D. DOMIRI GANJIProfessor of Mechanical EngineeringGet in touch
Methods/DTM
Intermediate · 10 minute learning note

Differential transform method

Compute a local power series through coefficient recurrences.

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.

Represent derivatives at an expansion point by scaled coefficients. Differentiation shifts the coefficient index, and multiplication becomes a convolution. Transforming the governing equation gives a recurrence that can be computed one coefficient at a time.

U(k)=u(k)(0)k!(k+1)U(k+1)=−∑j=0kU(j)U(k−j)\begin{gathered}U(k)=\frac{u^{(k)}(0)}{k!}\\(k+1)U(k+1)\\=-\sum_{j=0}^{k}U(j)U(k-j)\end{gathered}

Origins & connection to Ganji’s work

The engineering differential-transform formulation is associated with J. K. Zhou’s 1986 work on electrical circuits and is based on Taylor series. Ganji and collaborators applied it to flow, heat transfer and oscillations.

Applications with Ganji include thermal fins, micropolar channel flow, nanoparticle transport and nonlinear oscillation.

Before you begin: Taylor series · Derivatives · Finite sums · Initial-value problems.

A method you can follow.

  1. Set the expansion pointDefine U(k)=u⁽ᵏ⁾(t₀)/k! and translate the initial data into the first coefficients.
  2. Transform each operationThe coefficient of u′ is (k+1)U(k+1); the coefficient of u² is the convolution ΣU(j)U(k−j).
  3. Compute the recurrenceFor nonlinear decay, U(0)=1. Successive coefficients are U(1)=−1, U(2)=1 and U(3)=−1.
  4. Reconstruct and continue carefullyForm the truncated local series. If a longer interval is needed, use tested subinterval continuation or an explicitly described rational approximation and verify the result.
Worked example

Recover the nonlinear-decay Taylor series

u′+u²=0, u(0)=1Exact reference: u(t)=1/(1+t)
  1. At k=0, U(1)=−U(0)²=−1.
  2. At k=1, 2U(2)=−[U(0)U(1)+U(1)U(0)]=2, so U(2)=1.
  3. At k=2, 3U(3)=−[1+1+1]=−3, so U(3)=−1.
  4. Continuing gives U(k)=(−1)ᵏ, the same local geometric series produced by the HPM example.

Check the result

At t=0.5, retaining k=0…4 gives 0.6875; the exact value is 2/3.

DTM reorganizes Taylor-coefficient calculations. The nearest singularity still limits a local Taylor expansion; here the pole at t=−1 gives radius 1.

Explore the interactive notebook

Know the limits.

  • Ordinary Taylor DTM requires analyticity on its convergence domain.
  • A local polynomial need not represent long-time or far-field behavior.
  • Padé and multi-step extensions are additional procedures with their own convergence and error questions.

Where you will encounter it

Micropolar channel flowTemperature-dependent finsNonlinear vibration

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