Homotopy perturbation method

The idea, in plain language.

Introduce an embedding parameter p that continuously connects an easier equation to the target equation. Expand the unknown solution in powers of p, solve the resulting problems in order, and set p=1 only after examining the resulting series. An artificial parameter is a construction device, not proof of convergence.

Before you begin: First-order differential equations · Integration · Power series · Matching coefficients.

A method you can follow.

1. Set up a homotopyFor nonlinear decay, replace u′+u²=0 by u′+p u²=0. At p=0 the initial-value problem has the constant solution 1; at p=1 the target equation is recovered.
2. Expand and matchSubstitute u=Σpⁿuₙ and equate equal powers of p. Set u₀(0)=1 and uₙ(0)=0 for n≥1 to preserve the original initial condition.
3. Solve the hierarchyIntegrate each coefficient equation using terms already computed: u₀=1, u₁=−t, u₂=t² and u₃=−t³.
4. Check where it worksAt p=1 this is a geometric series. Inspect its convergence domain, compare against the exact solution, and measure the residual of each finite approximation.

A convergent series has a boundary

1. The coefficient equations are u₀′=0 and uₙ′=−Σ[j=0…n−1]uⱼuₙ₋₁₋ⱼ for n≥1.
2. They give uₙ(t)=(−t)ⁿ. Hence S_N(t)=1−t+t²−⋯+(−t)ᴺ.
3. The finite geometric identity gives S_N−u_exact=−(−t)ᴺ⁺¹/(1+t).
4. The series converges for |t|<1. At t=1 the partial sums alternate between 1 and 0, while the exact value is 1/2. For t>1 the terms grow.

A smooth exact solution can exist outside the convergence interval of an approximation. More terms can make an answer worse outside that interval.

Know the limits.

• The choice of initial guess and linear operator matters.
• No small physical parameter is required by the construction, but this does not imply convergence for all parameter values or times.
• Validate comparisons using exact solutions or independently converged numerical calculations.

Where you will encounter it

Read the originals.

• He (1999). Homotopy perturbation technique. ↗
  Foundational method · DOI 10.1016/S0045-7825(99)00018-3
• Ganji, Ganji, Karimpour & Ganji (2010). Numerical study of homotopy-perturbation method applied to Burgers equation in fluid. ↗
  Ganji research application · DOI 10.1002/num.20464
• Comparison of homotopy analysis method and homotopy perturbation method through an evolution equation (2009). ↗
  Critical convergence analysis · DOI 10.1016/j.cnsns.2009.02.016

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.