##  [Variational Iteration Method](/variational-iteration-method-0) 

 Definition

An iterative procedure that constructs successive approximations to the solution of differential or integral equations by applying variational correction functionals, typically using a Lagrange multiplier determined by variational theory.

 

 

 

 

 

 





## Principle

Principle

Formulate a correction functional that enforces the governing equation and determine a Lagrange multiplier so that each iteration corrects previous approximations toward the exact solution without linearizing the original operator.

 

 

 

 

 





## Demonstration

Demonstration

Solve a nonlinear ordinary differential equation u'(t)+u(t)+u(t)^2=f(t) by proposing an initial guess u0, forming a correction functional that includes an undetermined Lagrange multiplier λ(t), determine λ by stationary conditions, and iterate u_{n+1}=u_n+correction to obtain successive approximations.

 

 

 

 

## Misapplication

Misapplication

Picking an incorrect or nonstationary Lagrange multiplier, using a correction functional that does not reflect boundary conditions, or truncating iterations prematurely can produce diverging sequences or solutions that violate constraints.

 

 

 

 

 





## Consequence

Consequence

When applied correctly, VIM often yields rapidly convergent, closed-form or semi-analytical approximations that preserve the original nonlinearity and boundary conditions, reducing reliance on discretization.

 

 

 

 

## Reversal

Reversal

A direct fixed-point or simple Picard iteration that replaces variational correction with naive substitution; such reversal typically requires stronger contraction properties and may converge much slower or not at all.

 

 

 

 

 





## Boundary

Boundary

Applicable to a wide class of ordinary and partial differential and integral equations where a meaningful correction functional and variational multiplier can be derived; less suitable when no reasonable variational formulation exists or for extremely high-dimensional discrete systems without model reduction.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Competes with methods like Adomian Decomposition and Homotopy Analysis: all build successive approximations, but VIM centers a variationally determined correction functional rather than polynomial decompositions or homotopy embeddings.

 

 

 

 

 





## Synthesis

Synthesis

VIM is an iteration strategy that embeds the governing equation into a correction functional and uses a variationally chosen Lagrange multiplier to produce successive, often rapidly convergent, approximations while preserving nonlinearity and boundary structure.