Abstract
This article presents exact and approximate solutions of the seventh order time-fractional Lax’s Korteweg–de Vries (7TfLKdV) and Sawada–Kotera (7TfSK) equations using the modification of the homotopy analysis method called the q-homotopy analysis method. Using this method, we construct the solutions to these problems in the form of recurrence relations and present the graphical representation to verify all obtained results in each case for different values of fractional order. Error analysis is also illustrated in the present investigation.
Similar content being viewed by others
References
Adomian G (1994) Solving frontier problems of physics: the decomposition method. Kluwer, Boston
Chen D, Chen Y, Xue D (2013) Three fractional-order TV-models for image de-noising. J Comput Inf Syst 9(12):4773–80
Darvishi MT, Khani F, Kheybari S (2007) A numerical solution of the Laxs 7th-order KdV equation by pseudospectral method and darvishis preconditioning. Int J Comtep Math Sci 2:1097–1106
Das GC, Sarma J, Uberoi C (1997) Explosion of a soliton in a multicomponent plasma. Phys Plasmas 4(6):2095–2100
El-Sayed SM, Kaya D (2004) An application of the ADM to seven order Sawada-Kotera equations. Appl Math Comput 157:93–101
El-Tawil MA, Huseen SN (2012) The Q-homotopy analysis method (Q-HAM). Int J Appl Math Mech 8(15):51–75
El-Tawil MA, Huseen SN (2013) On convergence of the q -homotopy analysis method. Int J Contemp Math Sci 8:481–497
Huseen SN (2015) Solving the K(2,2) equation by means of the q-homotopy analysis method (q-HAM). Int J Innov Sci Eng Technol 2(8):805–817
Huseen SN (2016) Series solutions of fractional initial-value problems by q-homotopy analysis method. Int J Innov Sci Eng Technol 3(1):27–41
Iyiola OS (2013) A numerical study of ito equation and Sawada-Kotera equation both of time-fractional type. Adv Math Sci J 2(2):71–79
Iyiola OS (2015) On the solutions of non-linear time-fractional gas dynamic equations: an analytical approach. Int J Pure Appl Math 98(4):491–502
Iyiola OS (2016) Exact and approximate solutions of fractional diffusion equations with fractional reaction terms. Prog Fract Differ Appl 2(1):21–30
Iyiola OS, Zaman FD (2016) A note on analytical solutions of nonlinear fractional 2D heat equation with non-local integral terms. Pramana J Phys 87(4):51
Iyiola OS, Soh ME, Enyi CD (2013) Generalised homotopy analysis method (q-HAM) for solving foam drainage equation of time fractional type. Math Eng Sci Aerosp 4(4):105
Jafari H, Yazdani A, Vahidi J, Ganji DD (2008) Application of He’s variational iteration method for solving seventh order Sawada-Kotera equations. Appl Math Sci 2(9–12):471–477
Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. In: North-Holland Mathematics Studies, vol 204. Elsevier Science B.V., Amsterdam
Liao SJ (1992) The proposed homotopy analysis technique for the solution of nonlinear problems. Ph.D. Thesis, Shanghai Jiao Tong University
Liao SJ (1995) An approximate solution technique not depending on small parameters: a special example. Int J Non-linear Mech 30(3):371–380
Liao SJ (2003) Beyond perturbation: introduction to the homotopy analysis method. Chapman and Hall/CRC, Boca Raton
Liao SJ (2004) On the homotopy analysis method for nonlinear problems. Appl Math Comput 147(2):499–513
Liao SJ (2005) Comparison between the homotopy analysis method and homotopy perturbation method. Appl Math Comput 169(2):1186–1194
Luchko YF, Srivastava HM (1995) The exact solution of certain differential equations of fractional order by using operational calculus. Comput Math Appl 29:73–85
Podlubny I (1998) Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, vol 198. Academic press, New York
Pu YF (2007) Fractional differential analysis for texture of digital image. J Algorithms Comput Technol 1(03):357–80
Salas AH, Gomez CA (2010) Application of the Cole-Hopf transformation for finding exact solutions to several forms of the seventh-order KdV equation. Math Probl Eng 2010: 14 (Article ID 194329)
Şenol M, Tasbozan O, Kurt A (2018) Comparison of two reliable methods to solve fractional Rosenau-Hyman equation. Math Methods Appl Sci. https://doi.org/10.1002/mma.5497
Şenol M, Atpinar S, Zararsiz Z, Salahshour S, Ahmadian A (2019) Approximate solution of time-fractional fuzzy partial differential equations. Comput Appl Math 38(1):18
Sibatov RT, Svetukhin VV (2015) Subdiffusion kinetics of nanoprecipitate growth and destruction in solid solutions. Theor Math Phys 183(3):846–59
Soh ME, Enyi CD, Iyiola OS, Audu JD (2014) Approximate analytical solutions of strongly nonlinear fractional BBM-Burger’s equations with dissipative term. Appl Math Sci 8(155):7715–7726
Soliman AA (2006) A numerical simulation and explicit solutions of KdVBursers’ and Lax’s seventh-order KdV equations. Chaos Solitons Fractals 29(2):294–302
Tarasov VE, Tarasova VV (2017) Time-dependent fractional dynamics with memory in quantum and economic physics. Ann Phys 383:579–99
Ullah A, Chen W, Sun HG, Khan MA (2017) An efficient variational method for restoring images with combined additive and multiplicative noise. Int J Appl Comput Math 3(3):1999–2019
Yasar E, Yildirim Y, Khalique CM (2016) Lie symmetry analysis, conservation laws and exact solutions of the seventh-order time fractional Sawada-Kotera-Ito equation. Results Phys 6:322–8
Zhang J, Wei Z, Xiao L (2012a) Adaptive fractional multiscale method for image de-noising. J Math Imaging Vis 43:39–49
Zhang Y, Pu YF, Hu JR, Zhou JL (2012b) A class of fractional-order variational image in-painting models. Appl Math Inf Sci 06(02):299–306
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Vasily E. Tarasov.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Akinyemi, L. q-Homotopy analysis method for solving the seventh-order time-fractional Lax’s Korteweg–de Vries and Sawada–Kotera equations. Comp. Appl. Math. 38, 191 (2019). https://doi.org/10.1007/s40314-019-0977-3
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40314-019-0977-3
Keywords
- Lax’s seventh-order Korteweg–de Vries equation
- Sawada–Kotera seventh-order equation
- q-Homotopy analysis method
- Fractional derivative