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===Methods for non-linear equations===

===Methods for non-linear equations===

 

{{see also|nonlinear partial differential equation}}

{{see also|nonlinear partial differential equation}}

There are no generally applicable methods to solve nonlinear PDEs. Still, existence and uniqueness results (such as the Cauchy–Kowalevski theorem) are often possible, as are proofs of important qualitative and quantitative properties of solutions (getting these results is a major part of analysis). Computational solution to the nonlinear PDEs, the split-step method, exist for specific equations like nonlinear Schrödinger equation.

There are no generally applicable methods to solve nonlinear PDEs. Still, existence and uniqueness results (such as the Cauchy–Kowalevski theorem) are often possible, as are proofs of important qualitative and quantitative properties of solutions (getting these results is a major part of analysis). Computational solution to the nonlinear PDEs, the split-step method, exist for specific equations like nonlinear Schrödinger equation.

对于非线性偏微分方程，目前还没有普遍适用的求解方法。然而，存在性和唯一性结果(如柯西-科瓦列夫斯基定理)通常是可能的，解的重要定性和定量性质的证明(得到这些结果是分析的主要部分)也是可能的。非线性偏微分方程的计算解，即分步法，存在于一些特定的方程，比如非线性薛丁格方程式。

对于非线性偏微分方程，目前还没有普遍适用的求解方法。然而，存在性和唯一性(如柯西-科瓦列夫斯基定理)通常是可能得到的，解的重要定性和定量性质的证明(得到这些结果是分析的主要部分)也是可能得到的。非线性偏微分方程的计算解，即分步法，对一些特定的方程适用，比如非线性'''<font color="#ff8000">薛定谔方程 Schrödinger equation</font>'''。

Nevertheless, some techniques can be used for several types of equations. The -principle is the most powerful method to solve underdetermined equations. The Riquier–Janet theory is an effective method for obtaining information about many analytic overdetermined systems.

Nevertheless, some techniques can be used for several types of equations. The -principle is the most powerful method to solve underdetermined equations. The Riquier–Janet theory is an effective method for obtaining information about many analytic overdetermined systems.

然而，一些技巧可以用于几种类型的方程。- 原理是求解欠定方程最有效的方法。Riquier-Janet 理论是获得许多解析超定系统信息的有效方法。

然而，一些技巧可以用于几种类型的方程。h-原理是求解欠定方程最有效的方法。里基尔-珍妮特理论是一种可以获得许多解析超定系统信息的有效方法。

In some cases, a PDE can be solved via perturbation analysis in which the solution is considered to be a correction to an equation with a known solution. Alternatives are numerical analysis techniques from simple finite difference schemes to the more mature multigrid and finite element methods. Many interesting problems in science and engineering are solved in this way using computers, sometimes high performance supercomputers.

In some cases, a PDE can be solved via perturbation analysis in which the solution is considered to be a correction to an equation with a known solution. Alternatives are numerical analysis techniques from simple finite difference schemes to the more mature multigrid and finite element methods. Many interesting problems in science and engineering are solved in this way using computers, sometimes high performance supercomputers.

 

 

===Lie group method===

===Lie group method===