JOURNAL BROWSE
Search
Advanced SearchSearch Tips
VARIATION OF PARAMETERS METHOD FOR SOLVING SIXTH-ORDER BOUNDARY VALUE PROBLEMS
facebook(new window)  Pirnt(new window) E-mail(new window) Excel Download
 Title & Authors
VARIATION OF PARAMETERS METHOD FOR SOLVING SIXTH-ORDER BOUNDARY VALUE PROBLEMS
Mohyud-Din, Syed Tauseef; Noor, Muhammad Aslam; Waheed, Asif;
  PDF(new window)
 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.
VARIATION OF PARAMETERS METHOD FOR SOLVING A CLASS OF EIGHTH-ORDER BOUNDARY-VALUE PROBLEMS, International Journal of Computational Methods, 2012, 09, 02, 1240026  crossref(new windwow)
2.
An Efficient Method for Solving System of Third-Order Nonlinear Boundary Value Problems, Mathematical Problems in Engineering, 2011, 2011, 1  crossref(new windwow)
3.
An efficient algorithm on time-fractional partial differential equations with variable coefficients, QScience Connect, 2014, 2014, 7  crossref(new windwow)
4.
Effects on magnetic field in squeezing flow of a Casson fluid between parallel plates, Journal of King Saud University - Science, 2015  crossref(new windwow)
5.
Effects of Velocity Slip on MHD Flow of a Non-Newtonian Fluid in Converging and Diverging Channels, International Journal of Applied and Computational Mathematics, 2016, 2, 4, 469  crossref(new windwow)
6.
Variation of Parameters Method for Heat Diffusion and Heat Convection Equations, International Journal of Applied and Computational Mathematics, 2015  crossref(new windwow)
7.
Deficient discrete cubic spline solution for a system of second order boundary value problems, Numerical Algorithms, 2014, 66, 4, 793  crossref(new windwow)
8.
Analytical and numerical investigation of thermal radiation effects on flow of viscous incompressible fluid with stretchable convergent/divergent channels, Journal of Molecular Liquids, 2016, 224, 768  crossref(new windwow)
9.
On unsteady two-dimensional and axisymmetric squeezing flow between parallel plates, Alexandria Engineering Journal, 2014, 53, 2, 463  crossref(new windwow)
10.
MHD squeezing flow between two infinite plates, Ain Shams Engineering Journal, 2014, 5, 1, 187  crossref(new windwow)
11.
Heat Transfer Analysis of Third-Grade Fluid Flow Between Parallel Plates: Analytical Solutions, International Journal of Applied and Computational Mathematics, 2015  crossref(new windwow)
12.
Analysis of magnetohydrodynamic flow and heat transfer of Cu–water nanofluid between parallel plates for different shapes of nanoparticles, Neural Computing and Applications, 2016  crossref(new windwow)
13.
Variation of Parameters Solution for Two Dimensional Flow of a Viscous Fluid Between Dilating and Squeezing Channel with Permeable Walls, International Journal of Applied and Computational Mathematics, 2016  crossref(new windwow)
 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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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 crossref(new window)

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 crossref(new window)

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. crossref(new window)

10.
G. A. Glatzmaier, Numerical simulations of stellar convection dynamics at the base of the convection zone, geophysics. Fluid Dynamics 31 (1985), 137-150. crossref(new window)

11.
J. H. He, Variational approach to the sixth order boundary value problems, Appl. Math. Comput. 143 (2003), 235-236 crossref(new window)

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 crossref(new window)

13.
W. X. Ma and Y. You, Rational solutions of the Toda lattice equation in Casoratian form, Chaos, Solitons & Fractals 22 (2004), 395-406. crossref(new window)

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. crossref(new window)

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 crossref(new window)

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 crossref(new window)

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 crossref(new window)

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 crossref(new window)

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

31.
E. H. Twizell and A. Boutayeb, Numerical methods for the solution of special and general sixth-order boundary value problems, with applications to Benard layer Eigen value problem, Proc. Roy. Soc. Lond. A 431 (1990), 433-50 crossref(new window)

32.
A. M. Wazwaz, The numerical solution of sixth order boundary value problems by the modified decomposition method, Appl. Math. Comput. 118 (2001), 311-325. crossref(new window)