Variational iteration technique for solving higher order boundary value problems. (English) Zbl 1122.65374

Summary: We show that higher order boundary value problems can be written as a system of integral equations, which can be solved by using the variational iteration technique. The analytical results of the equations obtained in terms of convergent series with easily computable components. Several examples are given to illustrate the efficiency and implementation of the method. Comparisons are made to confirm the reliability of the technique. The variational iteration technique may be considered as alternative and efficient for finding the approximate solutions of boundary values problems.


65L10 Numerical solution of boundary value problems involving ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
Full Text: DOI


[1] Agarwal, R. P., Boundary-value Problems for Higher Order Differential Equations (1986), World Scientific: World Scientific Singapore · Zbl 0598.65062
[2] Caglar, H. N.; Caglar, S. H.; Twizell, E. E., The numerical solution of fifth order boundary-value problems with sixth degree B-spline functions, Appl. Math. Lett., 12, 25-30 (1999) · Zbl 0941.65073
[3] Chawla, M. M.; Katti, C. P., Finite difference methods for two-point boundary-value problems involving higher order differential equations, BIT, 19, 27-33 (1979) · Zbl 0401.65053
[4] Davies, A. R.; Karageoghis, A.; Philips, T. N., Spectral Galerkin methods for the primary two-point boundary-value problems in modeling viscoelastic flows, Int. J. Numer. Methods Eng., 26, 647-662 (1988) · Zbl 0635.73091
[5] Doedel, E., Finite difference methods for nonlinear two-point boundary-value problem, SIAM J. Numer. Anal., 16, 173-185 (1979) · Zbl 0438.65068
[6] Fyfe, D. J., Linear dependence relations connecting equal interval \(n\) th degree splines and their derivatives, J. Inst. Math. Appl., 7, 398-406 (1971) · Zbl 0219.65010
[7] He, J. H., Variational iteration method – a kind of nonlinear analytical technique: some examples, Int. J. Nonlinear Mech., 34, 699-708 (1999) · Zbl 1342.34005
[8] He, J. H., Variational method for autonomous ordinary differential equations, Appl. Math. Comput., 114, 115-123 (2000) · Zbl 1027.34009
[9] He, J. H., Variational theory for linear magneto-electro-elasticity, Int. J. Nonlinear Sci. Numer. Simul., 2, 4, 309-316 (2001) · Zbl 1083.74526
[10] He, J. H., Variational principle for some nonlinear partial differential equations with variable coefficients, Chaos Solitons Fract., 19, 4, 847-851 (2004) · Zbl 1135.35303
[11] Inokuti, M.; Sekine, H.; Mura, T., General use of the Lagrange multiplier in nonlinear mathematical physics, (Variational Method in the Mech. of Solids (1978), Pergamon Press: Pergamon Press New York), 156-162
[12] Karageoghis, A.; Philips, T. N.; Davies, A. R., Spectral collocation methods for the primary two-point boundary-value problems in modeling viscoelastic flows, Int. J. Numer. Methods Eng., 26, 805-813 (1998) · Zbl 0637.76008
[14] Moyud-Din, S. T.; Aslam Noor, M., Homotopy perturbation method for solving fourth order boundary value problems, Math. Problems Eng., 2007, 1-15 (2007), Article ID 98602
[15] Aslam Noor, M.; Mohyud-Din, S. T., An efficient algorithm for solving fifth order boundary value problems, Math. Comput. Model., 45, 954-964 (2007) · Zbl 1133.65052
[17] Wazwaz, A. M., The numerical solution of fifth-order boundary-value problems by Adomian decomposition method, J. Comput. Appl. Math., 136, 259-270 (2001) · Zbl 0986.65072
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.