Research Papers

Numerical Computation of a Fractional Model of Differential-Difference Equation

[+] Author and Article Information
Devendra Kumar

Department of Mathematics,
JECRC University,
Jaipur, Rajasthan 303905, India
e-mail: devendra.maths@gmail.com

Jagdev Singh

Department of Mathematics,
Jagan Nath University,
Jaipur, Rajasthan 303901, India
e-mail: jagdevsinghrathore@gmail.com

Dumitru Baleanu

Department of Mathematics,
Faculty of Arts and Sciences,
Cankaya University,
Etimesgut/Ankara 06790, Turkey;
Institute of Space Sciences,
Magurele-Bucharest 077125, Romania
e-mail: dumitru@cankaya.edu.tr

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received February 15, 2016; final manuscript received May 28, 2016; published online July 8, 2016. Assoc. Editor: Stefano Lenci.

J. Comput. Nonlinear Dynam 11(6), 061004 (Jul 08, 2016) (6 pages) Paper No: CND-16-1066; doi: 10.1115/1.4033899 History: Received February 15, 2016; Revised May 28, 2016

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.

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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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Liu, Y. , and He, J. H. , 2007, “ Bubble Electrospinning for Mass Production of Nanofibers,” Int. J. Nonlinear Sci. Numer. Simul., 8, pp. 393–396.
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. [CrossRef]
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. [CrossRef]
He, J. H. , and Zhu, S. D. , 2008, “ Differential-Difference Model for Nanotechnology,” J. Phys.: Conf. Ser., 96(1), p. 012189. [CrossRef]
Zhu, S. D. , 2007, “ Exp-Function Method for the Discrete mKdV Lattice,” Int. J. Nonlinear Sci. Numer. Simul., 8(3), pp. 465–469.
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. [CrossRef]
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.
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. [CrossRef]
Podlubny, I. , 1999, Fractional Differential Equations, Academic Press, New York.
Caputo, M. , 1969, Elasticita e Dissipazione, Zani-Chelli, Bologna, Italy.
Miller, K. S. , and Ross, B. , 1993, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York.
Baleanu, D. , Guvenc, Z. B. , and Machado, J. A. T. , eds., 2010, New Trends in Nanotechnology and Fractional Calculus Applications, Springer, New York.
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. [CrossRef]
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.
Magin, R. L. , 2006, Fractional Calculus in Bioengineering, Begell House, Redding, CT.
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.
Saxena, R. K. , Mathai, A. M. , and Haubold, H. J. , 2006, “ Fractional Reaction-Diffusion Equations,” Astrophys. Space Sci., 305(3), pp. 289–296. [CrossRef]
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. [CrossRef]
Kilbas, A. A. , Srivastava, H. M. , and Trujillo, J. J. , 2006, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands.
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. [CrossRef]
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. [CrossRef]
He, J. H. , 1999, “ Homotopy Perturbation Technique,” Comput. Method Appl. Mech. Eng., 178(3/4), pp. 257–262. [CrossRef]
Sheikholeslami, M. , and Ganji, D. D. , 2013, “ Heat Transfer of Cu–Water Nanofluid Flow Between Parallel Plates,” Powder Technol., 235, pp. 873–879. [CrossRef]
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. [CrossRef]
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).
Adomian, G. , 1994, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publishers, Boston.
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. [CrossRef]
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. [CrossRef]
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.
Sheikholeslami, M. , Ganji, D. D. , and Ashorynejad, H. R. , 2013, “ Investigation of Squeezing Unsteady Nanofluid Flow Using ADM,” Powder Technol., 239, pp. 259–265. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Liao, S. J. , 2003, Beyond Perturbation: Introduction to Homotopy Analysis Method, Chapman and Hall/CRC Press, Boca Raton, FL.
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. [CrossRef]
Liao, S. J. , 2012, Homotopy Analysis Method in Nonlinear Differential Equations, Springer-Verlag, Berlin.
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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.
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. [CrossRef]
Khuri, S. A. , 2001, “ A Laplace Decomposition Algorithm Applied to a Class of Nonlinear Differential Equations,” J. Appl. Math., 1(4), pp. 141–155. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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.
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. [CrossRef]
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. [CrossRef]
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.
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.
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.


Grahic Jump Location
Fig. 1

The comparative study between the HASTM and exact solution

Grahic Jump Location
Fig. 2

Plots of Un(n, t) versus n at t = 1 for distinct values of γ



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