On solving the initial-value problems using the dierential transformation method Ming-Jyi Jang a , Chieh-Li Chen b, * , Yung-Chin Liy b a Department of Mechanical Engineering, National Cheng Kung University, Tainan, Taiwan 70101, ROC b Institute of Aeronautics and Astronautics, National Cheng Kung University, Tainan, Taiwan 70101, ROC Abstract In this paper, initial-value problems are solved by the dierential transformation method. The dierential transformation method of ®xed grid size is used to approximate solutions of linear and nonlinear initial-value problems. An adaptive grid size mecha- nism based on the ®xed grid size technique is also proposed. The proposed adaptive grid size procedure provides concise adjustment policy and raises computational eciency of using dierential transformation method. Ó 2000 Elsevier Science Inc. All rights reserved. Keywords: Dierential transform; Adaptive step-size; Sti equation 1. Introduction Dierential equations are widely used to describe continuous time physical problems. In most cases, these problems may be too complicated to solve analytically. Alternatively, the numerical methods can provide approximate solutions rather than the analytic solutions of problems. In the literature [1], the Euler method, the Taylor method and the Runge±Kutta methods serve as an introduction to numerical methods for initial-value problems. The high- order Taylor method can also be applied to initial-value problems. However, the Taylor method requires the calculation of high-order derivatives, a dicult symbolic and complex problem. www.elsevier.com/locate/amc Applied Mathematics and Computation 115 (2000) 145±160 * Corresponding author. E-mail: jlchen@csun8.iaa.ncku.edu.tw 0096-3003/00/$ - see front matter Ó 2000 Elsevier Science Inc. All rights reserved. PII:S0096-3003(99)00137-X