An International Publications House

Albert Science International Organization

Connecting People With Pioneering Thought

Albert Science International Organization (ASIO) is international , peer-reviewed , open access , cum print version & online journals.
JOURNALS || ASIO Journal of Engineering & Technological Perspective Research (ASIO-JETPR) [ISSN: 2455-3794]
ON THE APPROXIMATE SOLUTION OF THE FORNBERG-WHITHAM EQUATION

Author Names : Shaheed N. Huseen
Page No. : 01-05  volume 3 issue 1
Article Overview

Abstract:

In this paper, an approximate solution for Fornberg-Whitham Equation by using the q-homotopy analysis method (q-HAM) was proposed. The q-homotopy analysis method contains the convergence control parameter n, which provides us with a simple way to adjust and control the convergence region of rate series solution. Comparison of the results with the exact solution shows the accuracy of the q-HAM.

Keywords: q-homotopy analysis method, Fornberg-Whitham   Equation.

Reference
  1. Zhou J. and Tian L., A type of bounded traveling wave solutions for the Fornberg Whitham equation, J. Math. Anal. Appl. 346 (2008) 255-261.
  2. El-Tawil M. A. and HuseenS.N., On Convergence of The q-Homotopy Analysis Method, Int. J. Contemp. Math. Sciences, Vol. 8, 2013, no. 10, 481 – 497.
  3. Whitham G.B., Variational methods and applications to water wave, Proc. R. Soc. Lond. A299 (1967) 6-25.
  4. Iyiola O. S., Ojo, G. O. and  Audu, J. D., A Comparison Results of Some Analytical Solutions of Model in Double Phase Flow through Porous Media,  Journal of Mathematics and System Science 4 (2014) 275-284.
  5. Liao S. J., Proposed homotopy analysis techniques for the solution of nonlinear problems, Ph.D. dissertation, Shanghai Jiao Tong University, Shanghai, 1992.
  6. Liao S. J., An approximate solution technique which does not depend upon small parameters: a special example, Int. J. Non-linear Mech. 30:371-380(1995).
  7. Liao S. J., An approximate solution technique which does not depend upon small parameters (Part 2): an application in fluid mechanics, Int. J. Non-linear Mech. 32:815- 822(1997).
  8. Liao S. J., An explicit, totally analytic approximation of Blasius viscous flow problem, Int. J. Non-Linear Mech. 34:759-778(1999).
  9. Liao S. J., A uniformly valid analytic solution of 2D viscous flow past a semi-infinite flat plate. J. Fluid Mech. 385:101-128(1999).
  10. Liao S. J., A. Campo, Analytic solutions of the temperature distribution in Blasius viscous flow problems, J. Fluid Mech. 453:411-425(2002).
  11. Liao S. J., An explicit analytic solution to the Thomas-Fermi equation, Appl. Math. Comput. 144: 495-506(2003).
  12. Liao S. J., On the analytic solution of magneto hydro dynamic flows of non Newtonian fluids over a stretching sheet, J. Fluid Mech. 488:189-212(2003)
  13. Liao S. J., On the homotopy analysis method for nonlinear problems, Appl. Math. Comput.147: 499-513(2004).
  14. Liao S. J. and Magyari E., Exponentially decaying boundary layers as limiting cases of families of algebraically decaying ones, Z. Angew. Math. Phys. 57:777-792(2006).
  15. Fornberg B. and Whitham G.B., A numerical and theoretical study of certain nonlinear wave phenomena,   Phil. Trans. R. Soc. Lond. A 289 (1978) 373-404.
  16. Huseen S. N. and Grace, S. R. 2013, Approximate Solutions of Nonlinear Partial Differential Equations by Modified q-Homotopy Analysis Method (mq-HAM), Hindawi Publishing Corporation, Journal of Applied Mathematics, Article ID 569674, 9 pages http:// dx.doi.org/10.1155/ 2013/ 569674.
  17. El-Tawil M. A. and Huseen S.N., The q-Homotopy Analysis Method (q-HAM), International Journal of Applied mathematics and mechanics, 8 (15): 51-75, 2012.
  18. Huseen S. N., Grace S. R. and El-Tawil M. A.  2013, The Optimal q-Homotopy Analysis Method (Oq-HAM), International Journal of Computers & Technology, Vol 11, No. 8.
  19. Huseen S. N., Solving the K(2,2) Equation by Means of the q-Homotopy Analysis Method (q-HAM), International Journal of Innovative Science, Engineering & Technology, Vol. 2 Issue 8, August 2015.
  20. Huseen S. N., Series Solutions of Fractional Initial-Value Problems by q-Homotopy Analysis Method, International Journal of Innovative Science, Engineering & Technology, Vol. 3 Issue 1, January 2016.
  21. Huseen Shaheed N. Application of optimal q-homotopy analysis method to second order initial and boundary value problems. Int J Sci Innovative Math Res (IJSIMR) 2015; 3(1):18–24.
  22. Huseen S. N., A Numerical Study of One-Dimensional HyperbolicTelegraph Equation, Journal of Mathematics and System Science 7 (2017) 62-72.
  23. Iyiola O. S., Soh, M. E. and  Enyi, C. D. 2013, Generalized Homotopy Analysis Method (q-HAM) For Solving Foam Drainage Equation of Time Fractional Type, Mathematics in Engineering, Science & Aerospace (MESA), Vol. 4, Issue 4, p. 429-440.
  24. Iyiola O. S., Ojo, G. O. and  Audu, J. D., A Comparison Results of Some Analytical Solutions of Model in Double Phase Flow through Porous Media, Journal of Mathematics and System Science 4 (2014) 275-284.
  25. Liao S. J. and Tan Y., A general approach to obtain series solutions of nonlinear differential equations, Stud. Appl. Math. 119:297-354(2007).
  26. Liao S. J., A general approach to get series solution of non-similarity boundary-layer flows, Commun. Nonlinear Sci. Numer. Simulat. 14:2144-2159(2009).
  27. Liao S. J., Beyond perturbation: Introduction to the homotopy analysis method, CRC pressLLC, Boca Raton, 2003.
  28. Mahmoudi Y. and Kazemian M., Some Notes on Homotopy Analysis Method for Solving the Fornberg-Whitham Equation, J. Basic. Appl. Sci. Res., 2(3)2985-2990, 2012.