The idea, in plain language.
Construct a correction functional containing the current residual and a Lagrange multiplier. Determine the multiplier from stationarity conditions for the chosen operator, then iterate. For a simple first-order initial-value problem, this can reduce to a familiar integral iteration.
Origins & connection to Ganji’s work
Developed by Ji-Huan He, with antecedents in earlier variational correction approaches. Ganji’s work applies and compares VIM; he did not originate it.
Ganji and collaborators investigated VIM alongside other analytical methods for thermal problems and nonlinear evolution equations.
Before you begin: Integration · Initial-value problems · Basic variational reasoning · Residuals.
A method you can follow.
- Identify the residualWrite the equation as L(u)+N(u)−f=0 and keep track of the prescribed data.
- Derive the multiplierConstruct a correction integral and apply the appropriate stationarity conditions. The multiplier depends on the operator; −1 is correct for this example, not for every equation.
- Iterate from an admissible guessStart with u₀=1. When each iterate satisfies uₙ(0)=1, the correction becomes uₙ₊₁(t)=1−∫₀ᵗuₙ(s)² ds.
- Test convergenceCheck differences between iterates together with the governing-equation residual and an independent reference.
Two corrections to nonlinear decay
- Choose u₀(t)=1 and the first-order multiplier λ=−1.
- The first correction is u₁(t)=1−∫₀ᵗ1 ds=1−t.
- The second is u₂(t)=1−∫₀ᵗ(1−s)² ds=1−t+t²−t³/3.
- Each iterate satisfies uₙ(0)=1. The iterate index is not the same as the number of retained HPM series terms.
Check the result
At t=0.25, u₂≈0.8072917 and u_exact=0.8.
For this problem VIM reduces to Picard iteration. Different analytical frameworks can lead to the same underlying iterative construction.
Explore the interactive notebookKnow the limits.
- A multiplier must be derived for the actual linear operator and boundary structure.
- Convergence of a correction sequence is conditional and may be local.
- A rapidly changing polynomial degree does not by itself establish greater accuracy.
Where you will encounter it
Read the originals.
- He (1999). Variational iteration method—a kind of non-linear analytical technique: some examples. ↗Foundational method · DOI 10.1016/S0020-7462(98)00048-1
- Ganji and collaborators (2009). Analytical solutions to nonlinear equations arising in heat transfer by variational iteration, homotopy perturbation, and Adomian decomposition methods. ↗Ganji research application · DOI 10.1002/num.20503
This is an original educational explanation, not a claim that all illustrated methods were invented by Professor Ganji. The worked model is deliberately shared across notes so the results and limitations can be compared.