In the pulsed eddy current testing of ferromagnetic casings, the engineering application of theoretical models often faces a critical dilemma. The classical continuous models suffer from low computational efficiency, while simplified models exhibit significant errors in quantitative measurements. This limitation hinders the widespread adoption of theoretical models in field applications. To address this issue, this article proposes a fast theoretical model based on the truncated region eigenfunction expansion method. First, the magnetic vector potential in the region of the ferromagnetic casing with a length of 3 m is discretely expanded to obtain a discrete summation expression, significantly accelerating the computation. Subsequently, a discrete complex-frequency domain expression for the transient induced voltage is established based on Faraday′s law. The numerical inverse Laplace transform fixed Talbot algorithm (FTA) is then employed to compute the time-domain transient response, further enhancing computational speed. Finally, experimental results show that the proposed fast theoretical model achieves an average relative error of only 1.63% compared with finite element simulations, and exhibits strong agreement with experimental measurements. Furthermore, the average computation time for a single point of the transient response in single-layer, double-layer, and triple-layer casings are 98, 107, and 368 ms, respectively. Compared with classical continuous models based on the FTA, the residue theorem (RT) and Fourier series (FS), the proposed fast model achieved a 17-fold improvement in computational speed while maintaining high accuracy. This advancement can promote the engineering application of pulsed eddy current theoretical models.