Editing Partial differential equation (section)

===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]].

A general approach to solving PDEs uses the symmetry property of differential equations, the continuous [[infinitesimal transformation]]s 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 pair]]s, recursion operators, [[Bäcklund transform]] and finally finding exact analytic solutions to the PDE.

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