Numerical Algorithms

, Volume 56, Issue 1, pp 93–106 | Cite as

An analytical approach for solving nonlinear boundary value problems in finite domains

  • Songxin Liang
  • David J. Jeffrey
Original Paper


Based on the homotopy analysis method (HAM), a general analytical approach for obtaining approximate series solutions to nonlinear two-point boundary value problems in finite domains is proposed. To demonstrate its effectiveness, this approach is applied to solve three nonlinear problems, and the analytical solutions obtained are more accurate than the numerical solutions obtained via the shooting method and the sinc-Galerkin method.


Boundary value problem Series solution Homotopy analysis method Symbolic computation Analytical solution 


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  1. 1.
    Doedel, E.: Finite difference methods for nonlinear two-point boundary-value problems. SIAM J. Numer. Anal. 16, 173–185 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Deacon, A.G., Osher, S.: Finite-element method for a boundary-value problem of mixed type. SIAM J. Numer. Anal. 16, 756–778 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Roberts, S.M., Shipman, J.S.: Two Point Boundary Value Problems: Shooting Methods. American Elsevier, New York (1972)zbMATHGoogle Scholar
  4. 4.
    Na, T.Y.: Computational Methods in Engineering Boundary Value Problems. Academic, New York (1979)zbMATHGoogle Scholar
  5. 5.
    El-Gamel, M., Behiry, S.H., Hashish, H.: Numerical method for the solution of special nonlinear fourth-order boundary value problems. Appl. Math. Comput. 145, 717–734 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Ha, S.N.: A nonlinear shooting method for two-point boundary value problems. Comput. Math. Appl. 42, 1411–1420 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Salama, A.A., Mansour, A.A.: Fourth-order finite-difference method for third-order boundary-value problems. Numer. Heat Transf. B 47, 383–401 (2005)CrossRefGoogle Scholar
  8. 8.
    Nayfeh, A.H.: Perturbation Methods. Wiley, New York (2000)zbMATHCrossRefGoogle Scholar
  9. 9.
    Zhou, J.K.: Differential Transformation and Its Applications for Electrical Circuits. Huazhong University Press, Wuhan (1986)Google Scholar
  10. 10.
    Adomian, G.: Solving Frontier Problems of Physics: the Decomposition Method. Kluwer Academic, Dordrecht (1994)zbMATHGoogle Scholar
  11. 11.
    Liao, S.J.: An approximate solution technique not depending on small parameters: a special example. Int. J. Non-Linear Mech. 30, 371–380 (1995)zbMATHCrossRefGoogle Scholar
  12. 12.
    Liao, S.J.: Beyond Perturbation: Introduction to the Homotopy Analysis Method. Chapman & Hall, Boca Raton (2003)CrossRefGoogle Scholar
  13. 13.
    Liao, S.J., Tan, Y.: A general approach to obtain series solutions of nonlinear differential equations. Stud. Appl. Math. 119, 297–354 (2007)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Abbasbandy, S., Shirzadi1, A.: Homotopy analysis method for multiple solutions of the fractional Sturm–Liouville problems. Numer. Algorithms. doi: 10.1007/s11075-009-9351-7
  15. 15.
    Hayat, T., Sajid, M.: On analytic solution for thin film flow of a fourth grade fluid down a vertical cylinder. Phys. Lett. A 361, 316–322 (2007)zbMATHCrossRefGoogle Scholar
  16. 16.
    Liao, S.J.: Series solution of nonlinear eigenvalue problems by means of the homotopy analysis method. Nonlinear Anal.: Real World Appl. 10, 2455–2470 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Zou, L., Zong, Z., Dong, G.H.: Generalizing homotopy analysis method to solve Lotka–Volterra equation. Comput. Math. Appl. 56, 2289–2293 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Liang, S., Jeffrey, D.J.: An efficient analytical approach for solving fourth order boundary value problems. Comput. Phys. Commun. 180, 2034–2040 (2009)zbMATHCrossRefGoogle Scholar
  19. 19.
    Liang, S., Jeffrey, D.J.: Approximate solutions to a parameterized sixth order boundary value problem. Comput. Math. Appl. 59, 247–253 (2010)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2010

Authors and Affiliations

  1. 1.Department of Applied MathematicsUniversity of Western OntarioLondonCanada

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