D. DOMIRI GANJIProfessor of Mechanical EngineeringGet in touch
Methods/ADM
Advanced · 12 minute learning note

Adomian decomposition method

Decompose the solution and its nonlinearity together.

u(0.5) = 2/3THE MODEL IN THIS NOTEu(t) = 1 / (1 + t)10.51tuEXACT REFERENCE · u′ + u² = 0, u(0) = 1
The exact reference solution used throughout the learning notes.

The idea, in plain language.

Express the solution as a sum of components and express a nonlinear term as a matching sum of Adomian polynomials. Applying the inverse linear operator yields a recursive way to construct the components while retaining the original nonlinear model.

u=∑n=0∞unAn=∑j=0nujun−jun+1(t)=−∫0tAn(s) ds\begin{gathered}u=\sum_{n=0}^{\infty}u_n\\A_n=\sum_{j=0}^{n}u_ju_{n-j}\\u_{n+1}(t)=-\int_0^t A_n(s)\,ds\end{gathered}

Origins & connection to Ganji’s work

Developed by George Adomian. Ganji and collaborators applied and compared ADM with other analytical methods.

Ganji’s coauthored studies use ADM in heat transfer, Jeffery–Hamel flow and nonlinear differential equations.

Before you begin: Integration as an inverse operator · Power series · Nonlinear functions · Coefficient matching.

A method you can follow.

  1. Invert the linear partIntegrate u′=−u² using the initial value to obtain u(t)=1−∫₀ᵗu(s)²ds.
  2. Decompose the nonlinearityFor a square, the Adomian polynomials are A₀=u₀², A₁=2u₀u₁, A₂=2u₀u₂+u₁², and so on.
  3. Build the componentsSet u₀=1 and use uₙ₊₁=−∫Aₙ. Previously calculated components determine each next term.
  4. Sum and assessForm a finite sum, evaluate its residual and compare against an exact or independently converged numerical reference.
Worked example

Decompose the quadratic reaction term

u′+u²=0, u(0)=1Exact reference: u(t)=1/(1+t)
  1. u₀=1 gives A₀=1 and u₁=−t.
  2. A₁=2u₀u₁=−2t gives u₂=t².
  3. A₂=2u₀u₂+u₁²=3t² gives u₃=−t³.
  4. The partial sum 1−t+t²−t³ agrees with the HPM and DTM constructions for this particular problem.

Check the result

At t=0.5, S₃=0.625 while u_exact=2/3; the absolute error is 1/24.

The nonlinear polynomials organize cross terms that would be lost by naively squaring each component separately. Identical final series have identical convergence limits.

Explore the interactive notebook

Know the limits.

  • Constructing the nonlinear polynomials can be expensive at high order.
  • Decomposition does not remove the need to establish convergence.
  • Boundary-value problems require careful treatment of integration constants and all boundary conditions.

Where you will encounter it

Jeffery–Hamel flowHeat-transfer modelsNonlinear wave equations

Read the originals.

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.

Back to all methods