Editing Partial differential equation (section)

===Separation of variables===
{{main|Separable partial differential equation}}
Linear PDEs can be reduced to systems of ordinary differential equations by the important technique of separation of variables. This technique rests on a feature of solutions to differential equations: if one can find any solution that solves the equation and satisfies the boundary conditions, then it is ''the'' solution (this also applies to ODEs). We assume as an [[ansatz]] that the dependence of a solution on the parameters space and time can be written as a product of terms that each depend on a single parameter, and then see if this can be made to solve the problem.<ref>{{cite book |last1=Gershenfeld |first1=Neil |title=The nature of mathematical modeling |url=https://archive.org/details/naturemathematic00gers_334 |url-access=limited|date=2000|publisher=Cambridge University Press|location=Cambridge|isbn=0521570956|page=[https://archive.org/details/naturemathematic00gers_334/page/n32 27]|edition=Reprinted (with corr.)}}</ref>

In the method of separation of variables, one reduces a PDE to a PDE in fewer variables, which is an ordinary differential equation if in one variable – these are in turn easier to solve.

This is possible for simple PDEs, which are called [[separable partial differential equation]]s, and the domain is generally a rectangle (a product of intervals). Separable PDEs correspond to [[diagonal matrices]] – thinking of "the value for fixed {{mvar|x}}" as a coordinate, each coordinate can be understood separately.

This generalizes to the [[method of characteristics]], and is also used in [[integral transform]]s.