Analytical solutions of the first and second model of Hristov fractional diffusion equations based on the non-singular Atangana-Baleanu derivative have been developed. The solutions are based on an integral method based on the consequent application of the Fourier and Laplace transforms. Particular cases of Hristov fractional diffusion equations considering operators with orders converging to unity have been analyzed, too.
REFERENCES
1.
Y.
Adjabi
, F.
Jarad
, and T.
Abdeljawad
, “On generalized fractional operators and a Gronwall type inequality with applications
,” Filomat
31
(17
), 5457
–5473
(2017
). 2.
A.
Atangana
and D.
Baleanu
, “New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model,” preprint arXiv:1602.03408 (2016).3.
A.
Atangana
and J.
Gómez-Aguilar
, “Fractional derivatives with no-index law property: Application to chaos and statistics
,” Chaos Solitons Fractals
114
, 516
–535
(2018
). 4.
B.
Baeumer
, M.
Kovács
, M. M.
Meerschaert
, and H.
Sankaranarayanan
, “Reprint of: Boundary conditions for fractional diffusion
,” J. Comput. Appl. Math.
339
, 414
–430
(2018
). 5.
M.
Caputo
and M.
Fabrizio
, “A new definition of fractional derivative without singular kernel
,” Prog. Fract. Differ. Appl
1
(2
), 1
–13
(2015
).6.
A.
Cetinkaya
and O.
Kiymaz
, “The solution of the time-fractional diffusion equation by the generalized differential transform method
,” Math. Comput. Model.
57
(9
), 2349
–2354
(2013
). 7.
E.
Franc
and D.
Goufo
, Chaotic processes using the two-parameter derivative with non-singular and non-local kernel: Basic theory and applications
,” Chaos
26
(8
), 084305
(2016
). 8.
R.
Garra
and R.
Garrappa
, “The prabhakar or three parameter mittag–leffler function: Theory and application
,” Commun. Nonlinear Sci. Numer. Simul.
56
, 314
–329
(2018
). 9.
J.
Hristov
, “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
,” Therm. Sci.
20
(2
), 757
–762
(2016
). 10.
J.
Hristov
, “Multiple integral-balance method basic idea and an example with Mullin’s model of thermal grooving
,” Therm. Sci
21
, 1555
–1560
(2017
). 11.
J.
Hristov
, “The non-linear Dodson diffusion equation: Approximate solutions and beyond with formalistic fractionalization
,” Math. Nat. Sci
1
(1
), 1
–17
(2017
). 12.
J.
Hristov
, “Space-fractional diffusion with a potential power-law coefficient: Transient approximate solution
,” Prog. Fract. Differ. Appl.
3
(1
), 19
–39
(2017
). 13.
J.
Hristov
, “Fourth-order fractional diffusion model of thermal grooving: Integral approach to approximate closed form solution of the Mullins model
,” Math. Model. Nat. Phenom.
13
(1
), 6
(2018
). 14.
J.
Hristov
, “Integral-balance solution to nonlinear subdiffusion equation
,” Front. Fract. Calculus
1
, 70
(2018
). 15.
J.
Hristov
, “The heat radiation diffusion equation: Explicit analytical solutions by improved integral-balance method
,” Therm. Sci.
22
(2
), 777
–788
(2018
). 16.
J.
Hristov
, “Integral balance approach to 1-d space-fractional diffusion models,” in Mathematical Methods in Engineering, (Springer, 2019), pp. 111–131.17.
J.
Hristov
, “On the Atangana-Baleanu derivative and its relation to the fading memory concept: The diffusion equation formulation,” in Trends in Theory and Applications of fractional derivatives with Mittag-Leffler kernel, edited by José Francisco Gómez, Lizeth Torres and Ricardo Escobar, (Springer, 2019). .18.
J.
Hristov
, “A transient flow of a non-newtonian fluid modelled by a mixed time-space derivative: An improved integral-balance approach,” in Mathematical Methods in Engineering, (Springer, 2019), pp. 153–174.19.
F.
Jarad
and T.
Abdeljawad
, “A modified laplace transform for certain generalized fractional operators
,” Results Nonlinear Anal.
2018
(2
), 88
–98
(2018
).20.
F.
Jarad
, E.
Uğurlu
, T.
Abdeljawad
, and D.
Baleanu
, “On a new class of fractional operators
,” Adv. Differ. Eq.
2017
(1
), 247
(2017
). 21.
U. N
Katugampola
, “A new approach to generalized fractional derivatives
,” Bull. Math. Anal. Appl.
6
(4
), 1
–15
(2014
).22.
T.
Langlands
and B. I.
Henry
, “The accuracy and stability of an implicit solution method for the fractional diffusion equation
,” J. Comput. Phys.
205
(2
), 719
–736
(2005
). 23.
Y.
Lin
and C.
Xu
, “Finite difference/spectral approximations for the time-fractional diffusion equation
,” J. Comput. Phys.
225
(2
), 1533
–1552
(2007
). 24.
Y.
Ma
, F.
Zhang
, and C.
Li
, “The asymptotics of the solutions to the anomalous diffusion equations
,” Comput. Math. Appl.
66
(5
), 682
–692
(2013
). 25.
S. L.
Mitchell
and T.
Myers
, “Improving the accuracy of heat balance integral methods applied to thermal problems with time dependent boundary conditions
,” Int. J. Heat Mass Transf.
53
(17-18
), 3540
–3551
(2010
). 26.
T. G.
Myers
, “Optimal exponent heat balance and refined integral methods applied to Stefan problems
,” Int. J. Heat Mass Transf.
53
(5-6
), 1119
–1127
(2010
). 27.
K. M.
Owolabi
and A.
Atangana
, “Robustness of fractional difference schemes via the caputo subdiffusion-reaction equations
,” Chaos Solitons Fractals
111
, 119
–127
(2018
). 28.
I.
Podlubny
, A.
Chechkin
, T.
Skovranek
, Y. Q.
Chen
, and B. M.
Vinagre Jara
, J. Comput. Phys.
228
(8
), 3137
–3153
(2009
). 29.
S.
Priyadharsini
, “Stability of fractional neutral and integrodifferential systems
,” J. Fract. Calculus Appl.
7
(1
), 87
–102
(2016
).30.
N.
Sene
, “Exponential form for Lyapunov function and stability analysis of the fractional differential equations
,” J. Math. Comput. Sci.
18
(4
), 388
–397
(2018
). 31.
N.
Sene
, “Lyapunov characterization of the fractional nonlinear systems with exogenous input
,” Fractal Fract.
2
(2
), 17
(2018
). 32.
N.
Sene
, “Solutions for some conformable differential equations
,” Prog. Fract. Differ. Appl.
4
(4
), 493
–501
(2018
).33.
N.
Sene
, “Stokes’ first problem for heated flat plate with Atangana–Baleanu fractional derivative
,” Chaos Solitons Fractals
117
, 68
–75
(2018
). 34.
N. H.
Sweilam
, A. M.
Nagy
, and A. A.
El-Sayed
, “On the numerical solution of space fractional order diffusion equation via shifted Chebyshev polynomials of the third kind
,” J. King Saud Univ. Sci.
28
(1
), 41
–47
(2016
). © 2019 Author(s).
2019
Author(s)
You do not currently have access to this content.
