非线性方程的自相似校正Pade逼近
Self-similarly corrected Pade approximants for nonlinear equations
摘要：
我们考虑的问题，寻找近似的解析解的非线性方程的物理应用的典型。重点是修改的方法Padé逼近已知提供最佳逼近类的合理的功能，但不提供足够的准确性或不能适用于所有的非线性问题，其解决方案表现出的行为特点是无理函数。为了提高精度，我们提出了一种自相似校正Padé逼近的方法，考虑到非理性的功能行为。该方法的思想是将所寻求的解表示为两个因子的乘积，其中一个因子由自相似根逼近给出，负责非理性函数行为，另一个因子是对应于有理函数的Padé逼近。通过构造非线性微分方程的精确解说明了该方法的有效性。一个深入的调查证明，建议的方法是更准确的比标准的Padé逼近的方法。
英文摘要：
We consider the problem of finding approximate analytical solutions for nonlinear equations typical of physics applications. The emphasis is on the modification of the method of Padé approximants that are known to provide the best approximation for the class of rational functions, but do not provide sufficient accuracy or cannot be applied at all for those nonlinear problems, whose solutions exhibit behaviour characterized by irrational functions. In order to improve the accuracy, we suggest a method of self-similarly corrected Padé approximants, taking into account irrational functional behaviour. The idea of the method is in representing the sought solution as a product of two factors, one of which is given by a self-similar root approximant, responsible for irrational functional behaviour, and the other being a Padé approximant corresponding to a rational function. The efficiency of the method is illustrated by constructing very accurate solutions for nonlinear differential equations. A thorough investigation is given proving that the suggested method is more accurate than the method of standard Padé approximants.
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