Adomian修正分解法的后处理方法比较

摘要

The theory of the Adomian modified decomposition method (AMDM) for solving nonlinear differential equations is well established.The main advantages of AMDM are computational simplicity and no involvement of any linearization or discretization.However,the accuracy of the AMDM solution depends on the convergence region.To extend the convergence region of the AMDM,several after-treatment techniques (such as Padé approximant,Laplace-Padé approximant and multistage method) have been proposed to improve the accuracy of the AMDM on a wide region.In this study,first,a brief review of the AMDM is given.Then these three after-treatment techniques are discussed.Finally,with examples of free and force Duffing oscillator problems,numerical results are presented to compare the drawbacks and advantages of these after-treatment techniques.It is shown that the multistage after-treatment technique offers an accurate and effective method for solving nonlinear differential equations in a wide applicable region.%Adomian修正分解法在求解非线性微分方程中得到广泛应用.Adomian修正分解法的主要特点在于计算简单快速,并且不需要进行线性化或离散化.但是Adomian修正分解法的计算精度取决于其收敛域.为了扩大Adomian修正分解法的收敛域,需要对所得解进行后处理,目前常见的后处理方法包括Padé近似、Laplace-Padé近似和多步迭代方法.本文首先简要回顾了Adomian修正分解法,然后讨论了这三种后处理方法,最后通过Duffing振子为例对这些后处理方法的优缺点进行讨论和分析.数值计算结果表明,多步迭代方法能够加速Adomian修正分解法解的收敛,并扩大其收敛域.