D. DOMIRI GANJIProfessor of Mechanical EngineeringGet in touch
Methods/HAM
Advanced · 14 minute learning note

Homotopy analysis method

Add an explicit parameter for controlling a series construction.

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.

HAM uses an embedding parameter, a freely chosen auxiliary linear operator and a nonzero convergence-control parameter ℏ. These choices change the sequence of linear subproblems and can improve a series approximation. They must be justified through analysis and residual or numerical checks.

(1−q)L[ϕ−u0]=qℏN[ϕ],ℏ≠0\begin{gathered}(1-q)\mathcal{L}[\phi-u_0]\\=q\hbar\mathcal{N}[\phi],\qquad\hbar\ne0\end{gathered}

Origins & connection to Ganji’s work

Originated by Shijun Liao. Ganji and collaborators are among researchers who have applied HAM to engineering and nonlinear wave problems.

Ganji coauthored applications including coupled Whitham–Broer–Kaup shallow-water equations.

Before you begin: Power series · Linear operators · Nonlinear differential equations · Residual evaluation.

A method you can follow.

  1. Define the nonlinear problemSpecify N[u]=0, the domain and every initial or boundary condition.
  2. Choose the auxiliary ingredientsSelect u₀, an invertible-on-the-constrained-space linear operator L and a nonzero parameter ℏ. Preserve the physical data in the deformation.
  3. Expand in the embedding parameterWrite φ=u₀+Σqᵐuₘ and match powers of q to obtain successive linear deformation equations.
  4. Select and verify ℏStudy residuals and stability over candidate values of ℏ and truncation orders. A flat parameter plot is a useful diagnostic, not a universal convergence proof.
Worked example

The first two deformation terms

N[u]=u′+u²=0, u(0)=1Exact reference: u(t)=1/(1+t)
  1. Choose u₀=1, L=d/dt and uₘ(0)=0 for m≥1.
  2. The q¹ equation gives u₁′=ℏ, so u₁=ℏt.
  3. The q² equation gives u₂′−u₁′=ℏ(u₁′+2u₁), so u₂=(ℏ+ℏ²)t+ℏ²t².
  4. At q=1, the two-correction approximation is 1+(2ℏ+ℏ²)t+ℏ²t². With ℏ=−1 this becomes 1−t+t².

Check the result

At ℏ=−1 and t=0.25, the approximation is 0.8125 and the exact value is 0.8.

An auxiliary parameter changes the finite approximation. Choosing a parameter is part of the numerical or analytical investigation, not a guarantee that every choice works.

Explore the interactive notebook

Know the limits.

  • Bad choices of auxiliary operator, basis, initial guess or ℏ can produce divergence.
  • The solution series must converge at q=1 and satisfy the target equation under justified termwise operations.
  • The HPM–HAM relationship depends on the particular homotopy and auxiliary choices.

Where you will encounter it

Shallow-water wave systemsNonlinear boundary layersThermal and fluid models

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