Recently, it was shown that the solution of the Helmholtz equation can be approximated by a series over the solutions of iterative parabolic equations (IPEs). An expansion of the fundamental solution of the Helmholtz equation over solutions of IPEs is considered. It is shown that the resulting Taylor-like series can be easily transformed into a Padé-type approximation. In practical propagation problems such iterative Padé approximations exhibit improved wide-angle capabilities and faster convergence to the solution of the Helmholtz equation in comparison to Taylor-like expansion over IPEs solutions. A Gaussian smoothing of the expansion terms gives insight into the derivation of initial conditions consistent for IPEs, which can be used for point source simulation. A correct point source model consistent with the wide-angle one-way propagation equations is important in many practical applications of the parabolic equations theory.

中文翻译：

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关于Helmholtz方程基本解对迭代抛物方程解的分解

最近，研究表明，在迭代抛物线方程（IPE）的解上，Helmholtz方程的解可以通过一系列来近似。考虑了亥姆霍兹方程的基本解相对于IPE解的扩展。结果表明，所得的类泰勒级数可以轻松地转换为Padé型逼近。在实际的传播问题中，与基于IPEs的泰勒式展开相比，这种迭代式Padé逼近具有改善的广角能力和对Helmholtz方程解的更快收敛。扩展项的高斯平滑可深入了解IPE的初始条件的派生，这些条件可用于点源仿真。