny, I., Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Elsevier, 1998.
  2. Baleanu, D., Machado, J. A. T. , CJ Luo,A., Fractional dynamics and control. Springer Science & Business Media, 2011.
  3. Baleanu, D., Kai, D., Enrico, S., Fractional calculus: models and numerical methods. Vol. 3. World Scientific, 2012.
  4. Elsaid, A., Abdel Latif, M. S. , Maneea, M., Similarity solutions of fractional order heat equations with variable coefficients. Miskolc Mathematical Notes 17.1 (2016) 245-254.
  5. Elsaid, A., Abdel Latif, M. S. , Maneea, M., Similarity solutions for multiterm time-fractional diffusion equation, Advances in Mathematical Physics 2016 (2016).
  6. Povstenko, Y. , Klekot, J., Fractional heat conduction with heat absorption in a sphere under Dirichlet boundary condition, Computational and Applied Mathematics 37.4 (2018) 4475-4483.
  7. Elsaid, A., Abdel Latif, M. S. , Maneea, M., Similarity solutions for solving Riesz fractional partial differential equations, Progr. Fract. Differ. Appl. 2 (4) (2016).293-298.
  8. Khan, M. F. , Some new hypergeometric transformations via fractional calculus technique, Appl . Math. Inf. Sci. 14 (2) (2020) 177-190.
  9. Cernea, A., Continuous family of solutions for fractional integro-differential inclusions of caputo-katugampola type, Progr. Fract. Differ. Appl. 5 (1) (2020) 37-42.
  10. Mouzakis, D. E., Lazopoulos, A. K., Fractional modelling and the leibniz (L-fractional) derivative as viscoelastic respondents in polymer biomaterials, Progr. Fract. Differ. Appl. 5 (1) (2020) 43-48.
  11. Srivastava, H. M., Saad, K. M., New approximate solution of the time-fractional Nagumo equation involving fractional integrals without singular kernel, Appl. Math. Inf. Sci. 14 (1) (2020) 1-8.
  12. Chatzarakis, G., Deepa, M., Nagajothi, N., Sadhasivam, V., Oscillatory properties of a certain class of mixed fractional differential equations, Appl. Math. Inf. Sci. 14 (1) (2020) 123-131.
  13. Abdel Kader, A. H., Abdel Latif, M. S. , Baleanu, D. Some exact solution of a variable coefficients fractional biological population model. Math. Meth. Appl. Sci. 44 (2021) 4701-4714.
  14. Abdel Latif, M. S., Abdel Kader, A. H., Baleanu, D., The invariant subspace method for solving nonlinear fractional partial differential equations with generalized fractional derivatives., Adv. Difference Equ. 2020 (2020) 119.
  15. He, J. H., A tutorial review on fractal spacetime and fractional calculus. Internat. J. Theoret. Phys. 53 (2014) 3698-3718.
  16. He, J. H., Fractal calculus and its geometrical explanation. Results Phys., 10 (2018) 272-276.
  17. He, J. H., Ain Q. T. New promises and future challenges of fractal calculus: from two-scale thermodynamics to fractal variational principle. Therm. Sci. 24 (2A) (2020) 659-681.
  18. Wang, K. L., Wang, K. J., He, C. H. Physical insight of local fractional calculus and its application to fractional Kdv-Burgers-Kuramoto equation. Fractals 27 (07) (2019) 1950122.
  19. He, J. H., & El-Dib, Y. O. Periodic property of the time-fractional Kundu-Mukherjee-Naskar equation. Results Phys., 19 (2020) 103345.
  20. Wang, K. L., & Liu, S. Y. He's fractional derivative and its application for fractional Fornberg-Whitham equation. Therm. Sci. 21 (5) (2017) 2049-2055.
  21. Baleanu, D., Fernandez, A., Akgül, A., On a Fractional Operator Combining Proportional and Classical Differintegrals. Mathematics 8 (3) (2020) 360
  22. Akgül, E. K., Akgül, A., Baleanu, D. Laplace Transform Method for Economic Models with Constant Proportional Caputo Derivative, Fractal Fract. 4 (3) (2020) 30.
  23. Asjad, M. I., Ikram, M. D., Akgül, A. Analysis of MHD viscous fluid flow through porous medium with novel power law fractional differential operator, Phys. Scr. 95 (11) (2020) 115209.
  24. Chu, Y. M., Ikram, M. D., Asjad, M. I., Ahmadian, A., Ghaemi, F. Influence of hybrid nanofluids and heat generation on coupled heat and mass transfer flow of a viscous fluid with novel fractional derivative. J. Therm. Anal. Calorim. (2021). doi.org/10.1007/s10973-021-10692-8
  25. Povstenko, Y., Linear fractional diffusion-wave equation for scientists and engineers. Switzerland: Springer International Publishing, 2015.
  26. Sandev, T. , Tomovski, Ž., Fractional Equations and Models: Theory and Applications. Vol. 61. Springer Nature, 2019.
  27. Tomovski, Z. , Hilfer, R. , Srivastava, H. M., Fractional and operational calculus with generalized fractional derivative operators and Mittag-Leffler type functions." Integral Transforms and Special Functions 21.11 (2010): 797-814.
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Studying heat conduction in a sphere considering hybrid fractional derivative operator

© 2024 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, National Institute of the Republic of Serbia, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International licence