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

Akbari–Ganji method

Turn a differential equation into a carefully chosen algebraic problem.

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.

Choose a finite trial function that reflects the expected shape of the solution. Its unknown coefficients are determined using the actual initial or boundary conditions and additional equations obtained from the differential equation and selected derivatives at chosen points. The original differential equation remains the test of whether the approximation is useful.

uN(t)=∑j=0NajtjRN(t)=uN′(t)+uN(t)2\begin{gathered}u_N(t)=\sum_{j=0}^{N}a_jt^j\\R_N(t)=u_N^{\prime}(t)+u_N(t)^2\end{gathered}

Origins & connection to Ganji’s work

Introduced in papers by M. R. Akbari, D. D. Ganji and collaborators in 2014. Credit belongs to the collaborating authors; it is not a universal exact solver.

A method associated directly with the collaborative work of Akbari and Ganji, with applications to nonlinear oscillators, structural response and thermal problems.

Before you begin: Ordinary differential equations · Initial and boundary conditions · Polynomial differentiation · Nonlinear algebraic equations.

A method you can follow.

  1. Choose the shapeSelect a polynomial, exponential or trigonometric trial function consistent with the expected solution and domain. Count its unknown coefficients.
  2. Impose the physical conditionsUse the stated initial or boundary conditions first. Do not invent extra physical boundary conditions.
  3. Close the algebraic systemUse the governing equation and, where the chosen AGM formulation requires them, its derivatives at selected points to obtain enough independent algebraic constraints.
  4. Solve and verifyFind the coefficients, select physically admissible branches, and evaluate the residual throughout the interval. Compare with an independent numerical solution and repeat with a richer trial function.
Worked example

A quadratic approximation to nonlinear decay

u′ + u² = 0, u(0) = 1Exact reference: u(t) = 1/(1+t)
  1. Let u₂(t)=a₀+a₁t+a₂t². The initial condition gives a₀=1.
  2. Enforce R₂(0)=0: a₁+1=0, so a₁=−1.
  3. Enforce R₂′(0)=0: 2a₂+2a₀a₁=0, so a₂=1.
  4. The resulting approximation is u₂(t)=1−t+t². Its residual is R₂(t)=3t²−2t³+t⁴, which is not zero away from the expansion point.

Check the result

At t=0.25, u₂=0.8125 and u_exact=0.8; the absolute error is 0.0125.

This original teaching example illustrates a polynomial residual-matching implementation of AGM. It also coincides with a Taylor truncation; a different name does not change the approximation or its limits.

Explore the interactive notebook

Know the limits.

  • Satisfying the equation at selected points does not guarantee a small residual between those points.
  • The trial family, truncation order, matching points and algebraic solution branch can change the answer.
  • Global convergence, uniqueness and high accuracy must be demonstrated for the particular problem; they do not follow from the method name.

Where you will encounter it

Nonlinear mechanical oscillatorsStructural deflectionFin and extended-surface heat transfer

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