In this paper, an analytical technique is proposed to determine the exact solution of fractional order modified Fornberg–Whitham equation. Since exact solution of fractional Fornberg–Whitham equation is unknown, first integral method has been applied to determine exact solutions. The solitary wave solution of fractional modified Fornberg–Whitham equation has been attained by using first integral method. The approximate solutions of fractional modified Fornberg–Whitham equation, obtained by optimal homotopy asymptotic method (OHAM), are compared with the exact solutions obtained by the first integral method. The obtained results are presented in tables to demonstrate the efficiency of these proposed methods. The proposed schemes are quite simple, effective, and expedient for obtaining solution of fractional modified Fornberg–Whitham equation.
Issue Section:
Research Papers
References
1.
Tavazoei
, M. S.
, and Haeri
, M.
, 2009
, “Describing Function Based Methods for Predicting Chaos in a Class of Fractional Order Differential Equations
,” Nonlinear Dyn.
, 57
(3
), pp. 363
–373
.2.
Podlubny
, I.
, 1999
, Fractional Differential Equations
, Academic Press
, New York
.3.
Bagley
, R. L.
, and Torvik
, P. J.
, 1984
, “On the Appearance of the Fractional Derivative in the Behavior of Real Materials
,” ASME J. Appl. Mech.
, 51
(2
), pp. 294
–298
.4.
Caputo
, M.
, 1969
, Elasticita` e Dissipazione
, Zanichelli
, Bologna, Italy
.5.
Suarez
, L. E.
, and Shokooh
, A.
, 1997
, “An Eigenvector Expansion Method for the Solution of Motion Containing Fractional Derivatives
,” ASME J. Appl. Mech.
, 64
(3
), pp. 629
–635
.6.
Miller
, K. S.
, and Ross
, B.
, 1993
, An Introduction to the Fractional Calculus and Fractional Differential Equations
, Wiley
, New York
.7.
Oldham
, K. B.
, and Spainer
, J.
, 1974
, The Fractional Calculus
, Academic Press
, New York
.8.
Caputo
, M.
, 1967
, “Linear Model of Dissipation Whose Q is Almost Frequency Independent, Part II
,” J. R. Astron. Soc.
, 13
(5
), pp. 529
–539
.9.
Das
, S.
, 2011
, Functional Fractional Calculus
, Springer-Verlag
, Berlin
.10.
Herrmann
, R.
, 2011
, Fractional Calculus an Introduction to Physicists
, World Scientific
, Singapore
.11.
Gupta
, A. K.
, and Saha Ray
, S.
, 2014
, “On the Solutions of Fractional Burgers–Fisher and Generalized Fisher's Equations Using Two Reliable Methods
,” Int. J. Math. Math. Sci.
, 2014
, p. 682910
.12.
Saha Ray
, S.
, 2012
, “On Haar Wavelet Operational Matrix of General Order and Its Application for the Numerical Solution of Fractional Bagley Torvik Equation
,” Appl. Math. Comput.
, 218
(9
), pp. 5239
–5248
.13.
Gupta
, A. K.
, and Saha Ray
, S.
, 2014
, “Comparison Between Homotopy Perturbation Method and Optimal Homotopy Asymptotic Method for the Soliton Solution of Boussinesq–Burgers Equation
,” Comput. Fluids
, 103
, pp. 34
–41
.14.
Saha Ray
, S.
, and Gupta
, A. K.
, 2014
, “A Two-Dimensional Haar Wavelet Approach for the Numerical Simulations of Time and Space Fractional Fokker–Planck Equations in Modelling of Anomalous Diffusion Systems
,” J. Math. Chem.
, 52
(8
), pp. 2277
–2293
.15.
Jiang
, H.
, Liu
, F.
, Turner
, I.
, and Burrage
, I.
, 2012
, “Analytical Solutions for the Multi-Term Time-Space Caputo–Riesz Fractional Advection-Diffusion Equations on a Finite Domain
,” J. Math. Anal. Appl.
, 389
(2
), pp. 1117
–1127
.16.
Yıldırım
, A.
, and Kocak
, H.
, 2009
, “Homotopy Perturbation Method for Solving the Space-Time Fractional Advection-Dispersion Equation
,” Adv. Water Resour.
, 32
(12
), pp. 1711
–1716
.17.
Saha Ray
, S.
, Chaudhuri
, K. S.
, and Bera
, R. K.
, 2006
, “Analytical Approximate Solution of Nonlinear Dynamic System Containing Fractional Derivative by Modified Decomposition Method
,” Appl. Math. Comput.
, 182
(1
), pp. 544
–552
.18.
Gorenflo
, R.
, Mainardi
, F.
, and Vivoli
, A.
, 2007
, “Continuous-Time Random Walk and Parametric Subordination in Fractional Diffusion
,” Chaos, Solitons Fractals
, 34
(1
), pp. 87
–103
.19.
He
, J. H.
, 1998
, “Nonlinear Oscillation With Fractional Derivative and Its Applications
,” International Conference on Vibrating Engineering’98
, Dalian, China, pp. 288
–291
.20.
He
, J. H.
, 1999
, “Some Applications of Nonlinear Fractional Differential Equations and Their Approximations
,” Bull. Sci. Technol.
, 15
(2
), pp. 86
–90
.21.
Bagley
, R. L.
, and Torvik
, P. J.
, 1983
, “A Theoretical Basis for the Application of Fractional Calculus to Viscoelasticity
,” J. Rheol.
, 27
(3
), pp. 201
–210
.22.
Bagley
, R. L.
, and Torvik
, P. J.
, 1983
, “Fractional Calculus—A Different Approach to the Analysis of Viscoelastically Damped Structures
,” AIAA J.
, 21
(5
), pp. 741
–748
.23.
Bagley
, R. L.
, and Torvik
, P. J.
, 1985
, “Fractional Calculus in the Transient Analysis of Viscoelastically Damped Structures
,” AIAA J.
, 23
(6
), pp. 918
–925
.24.
Sun
, H. H.
, Onaral
, B.
, and Tsao
, Y.
, 1984
, “Application of Positive Reality Principle to Metal Electrode Linear Polarization Phenomena
,” IEEE Trans. Biomed. Eng.
, 31
(10
), pp. 664
–674
.25.
Sun
, H. H.
, Abdelvahab
, A. A.
, and Onaral
, B.
, 1984
, “Linear Approximation of Transfer Function With a Pole of Fractional Order
,” IEEE Trans. Autom. Control
, 29
(5
), pp. 441
–444
.26.
Mandelbrot
, B.
, 1967
, “Some Noises With 1/f Spectrum, a Bridge Between Direct Current and White Noise
,” IEEE Trans. Inf. Theory
, 13
(2
), pp. 289
–298
.27.
Hartley
, T. T.
, 1995
, “Chaos in a Fractional Order Chua System
,” IEEE Trans. Circuits Syst.
, 42
(8
), pp. 485
–490
.28.
Inc
, M.
, 2008
, “Approximate Analytical Solution of the Space-and Time-Fractional Burgers Equations With Initial Conditions by Variational Iteration Method
,” J. Math. Anal. Appl.
, 345
(1
), pp. 476
–484
.29.
Saha Ray
, S.
, and Bera
, R. K.
, 2005
, “An Approximate Solution of a Nonlinear Fractional Differential Equation by Adomian Decomposition Method
,” Appl. Math. Comput.
, 167
(1
), pp. 561
–571
.30.
Saha Ray
, S.
, and Bera
, R. K.
, 2005
, “Analytical Solution of the Bagley Torvik Equation by Adomian Decomposition Method
,” Appl. Math. Comput.
, 168
(1
), pp. 398
–410
.31.
Odibat
, Z. M.
, 2011
, “On Legendre Polynomial Approximation With the VIM or HAM for Numerical Treatment of Nonlinear Fractional Differential Equations
,” J. Comput. Appl. Math.
, 235
(9
), pp. 2956
–2968
.32.
Gupta
, A. K.
, and Saha Ray
, S.
, 2015
, “The Comparison of Two Reliable Methods for Accurate Solution of Time-Fractional Kaup–Kupershmidt Equation Arising in Capillary Gravity Waves
,” Math. Methods Appl. Sci.
, 39
(3
), pp. 583
–592
.33.
Saha Ray
, S.
, 2013
, “Soliton Solutions for Time Fractional Coupled Modified KdV Equations Using New Coupled Fractional Reduced Differential Transform Method
,” J. Math. Chem.
, 51
(8
), pp. 2214
–2229
.34.
Wazwaz
, A. M.
, 2009
, Partial Differential Equations and Solitary Waves Theory
, Springer, Higher Education Press
, Berlin
.35.
Shang
, N.
, and Zheng
, B.
, 2013
, “Exact Solutions for Three Fractional Partial Differential Equations by the Method
,” Int. J. Appl. Math.
, 43
(3
), pp. 1
–6
.36.
Saha Ray
, S.
, and Gupta
, A. K.
, 2016
, “Numerical Solution of Fractional Partial Differential Equation of Parabolic Type With Dirichlet Boundary Conditions Using Two-Dimensional Legendre Wavelets Method
,” ASME J. Comput. Nonlinear Dyn.
, 11
(1
), p. 011012
.37.
Wang
, Y.
, and Fan
, Q.
, 2012
, “The Second Kind Chebyshev Wavelet Method for Solving Fractional Differential Equations
,” Appl. Math. Comput.
, 218
(17
), pp. 8592
–8601
.38.
Whitham
, G. B.
, 1967
, “Variational Methods and Applications to Water Wave
,” Proc. R. Soc. London, Ser. A
, 299
(1456
), pp. 6
–25
.39.
Fornberg
, B.
, and Whitham
, G. B.
, 1978
, “A Numerical and Theoretical Study of Certain Nonlinear Wave Phenomena
,” Philos. Trans. R. Soc. London Ser. A
, 289
(1361
), pp. 373
–404
.40.
He
, B.
, Meng
, Q.
, and Li
, S.
, 2010
, “Explicit Peakon and Solitary Wave Solutions for the Modified Fornberg–Whitham Equation
,” Appl. Math. Comput.
, 217
(5
), pp. 1976
–1982
.41.
Abidi
, F.
, and Omrani
, K.
, 2010
, “The Homotopy Analysis Method for Solving the Fornberg–Whitham Equation and Comparison With Adomian's Decomposition Method
,” Comput. Math. Appl.
, 59
(8
), pp. 2743
–2750
.42.
Zhou
, J.
, and Tian
, L.
, 2008
, “A Type of Bounded Traveling Wave Solutions for the Fornberg–Whitham Equation
,” J. Math. Anal. Appl.
, 346
(1
), pp. 255
–261
.43.
Gupta
, P. K.
, and Singh
, M.
, 2011
, “Homotopy Perturbation Method for Fractional Fornberg–Whitham Equation
,” Comput. Math. Appl.
, 61
(2
), pp. 250
–254
.44.
Sakar
, M. G.
, Erdogan
, F.
, and Yıldırım
, A.
, 2012
, “Variational Iteration Method for the Time-Fractional Fornberg–Whitham Equation
,” Comput. Math. Appl.
, 63
(9
), pp. 1382
–1388
.45.
Chen
, A.
, Li
, J.
, Deng
, X.
, and Huang
, W.
, 2009
, “Travelling Wave Solutions of the Fornberg–Whitham Equation
,” Appl. Math. Comput.
, 215
(8
), pp. 3068
–3075
.46.
Hesam
, S.
, Nazemi
, A.
, and Haghbin
, A.
, 2012
, “Reduced Differential Transform Method for Solving the Fornberg–Whitham Type Equation
,” Int. J. Nonlinear Sci.
, 13
(2
), pp. 158
–162
.47.
Lu
, J.
, 2011
, “An Analytical Approach to the Fornberg–Whitham Type Equations by Using the Variational Iteration Method
,” Comput. Math. Appl.
, 61
(8
), pp. 2010
–2013
.48.
Saha Ray
, S.
, and Gupta
, A. K.
, 2015
, “A Numerical Investigation of Time-Fractional Modified Fornberg–Whitham Equation for Analysing the Behaviour of Water Waves
,” Appl. Math. Comput.
, 266
, pp. 135
–148
.49.
Raslan
, K. R.
, 2008
, “The First Integral Method for Solving Some Important Nonlinear Partial Differential Equations
,” Nonlinear Dyn.
, 53
(4
), pp. 281
–286
.50.
Abbasbandy
, S.
, and Shirzadi
, A.
, 2010
, “The First Integral Method for Modified Benjamin–Bona–Mahony Equation
,” Commun. Nonlinear Sci. Numer. Simul.
, 15
(7
), pp. 1759
–1764
.51.
Lu
, B.
, 2012
, “The First Integral Method for Some Time Fractional Differential Equations
,” J. Math. Anal. Appl.
, 395
(2
), pp. 684
–693
.52.
Jafari
, H.
, Soltani
, R.
, Khalique
, C. M.
, and Baleanu
, D.
, 2013
, “Exact Solutions of Two Nonlinear Partial Differential Equations by Using the First Integral Method
,” Boundary Value Probl.
, 2013
(1
), p. 117
.53.
Bekir
, A.
, Güner
, Ö.
, and Ünsal
, Ö.
, 2015
, “The First Integral Method for Exact Solutions of Nonlinear Fractional Differential Equations
,” ASME J. Comput. Nonlinear Dyn.
, 10
(2
), p. 021020
.54.
Samko
, S. G.
, Kilbas
, A. A.
, and Marichev
, O. I.
, 1993
, Fractional Integrals and Derivatives: Theory and Applications
, Taylor and Francis
, London
.55.
Yang
, X. J.
, 2012
, Advanced Local Fractional Calculus and Its Applications
, World Science Publisher
, New York
.56.
Yang
, X. J.
, 2012
, “A Short Note on Local Fractional Calculus of Function of One Variable
,” J. Appl. Libr. Inf. Sci.
, 1
(1
), pp. 1
–13
.57.
Yang
, X. J.
, 2012
, “The Zero-Mass Renormalization Group Differential Equations and Limit Cycles in Non-Smooth Initial Value Problems
,” Prespacetime J.
, 3
(9
), pp. 913
–923
.58.
Hu
, M. S.
, Baleanu
, D.
, and Yang
, X. J.
, 2013
, “One-Phase Problems for Discontinuous Heat Transfer in Fractal Media
,” Math. Probl. Eng.
, 2013
, p. 358473
.59.
Agarwal
, O. P.
, 2004
, “A General Formulation and Solution Scheme for Fractional Optimal Control Problems
,” Nonlinear Dyn.
, 38
, pp. 323
–337
.60.
Su
, W. H.
, Yang
, X. J.
, Jafari
, H.
, and Baleanu
, D.
, 2013
, “Fractional Complex Transform Method for Wave Equations on Cantor Sets Within Local Fractional Differential Operator
,” Adv. Differ. Equations
, 2013
(97
), pp. 1
–8
.61.
Yang
, X. J.
, Baleanu
, D.
, and Srivastava
, H. M.
, 2015
, Local Fractional Integral Transforms and Their Applications
, Academic Press (Elsevier)
, London
.62.
Ding
, T. R.
, and Li
, C. Z.
, 1996
, Ordinary Differential Equations
, Peking University Press
, Peking, China
.63.
Bourbaki
, N.
, 1972
, Commutative Algebra
, Addison-Wesley
, Paris, France
.64.
Feng
, Z.
, and Wang
, X.
, 2001
, “Explicit Exact Solitary Wave Solutions for the Kundu Equation and the Derivative Schrödinger Equation
,” Phys. Scr.
, 64
(1
), pp. 7
–14
.65.
Feng
, Z.
, and Roger
, K.
, 2007
, “Traveling Waves to a Burgers–Korteweg–de Vries-Type Equation With Higher-Order Nonlinearities
,” J. Math. Anal. Appl.
, 328
(2
), pp. 1435
–1450
.66.
Marinca
, V.
, and Herisanu
, N.
, 2011
, Nonlinear Dynamical Systems in Engineering
, Springer-Verlag
, Berlin
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