Variational iteration method for autonomous ordinary differential systems. (English) Zbl 1027.34009

Summary: Here, a new iteration technique is proposed to solve autonomous ordinary differential systems. In this method, general Lagrange multipliers are introduced to construct correction functionals for the systems. The multipliers in the functionals can be identified by the variational theory. The initial approximations can be freely chosen with possible unknown constants, which can be determined by imposing boundary/initial conditions. Some examples are given. The results reveal that the method is very effective and convenient.


34A45 Theoretical approximation of solutions to ordinary differential equations
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
Full Text: DOI


[1] R.D. Driver, Ordinary and Delay Differential Equations, Springer, Berlin, 1977 · Zbl 0374.34001
[2] He, J.H., A new approach to nonlinear partial differential equations, Communications in nonlinear science and numerical simulation, 2, 4, 230-235, (1997)
[3] He, J.H., Variational iteration method for delay differential equations, Communications in nonlinear science and numerical simulation, 2, 4, 235-236, (1997)
[4] J.H. He, Nonlinear oscillation with fractional derivative and its approximation, in: International Conf. on Vibration Engineering ’98, 1998, Dalian, China
[5] J.H. He, Approximate analytical solution for seepage flow with fractional derivatives in porous media, Computer Methods in Applied Mechanics and Engineering 167 (1998) 57-68 · Zbl 0942.76077
[6] J.H. He, Approximate solution of nonlinear differential equations with convolution product nonlinearities, Computer Methods in Applied Mechanics and Engineering 167 (1998) 69-73 · Zbl 0932.65143
[7] He, J.H., A variational iteration method for nonlinearity and its applications (in Chinese), Mechanics and application, 20, 1, 30-32, (1998)
[8] He, J.H., Variational iteration approach to 2-spring system (in Chinese), Mechanical science and technology, 17, 2, 221-223, (1998)
[9] M. Inokuti et al., General use of the lagrange multiplier in nonlinear mathematical physics, in: Variational Method in the Mechanics of Solids, Pergamon Press, New York, 1978, pp. 156-162
[10] J.H. He, A new approach to establishing generalized variational principles in fluids and C.C. Liu constraints (in Chinese), Ph.D. Thesis, Shanghai University, 1996
[11] J.H. He, Generalized Variational Principle in Fluids (in Chinese), Shanghai University Press, Shangai, 1998
[12] D.R. Smith, Singular-Perturbation Theory. An Introduction with Applications, Cambridge University Press, Cambridge, 1985 · Zbl 0567.34055
[13] A.H. Nayfeh, Problems in Perturbation, Wiley, New York, 1985 · Zbl 0573.34001
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.