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q-Homotopy analysis method for solving the seventh-order time-fractional Lax’s Korteweg–de Vries and Sawada–Kotera equations

  • Lanre AkinyemiEmail author
Article
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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.

Keywords

Lax’s seventh-order Korteweg–de Vries equation Sawada–Kotera seventh-order equation q-Homotopy analysis method Fractional derivative 

Mathematics Subject Classification

26A33 34A12 35R11 35Q53 

Notes

References

  1. Adomian G (1994) Solving frontier problems of physics: the decomposition method. Kluwer, BostonCrossRefGoogle Scholar
  2. Chen D, Chen Y, Xue D (2013) Three fractional-order TV-models for image de-noising. J Comput Inf Syst 9(12):4773–80Google Scholar
  3. 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 2:1097–1106MathSciNetzbMATHGoogle Scholar
  4. Das GC, Sarma J, Uberoi C (1997) Explosion of a soliton in a multicomponent plasma. Phys Plasmas 4(6):2095–2100MathSciNetCrossRefGoogle Scholar
  5. El-Sayed SM, Kaya D (2004) An application of the ADM to seven order Sawada-Kotera equations. Appl Math Comput 157:93–101MathSciNetzbMATHGoogle Scholar
  6. El-Tawil MA, Huseen SN (2012) The Q-homotopy analysis method (Q-HAM). Int J Appl Math Mech 8(15):51–75Google Scholar
  7. El-Tawil MA, Huseen SN (2013) On convergence of the q -homotopy analysis method. Int J Contemp Math Sci 8:481–497MathSciNetCrossRefGoogle Scholar
  8. 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):805–817Google Scholar
  9. Huseen SN (2016) Series solutions of fractional initial-value problems by q-homotopy analysis method. Int J Innov Sci Eng Technol 3(1):27–41Google Scholar
  10. Iyiola OS (2013) A numerical study of ito equation and Sawada-Kotera equation both of time-fractional type. Adv Math Sci J 2(2):71–79MathSciNetGoogle Scholar
  11. Iyiola OS (2015) On the solutions of non-linear time-fractional gas dynamic equations: an analytical approach. Int J Pure Appl Math 98(4):491–502CrossRefGoogle Scholar
  12. Iyiola OS (2016) Exact and approximate solutions of fractional diffusion equations with fractional reaction terms. Prog Fract Differ Appl 2(1):21–30CrossRefGoogle Scholar
  13. 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):51CrossRefGoogle Scholar
  14. 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):105zbMATHGoogle Scholar
  15. 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):471–477MathSciNetzbMATHGoogle Scholar
  16. 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., AmsterdamGoogle Scholar
  17. Liao SJ (1992) The proposed homotopy analysis technique for the solution of nonlinear problems. Ph.D. Thesis, Shanghai Jiao Tong UniversityGoogle Scholar
  18. Liao SJ (1995) An approximate solution technique not depending on small parameters: a special example. Int J Non-linear Mech 30(3):371–380MathSciNetCrossRefGoogle Scholar
  19. Liao SJ (2003) Beyond perturbation: introduction to the homotopy analysis method. Chapman and Hall/CRC, Boca RatonCrossRefGoogle Scholar
  20. Liao SJ (2004) On the homotopy analysis method for nonlinear problems. Appl Math Comput 147(2):499–513MathSciNetzbMATHGoogle Scholar
  21. Liao SJ (2005) Comparison between the homotopy analysis method and homotopy perturbation method. Appl Math Comput 169(2):1186–1194MathSciNetzbMATHGoogle Scholar
  22. Luchko YF, Srivastava HM (1995) The exact solution of certain differential equations of fractional order by using operational calculus. Comput Math Appl 29:73–85MathSciNetCrossRefGoogle Scholar
  23. 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 YorkzbMATHGoogle Scholar
  24. Pu YF (2007) Fractional differential analysis for texture of digital image. J Algorithms Comput Technol 1(03):357–80CrossRefGoogle Scholar
  25. 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)Google Scholar
  26. 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 CrossRefGoogle Scholar
  27. 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):18MathSciNetCrossRefGoogle Scholar
  28. Sibatov RT, Svetukhin VV (2015) Subdiffusion kinetics of nanoprecipitate growth and destruction in solid solutions. Theor Math Phys 183(3):846–59MathSciNetCrossRefGoogle Scholar
  29. 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):7715–7726Google Scholar
  30. Soliman AA (2006) A numerical simulation and explicit solutions of KdVBursers’ and Lax’s seventh-order KdV equations. Chaos Solitons Fractals 29(2):294–302MathSciNetCrossRefGoogle Scholar
  31. Tarasov VE, Tarasova VV (2017) Time-dependent fractional dynamics with memory in quantum and economic physics. Ann Phys 383:579–99MathSciNetCrossRefGoogle Scholar
  32. 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):1999–2019MathSciNetCrossRefGoogle Scholar
  33. 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 6:322–8CrossRefGoogle Scholar
  34. Zhang J, Wei Z, Xiao L (2012a) Adaptive fractional multiscale method for image de-noising. J Math Imaging Vis 43:39–49CrossRefGoogle Scholar
  35. 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):299–306Google Scholar

Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019

Authors and Affiliations

  1. 1.Department of MathematicsOhio UniversityAthensUSA

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