Finding the one-loop soliton solution of the short-pulse equation by means of the homotopy analysis method

Numerical Methods for Partial Differential Equations

Volumn 25, Issue 2, 2009, Pages 401-408

  Parkes, E J ^a   Abbasbandy, S ^b  

^a UNIVERSITY OF STRATHCLYDE   (United Kingdom)

^b IMAM KHOMEINI INTERNATIONAL UNIVERSITY   (Iran)

Author keywords

Homotopy analysis method; Series solution; Short pulse equation; Solitary wave solution

Indexed keywords

EID: 63449121597     PISSN: 0749159X     EISSN: 10982426     Source Type: Journal    
DOI: 10.1002/num.20348     Document Type: Article

References (31)

1. 1 S. J. Liao, Beyond perturbation: introduction to the homotopy analysis method, Chapman & Hall/CRC Press, Boca Raton, 2003.
2. 2 S. Abbasbandy, The application of homotopy analysis method to nonlinear equations arising in heat transfer, Phys Lett A 360 (2006), 109–113.
3. 3 S. Abbasbandy, The application of homotopy analysis method to solve a generalized Hirota‐Satsuma coupled KdV equation, Phys Lett A 361 (2007), 478–483.
4. 4 S. Abbasbandy, Homotopy analysis method for heat radiation equations, Int Commun Heat Mass Trans 34 (2007), 380–387.
5. 5 S. Abbasbandy and F. Samadian Zakaria, Soliton solutions for the fifth‐order KdV equation with the homotopy analysis method, Nonlinear Dynam 51 (2008), 83–87.
6. 6 S. Abbasbandy, Homotopy analysis method for generalized Benjamin‐Bona‐Mahony equation, Z Angew Math Phys (ZAMP) 59 (2008), 51–62.
7. 7 S. Abbasbandy, Y. Tan, and S. J. Liao, Newton‐homotopy analysis method for nonlinear equations, Appl Math Comput 188 (2007), 1794–1800.
8. 8 S. Abbasbandy, Soliton solutions for the Fitzhugh‐Nagumo equation with the homotopy analysis method, Appl Math Model, to appear.
9. 9 S. J. Liao, A new branch of solutions of boundary‐layer flows over an impermeable stretched plate, Int J Heat Mass Transfer 48 (2005), 2529–2539.
10. 10 S. J. Liao, Series solutions of unsteady boundary‐layer flows over a stretching flat plate, Stud Appl Math 117 (2006), 239–264.
11. 11 S. J. Liao and E. Magyari, Exponentially decaying boundary layers as limiting cases of families of algebraically decaying ones, Z Angew Math Phys (ZAMP) 57 (2006), 777–792.
12. 12 S. J. Liao, J. Su, and A. T. Chwang, Series solutions for a nonlinear model of combined convective and radiative cooling of a spherical body, Int J Heat Mass Transfer 49 (2006), 2437–2445.
13. 13 Y. Tan, H. Xu, and S. J. Liao, Explicit series solution of travelling waves with a front of Fisher equation, Chaos, Solitons and Fractals 31 (2007), 462–472.
14. 14 W. Wu and S. J. Liao, Solving solitary waves with discontinuity by means of the homotopy analysis method, Chaos, Solitons and Fractals 26 (2005), 177–185.
15. 15 M. Sajid, A. M. Siddiqui, and T. Hayat, Wire coating analysis using MHD Oldroyd 8‐constant fluid, Int J Eng Sci 45 (2007), 381–392.
16. 16 T. Hayat and M. Sajid, Homotopy analysis of MHD boundary layer flow of an upper‐convected Maxwell fluid, Int J Eng Sci 45 (2007), 393–401.
17. 17 S. J. Liao and K. Cheung, Homotopy analysis of nonlinear progressive waves in deep water, J Eng Math 45 (2003), 105–116.
18. 18 M. Sajid, T. Hayat, and S. Asghar, On the analytic solution of the steady flow of a fourth grade fluid, Phys Lett A 355 (2006), 18–26.
19. 19 Y. Tan and S. Abbasbandy, Homotopy analysis method for quadratic Riccati differential equation, Commun Nonlinear Sci Numer Simul 13 (2008), 539–546.
20. 20 C. Wang, Analytic solutions for a liquid film on an unsteady stretching surface, Heat Mass Transfer 42 (2006), 759–766.
21. 21 S. Abbasbandy and E. J. Parkes, Solitary smooth‐hump solutions of the Camassa‐Holm equation by means of the homotopy analysis method, Chaos, Solitons and Fractals 36 (2008), 581–591.
22. 22 T. Hayat, Z. Abbas, and M. Sajid, On the analytic solution of magnetohydrodynamic flow of a second grade fluid over a shrinking sheet, ASME J Appl Mech 74 (2007), 1165–1171.
23. 23 T. Hayat, M. Sajid, and M. Ayub, A note on series solution for generalized Couette flow, Commun Nonlinear Sci Numer Simul 12 (2007), 1481–1487.
24. 24 T. Hayat, N. Ahmed, M. Sajid, and S. Asghar, On the MHD flow of a second grade fluid in a porous channel, Comput Math Appl 54 (2007), 407–414.
25. 25 T. Hayat, T. Javed, and M. Sajid, Analytic solution for rotating flow and heat transfer analysis of a third‐grade fluid, Acta Mech 191 (2007), 219–229.
26. 26 M. Sajid and T. Hayat, Non‐similar series solution for boundary layer flow of a third‐order fluid over a stretching sheet, Appl Math Comput 189 (2007), 1576–1585.
27. 27 T. Schäfer and C. E. Wayne, Propagation of ultra‐short optical pulses in cubic nonlinear media, Physica D 196 (2004), 90–105.
28. 28 A. Sakovich and S. Sakovich, The short pulse equation is integrable, J Phys Soc Jpn 74 (2005), 239–241.
29. 29 M. Wadati, K. Konno, and Y. H. Ichikawa, New integrable nonlinear evolution equations, J Phys Soc Jpn 47 (1979), 1698–1700.
30. 30 A. Sakovich and S. Sakovich, Solitary wave solutions of the short pulse equation, J Phys A Math Gen 39 (2006), 361–367.
31. 31 E. J. Parkes, Some perodic and solitary travelling‐wave solutions of the short‐pulse equation, Chaos, Solitons and Fractals, to appear.