VARIATION OF PARAMETERS METHOD FOR SOLVING SIXTH-ORDER BOUNDARY VALUE PROBLEMS

- Journal title : Communications of the Korean Mathematical Society
- Volume 24, Issue 4, 2009, pp.605-615
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/CKMS.2009.24.4.605

Title & Authors

VARIATION OF PARAMETERS METHOD FOR SOLVING SIXTH-ORDER BOUNDARY VALUE PROBLEMS

Mohyud-Din, Syed Tauseef; Noor, Muhammad Aslam; Waheed, Asif;

Mohyud-Din, Syed Tauseef; Noor, Muhammad Aslam; Waheed, Asif;

Abstract

In this paper, we develop a reliable algorithm which is called the variation of parameters method for solving sixth-order boundary value problems. The proposed technique is quite efficient and is practically well suited for use in these problems. The suggested iterative scheme finds the solution without any perturbation, discritization, linearization or restrictive assumptions. Moreover, the method is free from the identification of Lagrange multipliers. The fact that the proposed technique solves nonlinear problems without using the Adomian`s polynomials can be considered as a clear advantage of this technique over the decomposition method. Several examples are given to verify the reliability and efficiency of the proposed method. Comparisons are made to reconfirm the efficiency and accuracy of the suggested technique.

Keywords

variation of parameters;nonlinear problems;initial value problems;boundary value problems;error estimates;

Language

English

Cited by

1.

2.

3.

4.

5.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

References

1.

R. P. Agarwal, Boundary Value Problems for Higher Order Differential Equations, world scientific, Singapore, 1986.

2.

G. Akram and S. S. Siddiqi, Solution of sixth order boundary value problems using non-polynomial spline technique, Appl. Math. Comput. 181 (2006), no. 1, 708–720.

3.

P. Baldwin, Asymptotic estimates of the eigenvalues of a sixth-order boundary-value problem obtained by using global phase-integral methods, Phil, Trans. Roy. Soc. Lond. A 322 (1987), no. 1566, 281–305.

4.

P. Baldwin, Localized instability in a Benard layer, Appl. Aal. 24 (1987), no. 1-2, 117-156

5.

A. Boutayeb and E. H. Twizell, Numerical methods for the solution of special sixth-order boundary value problems, Int. J. Comput. Math. 45 (1992), 207–233.

6.

S. Chandrasekhar, Hydrodynamics and Hydromagntic Stability, Dover, New York, 1981.

7.

M. M. Chawla and C. P. Katti, Finite difference methods for two-point boundary-value problems involving higher order differential equations, BIT 19 (1979), 27-33

8.

Y. Cherrauault and G. Saccomandi, Some new results for convergence of G. Adomian's method applied to integral equations, Math. Comput. Modeling 16 (1992), no. 2, 85-93

9.

M. E. Gamel, J. R. Cannon, and A. I. Zayedm, Sinc-Galerkin method for solving linear sixth order boundary value problems, Appl. Math. Comput. 73 (2003), 1325-1343.

10.

G. A. Glatzmaier, Numerical simulations of stellar convection dynamics at the base of the convection zone, geophysics. Fluid Dynamics 31 (1985), 137-150.

11.

J. H. He, Variational approach to the sixth order boundary value problems, Appl. Math. Comput. 143 (2003), 235-236

12.

W. X. Ma and Y. You, Solving the Korteweg-de Vries equation by its bilinear form: Wronskian solutions, Trans. Amer. Math. Soc. 357 (2004), 1753-1778

13.

W. X. Ma and Y. You, Rational solutions of the Toda lattice equation in Casoratian form, Chaos, Solitons & Fractals 22 (2004), 395-406.

14.

S. T. Mohyud-Din and M. A. Noor, Homotopy perturbation method for solving partial differential equations, Zeitschrift fur Naturforschung A 64a (2009), 157–170

15.

S. T. Mohyud-Din, M. A. Noor, and K. I. Noor, On the coupling of polynomials with correction functional, Int. J. Mod. Phys. B (2009).

16.

S. T. Mohyud-Din, M. A. Noor, and K. I. Noor, Some relatively new techniques for nonlinear problems, Math. Prob. Eng. 2009 (2009), Article ID 234849, 25 pages, doi:10.1155/2009/234849.

17.

S. T. Mohyud-Din, M. A. Noor, and K. I. Noor, Travelling wave solutions of seventh-order generalized KdV equations using He's polynomials, Int. J. Nonlin. Sci. Num. Sim. 10 (2009), no. 2, 223–229

18.

S. T. Mohyud-Din, M. A. Noor, and K. I. Noor, Modified variation of parameters method for solving partial differential equations, Int. J. Mod. Phys. B (2009)

19.

S. T. Mohyud-Din, M. A. Noor, and K. I. Noor, Modified variation of parameters method for differential equations, Wd. Appl. Sci. J. 6 (2009), no. 10, 1372–1376

20.

S. T. Mohyud-Din, M. A. Noor, and K. I. Noor, Modified variation of parameter technique for Thomas-Fermi and fourth-order singular parabolic equations, Int. J. Mod. Phys. B (2009)

21.

S. T. Mohyud-Din, M. A. Noor, and K. I. Noor, Ma's variation of parameters method for Burger's and telegraph equations, Int. J. Mod. Phys. B (2009)

22.

S. T. Mohyud-Din, M. A. Noor, and K. I. Noor, Modified variation of parameters method for second-order integro-differential equations and coupled systems, Wd. Appl. Sci. J. 6 (2009), no. 9, 1298–1303

23.

M. A. Noor and S. T. Mohyud-Din, Homotopy perturbation method for solving sixthorder boundary value problems, Comput. Math. Appl. 55 (2008), no. 12, 2953–2972

24.

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. Sci. Num. Simul. 9 (2008), no. 2, 141-157.

25.

M. A. Noor and S. T. Mohyud-Din, Homotopy perturbation method for nonlinear higher-order boundary value problems, Int. J. Nonlin. Sci. Num. Simul. 9 (2008), no. 4, 395-408

26.

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. 2 (2008), 135-141.

27.

J. I. Ramos, On the variational iteration method and other iterative techniques for nonlinear differential equations, Appl. Math. Comput. 199 (2008), no. 1, 39-69

28.

S. S. Siddiqi and E. H. Twizell, Spline solutions of linear sixth-order boundary value problems, Int. J. Comput. Math. 60 (1996), 295-304

29.

J. Toomore, J. P. Zahn, J. Latour, and E. A Spiegel, Stellar convection theory II: singlemode study of the secong convection zone in A-type stars, Astrophs. J. 207 (1976), 545-563

30.

E. H. Twizell, Numerical methods for sixth-order boundary value problems, in: numerical Mathematics, Singapore, International Series of Numerical Mathematics, vol. 86, Birkhauser, Basel (1988), 495-506