Abstract: The Adomian Decomposition Method（ADM） is employed in this paper to solve three dimensional Helmholtz equations under arbitrary wave numbers. Based on the ADM, the three dimensional Helmholtz differential equation becomes a recursive algebraic equation. Furthermore, the boundary conditions become simple algebraic equations which are suitable for symbolic computation. By using boundary conditions, the closed-form series solution can be easily obtained. The main advantages of ADM are computational simplicity and do not involve any linearization or discretization. Finally, two numerical examples are presented to check the reliability of the proposed method for solving the three dimensional Helmholtz equations with different wave numbers. The numerical results on three dimensional problems with known analytic solutions demonstrate that the ADM is quite accurate and readily implemented. Furthermore, the good convergence and the excellent numerical stability of the solution based on the ADM can also be found for high wave numbers. It means that the ADM is quite efficient and is practically well suited for solving three dimensional Helmholtz equations at different wave numbers.

Key words: three dimensional Helmholtz equations, Adomian decomposition method, wave numbers

摘要： 提出通过Adomian分解法求解任意波数的三维Helmholtz方程。通过Adomian分解法可以把三维Helmholtz微分方程转换成递归代数公式，并进一步把其边界条件转换成适用符号计算的简单代数公式。利用边界条件可以很容易得到方程的解析解表达式。Adomian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。最后通过数值计算以验证Adomian分解法求解任意波数下三维Helmholtz方程的有效性。数值计算结果表明：Adomian分解法的计算结果非常接近精确解，并且该方法在大波数情况下还具有良好的收敛性。

关键词: 三维Helmholtz方程, Adomian分解法, 波数