标题： 循环算子分解法求解微分方程 显示英文标题

标题： Cyclic Operator Decomposition for Solving the Differential Equations

摘要： 我们提出一种方法，用于获得未知函数的任意线性算子方程的解。 特解可以通过无限算子级数（循环算子分解）表示，该级数作用于生成函数。 该方法允许我们根据初始问题选择循环算子和相应的生成函数，以便进行分析或数值研究。 我们的方法包括微扰理论作为特殊情况，但通常不需要任何小参数和未受扰解。 我们展示了该方法在数学物理中几个微分方程分析中的适用性，即经典振子、薛定谔方程和色散介质中的波动方程。 显示英文摘要

摘要： We present an approach how to obtain solutions of arbitrary linear operator equation for unknown functions. The particular solution can be represented by the infinite operator series (Cyclic Operator Decomposition), which acts the generating function. The method allows us to choose the cyclic operators and corresponding generating function selectively, depending on initial problem for analytical or numerical study. Our approach includes, as a particular case, the perturbation theory, but generally does not require inside any small parameters and unperturbed solutions. We demonstrate the applicability of the method to the analysis of several differential equations in mathematical physics, namely, classical oscillator, Schr\"odinger equation, and wave equation in dispersive medium.