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===Lie group method===

===Lie group method===

From 1870 [[Sophus Lie]]'s work put the theory of differential equations on a more satisfactory foundation. He showed that the integration theories of the older mathematicians can, by the introduction of what are now called [[Lie group]]s, be referred, to a common source; and that ordinary differential equations which admit the same [[infinitesimal transformation]]s present comparable difficulties of integration. He also emphasized the subject of [[contact transformation|transformations of contact]].

From 1870 [[Sophus Lie]]'s work put the theory of differential equations on a more satisfactory foundation. He showed that the integration theories of the older mathematicians can, by the introduction of what are now called [[Lie group]]s, be referred, to a common source; and that ordinary differential equations which admit the same [[infinitesimal transformation]]s present comparable difficulties of integration. He also emphasized the subject of [[contact transformation|transformations of contact]].

From 1870 Sophus Lie's work put the theory of differential equations on a more satisfactory foundation. He showed that the integration theories of the older mathematicians can, by the introduction of what are now called Lie groups, be referred, to a common source; and that ordinary differential equations which admit the same infinitesimal transformations present comparable difficulties of integration. He also emphasized the subject of transformations of contact.

From 1870 Sophus Lie's work put the theory of differential equations on a more satisfactory foundation. He showed that the integration theories of the older mathematicians can, by the introduction of what are now called Lie groups, be referred, to a common source; and that ordinary differential equations which admit the same infinitesimal transformations present comparable difficulties of integration. He also emphasized the subject of transformations of contact.

从1870年起，索弗斯 · 李的工作为微分方程理论奠定了一个较为令人满意的基础。他指出，通过引入现在所谓的李群，老一辈数学家的积分理论可以引用到一个共同的来源; 承认相同的无穷小变换的常微分方程在积分方面存在可比的困难。他还强调了接触的转变这一主题。

从1870年起，索弗斯·李的工作为微分方程理论奠定了一个较为令人满意的基础。他指出，通过引入现在所谓的李群，老一辈数学家的积分理论可以引用到一个共同的来源; 承认相同的无穷小变换的常微分方程在积分方面存在相当的困难。他还强调了接触的转变这一主题。

A general approach to solving PDEs uses the symmetry property of differential equations, the continuous infinitesimal transformations of solutions to solutions (Lie theory). Continuous group theory, Lie algebras and differential geometry are used to understand the structure of linear and nonlinear partial differential equations for generating integrable equations, to find its Lax pairs, recursion operators, Bäcklund transform and finally finding exact analytic solutions to the PDE.

A general approach to solving PDEs uses the symmetry property of differential equations, the continuous infinitesimal transformations of solutions to solutions (Lie theory). Continuous group theory, Lie algebras and differential geometry are used to understand the structure of linear and nonlinear partial differential equations for generating integrable equations, to find its Lax pairs, recursion operators, Bäcklund transform and finally finding exact analytic solutions to the PDE.

求解偏微分方程的一般方法是利用微分方程的对称性，即解的解的连续无穷小变换(李理论)。连续群论、李代数和微分几何理论被用来理解生成可积方程的线性和非线性偏微分方程的结构，找到它的 Lax 对、递归算子、 Bäcklund变换，最后找到偏微分方程的精确解析解。

求解偏微分方程的一般方法是利用微分方程的对称性，即解到解的连续无穷小变换(李理论)。连续群论、李代数和微分几何理论被用来理解生成可积方程的线性和非线性偏微分方程的结构，找到它的Lax对、递归算子、 贝克伦德变换，最后找到偏微分方程的精确解析解。

Symmetry methods have been recognized to study differential equations arising in mathematics, physics, engineering, and many other disciplines.

Symmetry methods have been recognized to study differential equations arising in mathematics, physics, engineering, and many other disciplines.

 

 

===Semianalytical methods===

===Semianalytical methods===