Application of the Hybrid Laplace Adomian Decomposition Method to the Nonlinear Oscillatory Systems

Laplace Adomian混合分解法應用於非線性振動系統之研究

蔡培毅(Pa-Yee Tsai) ； 陳朝光(Cha'o-Kuang Chen)

《中國機械工程學刊》 30卷6期 (2009/12) Pp. 493-501

https://doi.org/10.29979/JCSME.200912.0006

摘要

本研究提出一個新的演算法應用於求解非線性振動系統之近似解析解。Laplace Adomian分解法（簡稱LADM）結合了Laplace轉換運算法則與Adomian分解法（簡稱ADM）。利用ADM與LADM所求得的近似解析解為一個無窮多項式之片段解，當此系統之定義域增加時，其解會很快地發散而無法收斂，並且無法得到振動系統之週期性解。然而，利用Padé近似法可以解決此問題，進而使其片段解獲得更佳的精度與收斂值。因此，本文提出Laplace Adomian近似混合法，即利用LADM-Padé近似來克服利用ADM與LADM求解非線性系統所遇到之瓶頸，並利用四個範例加以說明，然後與四階Runge-Kutta數值解、修正ADM近似解與修正微分轉換近似解作比較，以證明LADM-Padé近似解析解之精確度。

並列摘要

In this study, a new algorithm is proposed to solve the nonlinear oscillatory systems. The Laplace Adomian decomposition method (LADM) combines the numerical Laplace transform algorithm and the Adomian decomposition method (ADM). The truncated series solutions solved by ADM and LADM both diverge rapidly as the applicable domain increases and do not exhibit periodicity. However, the Padé approximant extends the domain of the truncated series solution to obtain better accuracy and convergence, and the LADM-Padé approximant technique is introduced in this paper order to overcome the drawbacks of the ADM and LADM solutions. Four examples here in are given to show the accuracy in comparison with the fourth-order Runge-Kutta (RK4) solutions, the modified ADM solutions, and the modified differential transform solutions.

並列關鍵字

Adomian decomposition method ； Laplace transformation ； Padé approximant ； oscillatory system