DOI QR코드

DOI QR Code

SOLUTION OF TENTH AND NINTH-ORDER BOUNDARY VALUE PROBLEMS BY HOMOTOPY PERTURBATION METHOD

  • Received : 2009.11.23
  • Accepted : 2009.12.16
  • Published : 2010.03.25

Abstract

In this paper, we apply homotopy perturbation method (HPM) for solving ninth and tenth-order boundary value problems. The suggested algorithm is quite efficient and is practically well suited for use in these problems. The proposed iterative scheme finds the solution without any discretization, linearization or restrictive assumptions. Several examples are given to verify the reliability and efficiency of the method. The fact that the proposed homotopy perturbation method solves nonlinear problems without using Adomian's polynomials can be considered as a clear advantage of this technique over the decomposition method.

Keywords

References

  1. S. Abbasbandy, Numerical solutions of nonlinear Klein-Gordon equation by variational iteration method, Internat. J. Numer. Mech. Engg. 70 (2007), 876-881. https://doi.org/10.1002/nme.1924
  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. https://doi.org/10.1016/j.cam.2006.07.012
  3. M. A. Abdou and A. A. Soliman, New applications of variational iteration method, Phys. D 211 (1-2) (2005), 1-8. https://doi.org/10.1016/j.physd.2005.08.002
  4. R. P. Agarwal, Boundary value problems for higher order differential equations, world scientific, Singapore (1986).
  5. S. Chandrasekhar, Hydrodynamic and hydro magnetic stability, Dover, New York 1981.
  6. K. Djidjeli, E. H. Twizell and A. Boutayeb, Numerical methods for special nonlinear boundary value problems of order 2m, J. Comput. Appl. Math. 47 (1993), 35-45. https://doi.org/10.1016/0377-0427(93)90088-S
  7. 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. https://doi.org/10.1515/IJNSNS.2007.8.2.229
  8. A. Ghorbani, Beyond Adomian's polynomials: He polynomials, Chas. Soltn. Fract. (2007), in press.
  9. J. H. He, Homotopy perturbation method for solving boundary value problems, Phys. Lett. A 350(2006), 87-88. https://doi.org/10.1016/j.physleta.2005.10.005
  10. J. H. He, Comparison of homotopy perturbation method and homotopy analysis method, Appl. Math. Comput. 156 (2004), 527-539. https://doi.org/10.1016/j.amc.2003.08.008
  11. J. H. He, Homotopy perturbation method for bifurcation of nonlinear problems, Int. J. Nonlin. Scne. Numer. Simul. 6 (2) (2005), 207-208. https://doi.org/10.1515/IJNSNS.2005.6.2.207
  12. J. H. He, The homotopy perturbation method for nonlinear oscillators with discontinuities, Appl. Math. Comput. 151 (2004), 287-292. https://doi.org/10.1016/S0096-3003(03)00341-2
  13. J. H. He, A coupling method of homotopy technique and perturbation technique for nonlinear problems, Int. J. Nonlinear Mech. 35 (1) (2000), 115-123.
  14. J. H. He, Some asymptotic methods for strongly nonlinear equation, Int. J. Mod. Phys. 20 (20)10 (2006), 1144-1199.
  15. J. H. He, Variational iteration method- Some recent results and new interpretations, J. Comput. Appl. Math. 207 (2007), 3-17. https://doi.org/10.1016/j.cam.2006.07.009
  16. J. H. He and X. Wu, Variational iteration method: New developments and applications, Comput. Math. Appl. 54(2007), 881-894. https://doi.org/10.1016/j.camwa.2006.12.083
  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. https://doi.org/10.1016/S0020-7462(98)00048-1
  18. J. H. He, Variational iteration method for autonomous ordinary differential systems, Appl. Math. Comput. 114 (2-3) (2000), 115-123. https://doi.org/10.1016/S0096-3003(99)00104-6
  19. J. H. He and X. H. Wu, Construction of solitary solution and compaction-like solution by variational iteration method, Chas. Soltn. Fract. 29 (1) (2006), 108-113. https://doi.org/10.1016/j.chaos.2005.10.100
  20. J. H. He, The variational iteration method for eighth-order initial boundary value problems, Phys. Scr. 76(6) (2007), 680-682. https://doi.org/10.1088/0031-8949/76/6/016
  21. S. Momani and Z. Odibat, Application of He's variational iteration method to Helmholtz equations, Chas. Soltn. Frctls. 27 (5) (2006), 1119-1123.
  22. S. T. Mohyud-Din and M. A. Noor, Homotopy perturbation method for solving fourth-order boundary value problems, Math. Prob. Engg. (2007), 1-15, Article ID 98602, doi:10.1155/2007/98602.
  23. M. A. Noor and S. T. Mohyud-Din, An efficient algorithm for solving fifth order boundary value problems, Math. Comput. Modl. 45 (2007), 954-964. https://doi.org/10.1016/j.mcm.2006.09.004
  24. M. A. Noor and S. T. Mohyud-Din, Homotopy perturbation method for solving sixth-order boundary value problems, Comput. Math. Appl. 55 (12) (2008), 2953-2972. https://doi.org/10.1016/j.camwa.2007.11.026
  25. M. A. Noor and S. T. Mohyud-Din, Variational iteration method for solving higher-order nonlinear boundary value problems using He's polynomials, Int. J. Nonlin. Scne. Num. Simul. 9 (2) (2008).
  26. M. A. Noor and S. T. Mohyud-Din, Variational iteration method for solving fifth-order boundary value problems using He's polynomials, Math. Prob. Engg. (2008), Article ID 954794, doi: 10:1155/2008/954794.
  27. M. A. Noor and S. T. Mohyud-Din, Variational iteration method for unsteady flow of gas through a porous medium using He's polynomials and Pade approximants, Comput. Math. Appl. (2008).
  28. M. A. Noor and S. T. Mohyud-Din, Variational homotopy perturbation method for solving higher dimensional initial boundary value problems, Math. Prob. Engg. (2008), in press.
  29. M. A. Noor and S. T. Mohyud-Din, Solution of singular and nonsingular initial and boundary value problems by modified variational iteration method, Math. Prob. Engg. (2008), in press.
  30. M. A. Noor and S. T. Mohyud-Din, Solution of twelfth-order boundary value problems by variational iteration technique, J. Appl. Math. Computg (2008), DOI: 10.1007/s12190-008-0081-0.
  31. 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. https://doi.org/10.1016/j.amc.2006.12.071
  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. https://doi.org/10.1016/j.camwa.2006.12.057
  33. M. A. Noor, S. T. Mohyud-Din and M. Tahir, Variational iteration decomposition method for solving eighth-order boundary value problems, Diff. Eqns. Nonlin. Mech. (2007), Article ID, 19529, doi: 10.1155/2007/19529.
  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, A reliable algorithm for solving tenth-order boundary value problems, Appl. Math. Inf. Sci. (2008).
  36. M. A. Noor, S. T. Mohyud-Din and A. Waheed, Variation of parameters method for solving fifth-order boundary value problems, Appl. Math. Inf. Sci. (2008).
  37. M. A. Noor, S. T. Mohyud-Din and A. Waheed, Exp-function method for solving Kuramoto-Sivashinsky and Boussinesq equations, J. Appl. Math. Computg. (2008), DOI: 10.1007/s12190-008-0083-y.
  38. M. A. Noor, S. T. Mohyud-Din and A. Waheed, Exp-function method for generalized travelling solutions of master partial differential equations, Acta Applnda. Mathmtce. (2008), DOI: 10.1007/s10440-008-9245-z.
  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. https://doi.org/10.1016/S0898-1221(00)00187-5
  40. A. M. Wazwaz, The modified decomposition method for solving linear and nonlinear boundary value problems of tenth-order and twelfth-order, Int. J. Nonlin. Scie. Num. Sim. 1 (2000), 17-24. https://doi.org/10.1515/IJNSNS.2000.1.1.17
  41. L. Xu, The variational iteration method for fourth-order boundary value problems, Chas. Soltn. Fract. (2007), in press.
  42. L. Xu, He's homotopy perturbation method for a boundary layer equation in unbounded domain, Comput. Math. Appl. 54 (2007), 1067-1070. https://doi.org/10.1016/j.camwa.2006.12.052