Variational iteration method for solving higher-order nonlinear boundary value problems using He's polynomials

International Journal of Nonlinear Sciences and Numerical Simulation

Volumn 9, Issue 2, 2008, Pages 141-156

  Noor, Muhammad Aslam ^a   Mohyud Din, Syed Tauseef ^a  

^a COMSATS Institute of Information Technology ^*  (Pakistan)

Author keywords

Boussinesq equations; Error estimates; He's polynomials; Higher order boundary value problems; Nonlinear problems; Variational iteration method

Indexed keywords

EID: 43749107936     PISSN: 15651339     EISSN: None     Source Type: Journal    
DOI: 10.1515/IJNSNS.2008.9.2.141     Document Type: Article

References (42)

1. S. Abbasbandy, Numerical solutions of nonlinear Klein-Gordon equation by variational iteration method, Internat. J. Numer. Mech. Engg. 70 (2007), 876-881.
2. S. Abbasbandy, A new application of He's variational iteration method for quadratic Riccati differential equation by using Adomian's polynomials, J. Comput. Appl. Math. 207 (2007), 59-63.
3. R. P. Agarwal, Boundary value problems for higher order differential equations, world scientific, Singapore (1986).
4. A. Boutayeb and E. H. Twizell, Finite-difference methods for the solution of special eighth-order boundary-value problems, Int. J. Comput. Math., 48 (1993), 63-75.
5. R. E. D. Bishop, S. M. Cannon and S. Miao, On coupled bending and torsional vibration of uniform beams, J. Sod. Vibn. 131 (1989), 457-464.
6. S. Chandrasekhar, Hydrodynamic and hydro magnetic stability, Dover, New York, (1981).
7. K. Djidjeli, E. H. Twizell and A. Boutayeb, Numerical methods for special nonlinear boundary-value problems of order 2 m, J. Comput. Appl. Math. 47 (1993), 35-45.
8. M. A. Haji and K. Al-Khaled, Analytic studies and numerical simulation of the generalized Boussinesq equation, Appl. Math. Comput. 191 (2007), 320-333.
9. A. Ghorbani and J. S. Nadjfi, He's homotopy perturbation method for calculating Adomian's polynomials, Int. J. Nonlin. Scne. Num. Simul. 8 (2) (2007), 229-332.
10. A. Ghorbani, Beyond Adomian's polynomials: He polynomials, Chas. Solton. Fract. (2007), in press.
11. J. H. He, Homotopy perturbation method for solving boundary value problems, Phy.Lett. A 350(2006), 87-88.
12. J. H. He, Comparison of homotopy perturbation method and homotopy analysis method, Appl. Math. Comput. 156 (2004), 527-539.
13. J. H. He, Homotopy perturbation method for bifurcation of nonlinear problems, Int. J. Nonlin. Sci Numer. Simul. 6 (2) (2005), 207-208.
14. J. H. He, The homotopy perturbation method for nonlinear oscillators with discontinuities, Appl. Math. Comput. 151 (2004), 287-292.
15. J. H. He, A coupling method of homotopy technique and perturbation technique for nonlinear problems, Int. J. Nonlinear Mech. 35 (1) (2000), 115-123.
16. J. H. He, Variational iteration method- Some recent results and new interpretations, J. Comp. Appl. Math. 207 (2007), 3-17.
17. J. H. He, Variational iteration method, A kind of non-linear analytical technique, some examples, Internat. J. Nonlinear Mech. 34 (4) (1999), 699-708.
18. J. H. He, Variational iteration method for autonomous ordinary differential systems, Appl. Math. Comput. 114 (2-3) (2000), 115-123.
19. J. H. He and X. H. Wu, Construction of solitary solution and compaction-like solution by variational iteration method, Chas. Solton. Fract. 29 (1) (2006), 108-113.
20. J. H. He, Some asymptotic methods for strongly nonlinear equation, Int. J. Mod. Phy. 20 (20)10 (2006), 1144-1199.
21. J. H. He, The variational iteration method for eighth-order initial boundary value problems, Phys. Scr. 76(6) (2007), 680-682.
22. M. Inokuti, H. Sekine and T. Mura, General use of the Lagrange multiplier in nonlinear mathematical physics, in: S. Nemat-Naseer (Ed.). Variational method in the Mechanics of solids, Pergamon press, New York, (1978), 156-162.
23. S. Momani and Z. Odibat, Application of He's Variational iteration method to Helmholtz equations, Chas. Soltns. Fract. 27 (5) (2006), 1119-1123.
24. S. T. Mohyud-Din, Variational decomposition method for unsteady flow of gas through a porous medium, J. Appl. Math. Comutg. (2008).
25. S. T. Mohyud-Din, A reliable algorithm for Blasius equation, Proceedings of ICMS (2007), 616-626.
26. S. T. Mohyud-Din and M. A. Noor, Homotopy perturbation method for solving fourth order boundary value problems, Math. Prob. Engg. (2006), 1-15.
27. M. A. Noor and S. T. Mohyud-Din, An efficient algorithm for solving fifth order boundary value problems, Math. Comput. Modi. 45 (2007), 954-964.
28. M. A. Noor and S. T. Mohyud-Din, Homotopy perturbation method for solving sixth-order boundary value problems, Comput. Math. Appl. (2008), in press.
29. M. A. Noor and S. T. Mohyud-Din, Homotopy method for solving eighth order boundary value problem, J. Math. Anal. Appl. Th. 1(2) (2006), 161-169.
30. M. A. Noor and S. T. Mohyud-Din, Approximate solutions of Flieral Petviashivili equation and its variants, Int. J. Math. Comput. Scne. 2 (4) (2007), 345-360.
31. M. A. Noor and S. T. Mohyud-Din, A reliable approach for solving linear and nonlinear sixth-order boundary value problems, Int. J. Comput. Appl. Math. (2007).
32. M. A Noor and S.T. Mohyud-Din, An efficient method for fourth order boundary value problems, Comput. Math. Appl. 54 (2007), 1101-1111.
33. M. A. Noor and S.T. Mohyud-Din, Variational iteration technique for solving higher order boundary value problems, Appl. Math. Comput. 189(2007), 1929-1942.
34. M. A. Noor and S. T. Mohyud-Din, Variational iteration technique for solving tenth-order boundary value problems, A. J. Math. Mathl. Scne. (2008).
35. M. A. Noor and S. T. Mohyud-Din, Variational iteration decomposition method for solving eighth-order boundary value problems, Diff. Eqns. Nonlin. Mech. (2008), in press.
36. M. A. Noor and S. T. Mohyud-Din, Variational decomposition method for solving singular initial value problems, Int. J. P. Appl. Math. (2008).
37. J. I. Ramos, On the Variational iteration method and other iterative techniques for nonlinear differential equations, Appl. Math. Comput. (2007), in press.
38. S. S. Siddiqi and E. H. Twizell, Spline solution of linear eighth-order boundary value problems, Comput. Math. Appl. Mech. Engg., 131 (1996), 309-325.
39. A. M. Wazwaz, Approximate solutions to boundary value problems of higher-order by the modified decomposition method, Comput. Math. Appl. 40(2000), 679-691.
40. A. M. Wazwaz, The modified decompoition method for solving linear and nonlinear boundary value problems of tenth-order and twelfth-order, Int. J. Nonlinear. Sci. Num. Simul. 1 (2000), 17-24.
41. A. M. Wazwaz, The numerical solutions of special eighth-order boundary value problems by the modified decomposition method, Neu. Parll. Sci. Comput. 8(2) (2000), 133-146.
42. L. Xu, The variational iteration method for fourth-order boundary value problems, Chaos Solton. Fract. (2007), in press.