In the present article, we apply a numerical scheme, namely, homotopy analysis Sumudu transform algorithm, to derive the analytical and numerical solutions of a nonlinear fractional differential-difference problem occurring in nanohydrodynamics, heat conduction in nanoscale, and electronic current that flows through carbon nanotubes. The homotopy analysis Sumudu transform method (HASTM) is an inventive coupling of Sumudu transform algorithm and homotopy analysis technique that makes the calculation very easy. The fractional model is also handled with the aid of Adomian decomposition method (ADM). The numerical results derived with the help of HASTM and ADM are approximately same, so this scheme may be considered an alternative and well-organized technique for attaining analytical and numerical solutions of fractional model of discontinued problems. The analytical and numerical results derived by the application of the proposed technique reveal that the scheme is very effective, accurate, flexible, easy to apply, and computationally very appropriate for such type of fractional problems arising in physics, chemistry, biology, engineering, finance, etc.

References

References
1.
Sheikholeslami
,
M.
,
Ganji
,
D. D.
,
Javed
,
M. Y.
, and
Ellahi
,
R.
,
2015
, “
Effect of Thermal Radiation on Magnetohydrodynamics Nanofluid Flow and Heat Transfer by Means of Two Phase Model
,”
J. Magn. Magn. Mater.
,
374
(
15
), pp.
36
43
.
2.
Sheikholeslami
,
M.
, and
Ganji
,
D. D.
,
2015
, “
Entropy Generation of Nanofluid in Presence of Magnetic Field Using Lattice Boltzmann Method
,”
Phys. A
,
417
, pp.
273
286
.
3.
Sheikholeslami
,
M.
, and
Rashidi
,
M. M.
,
2015
, “
Effect of Space Dependent Magnetic Field on Free Convection of Fe3O4–Water Nanofluid
,”
J. Taiwan Inst. Chem. Eng.
,
56
, pp.
6
15
.
4.
Sheikholeslami
,
M.
,
Hayat
,
T.
, and
Alsaedi
,
A.
,
2016
, “
MHD Free Convection of Al2O3–Water Nanofluid Considering Thermal Radiation: A Numerical Study
,”
Int. J. Heat Mass Transfer
,
96
, pp.
513
524
.
5.
Sheikholeslami
,
M.
,
Rashidi
,
M. M.
, and
Ganji
,
D. D.
,
2015
, “
Effect of Non-Uniform Magnetic Field on Forced Convection Heat Transfer of Fe3O4–Water Nanofluid
,”
Comput. Methods Appl. Mech. Eng.
,
294
, pp.
299
312
.
6.
Liu
,
Y.
, and
He
,
J. H.
,
2007
, “
Bubble Electrospinning for Mass Production of Nanofibers
,”
Int. J. Nonlinear Sci. Numer. Simul.
,
8
, pp.
393
396
.
7.
He
,
J. H.
,
Wan
,
Y. Q.
, and
Xu
,
L.
,
2007
, “
Nano-Effects, Quantum-Like Properties in Electrospun Nanofibers
,”
Chaos, Solitons Fractals
,
33
(
1
), pp.
26
37
.
8.
He
,
J. H.
,
Liu
,
Y. Y.
,
Xu
,
L.
, and
Yu
,
J. Y.
,
2007
, “
Micro Sphere With Nanoporosity by Electrospinning
,”
Chaos, Solitons Fractals
,
32
(
3
), pp.
1096
1100
.
9.
He
,
J. H.
, and
Zhu
,
S. D.
,
2008
, “
Differential-Difference Model for Nanotechnology
,”
J. Phys.: Conf. Ser.
,
96
(
1
), p.
012189
.
10.
Zhu
,
S. D.
,
2007
, “
Exp-Function Method for the Discrete mKdV Lattice
,”
Int. J. Nonlinear Sci. Numer. Simul.
,
8
(
3
), pp.
465
469
.
11.
Zhu
,
S. D.
,
Chu
,
Y. M.
, and
Qiu
,
S. L.
,
2009
, “
The Homotopy Perturbation Method for Discontinued Problems Arising in Nanotechnology
,”
Comput. Math. Appl.
,
58
, pp.
2398
2401
.
12.
Singh
,
J.
,
Kumar
,
D.
, and
Kumar
,
S.
,
2013
, “
A Reliable Algorithm for Solving Discontinued Problems Arising in Nanotechnology
,”
Sci. Iran.
,
20
(
3
), pp.
1059
1062
.
13.
Shah
,
K.
, and
Singh
,
T.
,
2015
, “
The Mixture of New Integral Transform and Homotopy Perturbation Method for Solving Discontinued Problems Arising in Nanotechnology
,”
Open J. Appl. Sci.
,
5
(
11
), pp.
688
695
.
14.
Podlubny
,
I.
,
1999
,
Fractional Differential Equations
,
Academic Press
,
New York
.
15.
Caputo
,
M.
,
1969
,
Elasticita e Dissipazione
,
Zani-Chelli
,
Bologna, Italy
.
16.
Miller
,
K. S.
, and
Ross
,
B.
,
1993
,
An Introduction to the Fractional Calculus and Fractional Differential Equations
,
Wiley
,
New York
.
17.
Baleanu
,
D.
,
Guvenc
,
Z. B.
, and
Machado
,
J. A. T.
, eds.,
2010
,
New Trends in Nanotechnology and Fractional Calculus Applications
,
Springer
,
New York
.
18.
Tarasov
,
V. E.
,
2016
, “
Three-Dimensional Lattice Models With Long-Range Interactions of Grünwald–Letnikov Type for Fractional Generalization of Gradient Elasticity
,”
Meccanica
,
51
(
1
), pp.
125
138
.
19.
Sierociuk
,
D.
,
Skovranek
,
T.
,
Macias
,
M.
,
Podlubny
,
I.
,
Petras
,
I.
,
Dzielinski
,
A.
, and
Ziubinski
,
P.
,
2015
, “
Diffusion Process Modeling by Using Fractional-Order Models
,”
Appl. Math. Comput.
,
257
, pp.
2
11
.
20.
Magin
,
R. L.
,
2006
,
Fractional Calculus in Bioengineering
,
Begell House
,
Redding, CT
.
21.
Garra
,
R.
,
Giusti
,
A.
,
Mainardi
,
F.
, and
Pagnini
,
G.
,
2014
, “
Fractional Relaxation With Time-Varying Coefficient
,”
Fractional Calculus Appl. Anal.
,
17
(
2
), pp.
424
439
.
22.
Saxena
,
R. K.
,
Mathai
,
A. M.
, and
Haubold
,
H. J.
,
2006
, “
Fractional Reaction-Diffusion Equations
,”
Astrophys. Space Sci.
,
305
(
3
), pp.
289
296
.
23.
Nigmatullin
,
R. R.
,
Ceglie
,
C.
,
Maione
,
G.
, and
Striccoli
,
D.
,
2015
, “
Reduced Fractional Modeling of 3D Video Streams: The FERMA Approach
,”
Nonlinear Dyn.
,
80
(
4
), pp.
1869
1882
.
24.
Kilbas
,
A. A.
,
Srivastava
,
H. M.
, and
Trujillo
,
J. J.
,
2006
,
Theory and Applications of Fractional Differential Equations
,
Elsevier
,
Amsterdam, The Netherlands
.
25.
Razminia
,
K.
,
Razminia
,
A.
, and
Machado
,
J. A. T.
,
2016
, “
Analytical Solution of Fractional Order Diffusivity Equation With Wellbore Storage and Skin Effects
,”
ASME J. Comput. Nonlinear Dyn.
,
11
(
1
), p.
011006
.
26.
Krishnasamy
,
V. S.
, and
Razzaghi
,
M.
,
2016
, “
The Numerical Solution of the Bagley–Torvik Equation With Fractional Taylor Method
,”
ASME J. Comput. Nonlinear Dyn.
,
11
(
5
), p.
051010
.
27.
He
,
J. H.
,
1999
, “
Homotopy Perturbation Technique
,”
Comput. Method Appl. Mech. Eng.
,
178
(
3/4
), pp.
257
262
.
28.
Sheikholeslami
,
M.
, and
Ganji
,
D. D.
,
2013
, “
Heat Transfer of Cu–Water Nanofluid Flow Between Parallel Plates
,”
Powder Technol.
,
235
, pp.
873
879
.
29.
Sheikholeslami
,
M.
, and
Ganji
,
D. D.
,
2015
, “
Nanofluid Flow and Heat Transfer Between Parallel Plates Considering Brownian Motion Using DTM
,”
Comput. Methods Appl. Mech. Eng.
,
283
, pp.
651
663
.
30.
Sheikholeslami
,
M.
,
Rashidi
,
M. M.
,
Saad
,
D. M. A.
,
Firouzi
,
F.
,
Rokni
,
H. B.
, and
Domairry
,
G.
, “
Steady Nanofluid Flow Between Parallel Plates Considering Thermophoresis and Brownian Effects
,”
J. King Saud Univ. Sci.
(in press).
31.
Adomian
,
G.
,
1994
,
Solving Frontier Problems of Physics: The Decomposition Method
,
Kluwer Academic Publishers
,
Boston
.
32.
Wazwaz
,
A. M.
,
Rach
,
R.
, and
Duan
,
J. S.
,
2015
, “
Solving New Fourth-Order Emden–Fowler-Type Equations by the Adomian Decomposition Method
,”
Int. J. Comput. Methods Eng. Sci. Mech.
,
16
(
2
), pp.
121
131
.
33.
Duan
,
J. S.
,
Rach
,
R.
, and
Wazwaz
,
A. M.
,
2014
, “
A Reliable Algorithm for Positive Solutions of Nonlinear Boundary Value Problems by the Multistage Adomian Decomposition Method
,”
Open Eng.
,
5
(
1
), pp.
59
74
.
34.
Bobolian
,
E.
,
Vahidi
,
A. R.
, and
Shoja
,
A.
,
2014
, “
An Efficient Method for Nonlinear Fractional Differential Equations: Combination of the Adomian Decomposition Method and Spectral Method
,”
Int. J. Pure Appl. Math.
,
45
(
6
), pp.
1017
1028
.
35.
Sheikholeslami
,
M.
,
Ganji
,
D. D.
, and
Ashorynejad
,
H. R.
,
2013
, “
Investigation of Squeezing Unsteady Nanofluid Flow Using ADM
,”
Powder Technol.
,
239
, pp.
259
265
.
36.
Ashorynejad
,
H. R.
,
Javaherdeh
,
K.
,
Sheikholeslami
,
M.
, and
Ganji
,
D. D.
,
2014
, “
Investigation of the Heat Transfer of a Non-Newtonian Fluid Flow in an Axisymmetric Channel With Porous Wall Using Parameterized Perturbation Method (PPM)
,”
J. Franklin Inst.
,
351
(
2
), pp.
701
712
.
37.
Fakour
,
M.
,
Vahabzadeh
,
A.
,
Ganji
,
D. D.
, and
Hatami
,
M.
,
2015
, “
Analytical Study of Micropolar Fluid Flow and Heat Transfer in a Channel With Permeable Walls
,”
J. Mol. Liq.
,
204
, pp.
198
204
.
38.
Malvandi
,
A.
,
Moshizi
,
S. A.
, and
Ganji
,
D. D.
,
2014
, “
An Analytical Study on Unsteady Motion of Vertically Falling Spherical Particles in Quiescent Power-Law Shear-Thinning Fluids
,”
J. Mol. Liq.
,
193
, pp.
166
173
.
39.
Liao
,
S. J.
,
2003
,
Beyond Perturbation: Introduction to Homotopy Analysis Method
,
Chapman and Hall/CRC Press
,
Boca Raton, FL
.
40.
Liao
,
S. J.
,
1995
, “
An Approximate Solution Technique Not Depending on Small Parameters: A Special Example
,”
Int. J. Non-Linear Mech.
,
30
(
3
), pp.
371
380
.
41.
Liao
,
S. J.
,
2012
,
Homotopy Analysis Method in Nonlinear Differential Equations
,
Springer-Verlag
,
Berlin
.
42.
Kumar
,
S.
,
Kumar
,
D.
, and
Singh
,
J.
,
2014
, “
Numerical Computation of Fractional Black–Scholes Equation Arising in Financial Market
,”
Egypt. J. Basic Appl. Sci.
,
1
(
3–4
), pp.
177
183
.
43.
Zou
,
K.
, and
Nagarajaiah
,
S.
,
2015
, “
An Analytical Method for Analyzing Symmetry-Breaking Bifurcation and Period-Doubling Bifurcation
,”
Commun. Nonlinear Sci. Numer. Simul.
,
22
(
1–3
), pp.
780
792
.
44.
Odibat
,
Z.
, and
Bataineh
,
A. S.
,
2015
, “
An Adaptation of Homotopy Analysis Method for Reliable Treatment of Strongly Nonlinear Problems: Construction of Homotopy Polynomials
,”
Math. Methods Appl. Sci.
,
38
(
5
), pp.
991
1000
.
45.
Miandoab
,
E. M.
,
Tajaddodianfar
,
F.
,
Pishkenari
,
H. N.
, and
Ouakad
,
H. M.
,
2015
, “
Analytical Solution for the Forced Vibrations of a Nano-Resonator With Cubic Nonlinearities Using Homotopy Analysis Method
,”
Int. J. Nanosci. Nanotechnol.
,
11
(
3
), pp.
159
166
.
46.
Freidoonimehr
,
N.
,
Rostami
,
B.
, and
Rashidi
,
M. M.
,
2015
, “
Predictor Homotopy Analysis Method for Nanofluid Flow Through Expanding or Contracting Gaps With Permeable Walls
,”
Int. J. Biomath.
,
8
(
4
), p.
1550050
.
47.
Khuri
,
S. A.
,
2001
, “
A Laplace Decomposition Algorithm Applied to a Class of Nonlinear Differential Equations
,”
J. Appl. Math.
,
1
(
4
), pp.
141
155
.
48.
Ramswroop
,
Singh
,
J.
, and
Kumar
,
D.
,
2015
, “
Numerical Computation of Fractional Lotka–Volterra Equation Arising in Biological Systems
,”
Nonlinear Eng.
,
4
(
2
), pp.
117
125
.
49.
Ramswroop
,
Singh
,
J.
, and
Kumar
,
D.
,
2014
, “
Numerical Study for Time-Fractional Schrödinger Equations Arising in Quantum Mechanics
,”
Nonlinear Eng.
,
3
(
3
), pp.
169
177
.
50.
Gupta
,
S.
,
Kumar
,
D.
, and
Singh
,
J.
,
2015
, “
Numerical Study for Systems of Fractional Differential Equations Via Laplace Transform
,”
J. Egypt. Math. Soc.
,
23
(
2
), pp.
256
262
.
51.
Kumar
,
D.
,
Singh
,
J.
, and
Kumar
,
S.
,
2015
, “
Analytical Modeling for Fractional Multi-Dimensional Diffusion Equations by Using Laplace Transform
,”
Commun. Numer. Anal.
,
2015
(
1
), pp.
16
29
.
52.
Rathore
,
S.
,
Kumar
,
D.
,
Singh
,
J.
, and
Gupta
,
S.
,
2012
, “
Homotopy Analysis Sumudu Transform Method for Nonlinear Equations
,”
Int. J. Ind. Math.
,
4
(
4
), pp.
301
314
.
53.
Watugala
,
G. K.
,
1993
, “
Sumudu Transform—A New Integral Transform to Solve Differential Equations and Control Engineering Problems
,”
Int. J. Math. Educ. Sci. Tech.
,
24
(
1
), pp.
35
43
.
54.
Belgacem
,
F. B. M.
,
Karaballi
,
A. A.
, and
Kalla
,
S. L.
,
2003
, “
Analytical Investigations of the Sumudu Transform and Applications to Integral Production Equations
,”
Math. Probl. Eng.
,
2003
(
3
), pp.
103
118
.
55.
Khalaf
,
R. F.
, and
Belgacem
,
F. B. M.
,
2014
, “
Extraction of the Laplace, Fourier, and Mellin Transforms From the Sumudu Transform
,”
AIP Conf. Proc.
,
1637
, p.
1426
.
56.
Srivastava
,
H. M.
,
Golmankhaneh
,
A. K.
,
Baleanu
,
D.
, and
Yang
,
X. J.
,
2014
, “
Local Fractional Sumudu Transform With Application to IVPs on Cantor Sets
,”
Abstr. Appl. Anal.
,
2014
, p.
620529
.
57.
Singh
,
J.
,
Kumar
,
D.
, and
Kilicman
,
A.
,
2014
, “
Numerical Solutions of Nonlinear Fractional Partial Differential Equations Arising in Spatial Diffusion of Biological Populations
,”
Abstr. Appl. Anal.
,
2014
, p.
535793
.
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