求任意阶非线性偏微分方程第一积分的分解方法和Maple程序
The decomposition method and Maple procedure for finding first integrals of nonlinear PDEs of any order with any number of independent variables
摘要：
本文提出了一种新的方法来求解某些类型的非线性偏微分方程的封闭形式的解。该方法基于非线性算子在低阶算子序列上的分解。结果表明，分解过程可以通过迭代过程来完成，每一步都归结为对一个因变量求解一些辅助偏微分方程组。此外，还得到了可分解初始偏微分方程的首次积分的一阶偏微分方程的显式表达式。值得注意的是，如果初始PDE在其最高导数中是线性的，则该一阶PDE是线性的。该方法在Maple程序中得以实现，可以求解多个不同阶数的独立变量的偏微分方程。算例表明该方法具有自动求解非线性偏微分方程的潜力。
英文摘要：
In present paper we propose seemingly new method for finding solutions of some types of nonlinear PDEs in closed form. The method is based on decomposition of nonlinear operators on sequence of operators of lower orders. It is shown that decomposition process can be done by iterative procedure(s), each step of which is reduced to solution of some auxiliary PDEs system(s) for one dependent variable. Moreover, we find on this way the explicit expression of the first-order PDE(s) for first integral of decomposable initial PDE. Remarkably that this first-order PDE is linear if initial PDE is linear in its highest derivatives. The developed method is implemented in Maple procedure, which can really solve many of different order PDEs with different number of independent variables. Examples of PDEs with calculated their general solutions demonstrate a potential of the method for automatic solving of nonlinear PDEs.
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