Time-Fractional Fornberg-Whitham Equation Solved by Fractional Homotopy Perturbation Transform Method

International Journal of Mathematics Trends and Technology (IJMTT)
© 2022 by IJMTT Journal
Volume-68 Issue-6
Year of Publication : 2022
Authors : Subodh Pratap Singh, Amardeep Singh

How to Cite?

Subodh Pratap Singh, Amardeep Singh, "Time-Fractional Fornberg-Whitham Equation Solved by Fractional Homotopy Perturbation Transform Method ," International Journal of Mathematics Trends and Technology, vol. 68, no. 6, pp. 8-16, 2022. Crossref, https://doi.org/10.14445/22315373/IJMTT-V68I6P502

The paper presents the solution of time fractional fornberg-whitham equation by a fractional homotopy perturbation method (FHPTM). The traditional Adomian Decomposition method (ADM) for solving time-fractional fornberg-whitham equation gives good approximation in the neighborhood of initial conditions only. Here, we have presented the solution with the method FHPTM. The analysis suggests that FHPTM shows more accuracy as compared to Homotopy Analysis Method (HAM) and Adomian Decomposition Method (ADM). Obtained numerical results shows the efficacy of the method..

Keywords : Fractional homotopy perturbation transform method, Fornberg-whitham equation, Adomian Decomposition method, Homotopy analysis method, Partial differential equation, Perturbation methods.


[1] K.B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York, (1974).
[2] K. Diethelm, Analysis of Fractional Differential Equations, Springer-Verlag, Berlin, (2010).
[3] I. Podlubny, Fractional Differential Equations, Academic Press, New York, (1999).
[4] V.E. Lynch, B.A. Carreras, D. Del-Castillo-Negrete, K.M. Ferriera-Mejias, H.R. Hicks, Numerical Methods for the Solution of Partial Differential Equations of Fractional Order, J. Comput. Phys. 192 (2003) 406–421.
[5] M.M. Meerschaert, C. Tadjeran, Finite Difference Approximations for Two-Sided Space-Fractional Partial Differential Equations, Appl. Numer. Math, 56 (2006) 80–90.
[6] C. Tadjeran, M.M. Meerschaert, A Second-Order Accurate Numerical Method for the Two-Dimensional Fractional Diffusion Equation, J. Comput. Phys, 220 (2007) 813–823.
[7] K. Diethelm, N.J. Ford, A.D. Freed, Detailed Error Analysis for A Fractional Adams Method, Numer. Algorithms, 36 (2004) 31–52.
[8] Q. Wang, Homotopy Perturbation Method for Fractional Kdv Equation, Appl. Math. Comput,190 (2007) 1795–1802.
[9] H. Bulut, Comparing Numerical Methods for Boussinesq Equation Model Problem, Numer. Methods Partial Differ. Equ,. 25 (4) (2009) 783–796.
[10] P.K. Gupta, M. Singh, Homotopy Perturbation Method for Fractional Fornberg–Whitham Equation, Comput. Math. Appl, 61 (2011) 250–254.
[11] S. Momani, Z. Odibat, A Novel Method for Nonlinear Fractional Partial Differential Equations: Combination of Dtm and Generalized Taylor’s Formula, J. Comput. Appl. Math, 220 (1–2) (2008) 85–95.
[12] J. Liu, G. Hou, Numerical Solutions of the Space- and Time-Fractional Coupled Burgers Equations By Generalized Differential Transform Method, Appl. Math. Comput, 217 (16) (2011) 7001–7008.
[13] R. Yulita Molliq, M.S.M. Noorani, I. Hashim, R.R. Ahmad, Approximate Solutions of Fractional Zakharov–Kuznetsov Equations By Vim, J. Comput. Appl. Math, 233 (2) (2009) 103–108.
[14] M.G. Sakar, F. Erdogan, A. Yildirim, Variational Iteration Method for the Time-Fractional Fornberg–Whitham Equation, Comput. Math. Appl, 63 (9) (2012) 1382–1388.
[15] M. Inc, the Approximate and Exact Solutions of the Space- and Time-Fractional Burgers Equations With Initial Conditions By Variational Iteration Method, J. Math. Anal. Appl, 345 (1) (2008) 476–484.
[16] H. Jafari, V.D. Gejji, Solving A System of Nonlinear Fractional Differential Equations Using Adomian Decomposition, Appl. Math. Comput, 196 (2006) 644– 651.
[17] S. Momani, Z. Odibat, Analytical Solution of A Time-Fractional Navier–Stokes Equation By Adomian Decomposition Method, Appl. Math. Comput, 177 (2006) 488–494.
[18] S.J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman and Hall/Crc Press, Boca Raton, (2003).
[19] S.J. Liao, An Approximate Solution Technique Which Does Not Depend Upon Small Parameters: A Special Example, Int. J. NonLinear Mech,30 (1995) 371– 380.
[20] S.J. Liao, on the Homotopy Analysis Method for Nonlinear Problems, Appl. Math. Comput, 147 (2004) 499–513.
[21] S.J. Liao, Notes on the Homotopy Analysis Method: Some Definitions and Theorems, Commun. Nonlinear Sci. Numer. Simul, 14 (2009) 983–997.
[22] S.J. Liao, Comparison Between the Homotopy Analysis Method and the Homotopy Perturbation Method, Appl. Math. Comput, 169 (2005) 1186–1194.
[23] S.J. Liao, An Analytic Approach to Solve Multiple Solutions of A Strongly Nonlinear Problem, Appl. Math. Comput, 169 (2005) 854– 865.
[24] L. Song, H. Zhang, Application of Homotopy Analysis Method to Fractional Kdv–Burgers–Kuramoto Equation, Phys. Lett, A 367 (1– 2) (2007) 88–94.
[25] M. Ganjiani, Solution of Nonlinear Fractional Differential Equations Using Homotopy Analysis Method, Appl. Math. Model, 34 (6) (2010) 1634–1641.
[26] F. Abidi, K. Omrani, the Homotopy Analysis Method for Solving the Fornberg–Whitham Equation and Comparison With Adomian’s Decomposition Method, Comput. Math. Appl, 59 (2010) 2743–2750.
[27] I. Podlubny, Geometric and Physical Interpretation of Fractional Integration and Fractional Differentiation, Fract. Calc. Appl. Anal, 5 (4) (2002) 367–386.
[28] G. Adomian, A Review of the Decomposition Method In Applied Mathematics, J. Math. Anal. Appl,135 (1988) 44–501.
[29] G. Adomian, Solving Frontier Problems of Physics: the Decomposition Method, Kluwer Academic Publishers, Boston, (1999).
[30] K. Abbaoui, Y. Cherruault, New Ideas for Proving Convergence of Decomposition Methods, Comput. Math. Appl, 29 (7) (1995) 103– 108.